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Lecture Notes on Data Engineering and Communications Technologies Volume 146

Series Editor Fatos Xhafa, Technical University of Catalonia, Barcelona, Spain

The aim of the book series is to present cutting edge engineering approaches to data technologies and communications. It will publish latest advances on the engineering task of building and deploying distributed, scalable and reliable data infrastructures and communication systems. The series will have a prominent applied focus on data technologies and communications with aim to promote the bridging from fundamental research on data science and networking to data engineering and communications that lead to industry products, business knowledge and standardisation. Indexed by SCOPUS, INSPEC, EI Compendex. All books published in the series are submitted for consideration in Web of Science.

More information about this series at https://link.springer.com/bookseries/15362

Liang-Yee Cheng Editor

ICGG 2022 - Proceedings of the 20th International Conference on Geometry and Graphics

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Editor Liang-Yee Cheng Department of Construction Engineering University of São Paulo São Paulo, Brazil

ISSN 2367-4512 ISSN 2367-4520 (electronic) Lecture Notes on Data Engineering and Communications Technologies ISBN 978-3-031-13587-3 ISBN 978-3-031-13588-0 (eBook) https://doi.org/10.1007/978-3-031-13588-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Geometry and Graphics are the bedrock of technical communication, analysis of spatial properties, problem modeling and solving, and artistic creations. It is also an ever-expanding ﬁeld on both analogical and digital fronts with remarkable achievements in theoretical foundations, covering a broad range of applications, and innovative teaching methodologies and tools. These Proceedings are from the 20th edition of the International Conference on Geometry and Graphics (ICGG2022), a series of conferences started 44 years ago and promoted by the International Society for Geometry and Graphics (ISGG) to provide a high-quality forum for the exchange of academic and industrial research on theoretical and applied Geometry and Graphics, as well as the historical aspects and educational experiences. In the last four decades, through the ICGGs, the ideals of a group of enthusiasts cultivated a vigorous community that promotes synergy networking through international collaborations. The highly multidisciplinary background of the participants, as well as the immense potential of such a fertile environment, reflects on the diversity of the themes and high-quality studies shown in the present proceedings. Especially in this edition, we are commemorating the 30th anniversary of the foundation of ISGG. In this way, the contents of the Proceedings are organized into seven parts. In the ﬁrst part, we are pleased to show the reflections proposed by Prof. Luigi Cocchiarella, President of ISGG, on being a Scientiﬁc Society on Geometry and Graphics. It is followed by ﬁve parts associated, respectively, with the traditional topics of the conference that depicture the wide spectrum of the related ﬁelds with remarkably different subjects, approaches, and formats: • • • • •

Theoretical Graphics and Geometry, Applied Geometry and Graphics, Engineering Computer Graphics, Graphics Education, Geometry and Graphics in History. Then, the last part is composed by the poster papers. v

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Preface

Besides expanding the conference topic on ‘Engineering Computer Graphics’ to include Data Engineering, concerning both pictorial representation of data and the design and building of Graphics data manipulation systems, this edition is also a milestone by joining the Springer Series Lecture Notes on Data Engineering and Communications Technologies (DECT). Moreover, the workflow from the paper submission to the conference proceeding production has been modiﬁed to provide a smoother reviewing process and to ensure the quality of the publications within a relatively short time frame. At ﬁrst, the submission of abstracts was replaced by the direct submission of the full paper or poster paper. Then, a two-phase peer review with at least two reviewers was introduced. In some controversial cases, a third or fourth reviewer has been assigned. This is required considering the vast scope of subjects related to Geometry and Graphics and the divergent view of the reviewers with distinct backgrounds. As a result, from 119 submissions from 28 countries, 50 have been approved in the ﬁrst review, and, after the second phase, a total of 95 have been accepted for publication. We hope the content of these Proceedings and the various aspects of Geometry and Graphics reported herein be inspiring and helpful to catalyze synergic international interactions among different areas and to foster new interdisciplinary research toward novel ﬁndings or innovative solution for a better future. On behalf of the Conference Steering Committee, I would like to thank all authors for their contributions and express our gratitude to ISGG, the Japan Society for Geometry and Graphics (JSGG), and the Brazilian Association for Graphics Expression (ABEG) for their sponsorship and support. I would like to express my sincere appreciation to the highly competent and dedicated International Program Committee, which accomplished the heavy-load task within a very tight schedule, and the Local Organizing Committee, for valuable collaboration. Finally, special thanks to the staff, which worked extremely hard, even in the difﬁcult and uncertain times. Liang-Yee Cheng

Organization

Honorary Chair Hellmuth Stachel

Vienna University of Technology, Austria

Program Chairs Liang-Yee Cheng Aura Conci Luigi Cocchiarella Gilson Braviano

University of São Paulo, Brazil Universidade Federal Fluminense, Brazil Politecnico di Milano, Italy Federal University of Santa Catarina, Brazil

Steering Committee: Asia/Australia/Oceania Baoling Han Emiko Tsutsumi Hirotaka Suzuki Kenjiro Suzuki Kumiko Shiina Peeraya Sripian Yasushi Yamaguchi

Beijing Institute of Technology, China Otsuma Women’s University, Japan Kobe University, Japan The University of Tokyo, Japan National Center for University Entrance Examinations, Japan Shibaura Institute of Technology, Japan The University of Tokyo, Japan

Steering Committee: Europe/Near East/Africa Cornelie Leopold Ema Jurkin Gunter Weiss Hans-Peter Schröcker Miklós Hoffmann

TU Kaiserslautern, Germany University of Zagreb, Croatia Technical University of Dresden, Germany University of Innsbruck, Austria Esterházy Károly University, Hungary

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Milena Stavric Otto Röschel

Organization

Graz University of Technology, Austria Graz University of Technology, Austria

Steering Committee: North America/South America Eduardo Toledo Santos Frank Maxﬁeld Croft Paul Zsombor-Murray Ted Branoff

University of São Paulo, Brazil The Ohio State University, USA McGill University, Canada Illinois State University, USA

Program Committee Naomi Ando Leônidas Brandão Ted Branoff Gilson Braviano Hongming Cai Juan Manuel Villa Carrero Patrícia Biasi Cavalcanti João Carlos de Oliveira Cesar Érica de Sousa Checcucci Liang-Yee Cheng Luigi Cocchiarella Aura Conci Denise Dantas Magdalena Dragovic Douglas Dunham Priscila Farias Sérgio Leal Ferreira Wilson Florio Sande Gao Sergio Gavino Marcelo Eduardo Giacaglia Andrea Giordano Baoling Han Marco Hemmerling Miklós Hoffmann Manfred Husty Biljana Jovic Ema Jurkin Ashraf Khattab

Hosei University, Japan University of São Paulo, Brazil Illinois State University, USA Federal University of Santa Catarina, Brazil Shanghai Jiao Tong University, China University Francisco de Paula Santander, Colombia Universidade Federal de Santa Catarina, Brazil University of São Paulo, Brazil Universidade Federal da Bahia, Brazil University of São Paulo, Brazil Politecnico di Milano, Italy Universidade Federal Fluminense, Brazil University of São Paulo, Brazil Faculty of Civil Engineering, University of Belgrade, Serbia University of Minnesota Duluth, USA University of São Paulo, Brazil University of São Paulo, Brazil Mackenzie University, Brazil Meisei University, Japan Universidad Nacional de La Plata, Argentina University of São Paulo, Brazil University of Padova, Italy Beijing Institute of Technology, China Cologne University of Applied Sciences, Germany Esterházy Károly University, Hungary University of Innsbruck, Austria University of Belgrade, Serbia University of Zagreb, Croatia Ain Shams University, Egypt

Organization

Kunio Kondo Sonja Krasić Cornelie Leopold Daniel Lordick Daiva Makuteniene Ivan Luiz de Medeiros Luciano Migliacco Jun Mitani Zhendong Niu Boris Odehnal Regiane Pupo Augusto Righetti Dina Rochman Daniel Wyllie Lacerda Rodrigues José Ignacio Rojas Sola Roberto Rosso Eduardo Toledo Santos Hans-Peter Schröcker Rodrigo Duarte Seabra Yukio Shigaki Kumiko Shiina André Tavares da Silva Paulo Siqueira Roberta Spallone Peeraya Sripian Monika Sroka-Bizoń Hellmuth Stachel Milena Stavric Hirotaka Suzuki Tomohiro Tachi Emiko Tsutsumi Marcos de Sales Guerra Tsuzuki Vera Viana Krassimira Vlachkova Thaís Regina Ueno Yamada Yasushi Yamaguchi Kensuke Yasufuku Haiyan Yu

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Tokyo University of Technology, Japan University of Nis, Serbia TU Kaiserslautern, Germany Technical University of Dresden, Germany Vilnius Gediminas Technical University, Lithuania Universidade Federal de Santa Catarina, Brazil University of São Paulo, Brazil University of Tsukuba, Japan Beijing Institute of Technology, China University of Applied Arts Vienna, Austria Universidade Federal de Santa Catarina, Brazil Universidade Federal Fluminense, Brazil Metropolitan Autonomous University, Mexico Universidade Federal do Rio de Janeiro, Brazil Universidad de Jaén, Spain Universidade do Estado de Santa Catarina, Brazil University of São Paulo, Brazil University of Innsbruck, Austria Universidade Federal de Itajubá, Brazil Centro Federal de Educação Tecnológica de Minas Gerais, Brazil National Center for University Entrance Examinations, Japan Universidade do Estado de Santa Catarina, Brazil Universidade Federal do Paraná, Brazil Politecnico di Torino, Italy Shibaura Institute of Technology, Japan Silesian University of Technology, Poland Vienna University of Technology, Austria Graz University of Technology, Austria Kobe University, Japan The University of Tokyo, Japan Otsuma Women’s University, Japan University of São Paulo, Brazil Universidade do Porto, Portugal University of Soﬁa St. Kliment Ohridski, Bulgaria São Paulo State University, Brazil The University of Tokyo, Japan Osaka University, Japan Donghua University, China

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Pengfei Zheng Aleksandar Čučaković Gang Zhao Marcelo da Silva Hounsell Mirjana Devetaković Nobuhiro Yamahata Zorana Jeli

Organization

East China University of Science and Technology, China Faculty of Civil Engineering, University Belgrade, Serbia Beihang University, China Universidade do Estado de Santa Catarina, Brazil University of Belgrade, Serbia Tohoku University of Art and Design, Japan University of Belgrade, Serbia

Local Organizing Committee Arivaldo Leão Amorim Brenda Chaves Coelho Leite Elsa Vásquez Alvarez Fabiano Rogerio Corrêa Fabio Kenji Motezuki Fernando Akira Kurokawa João Roberto Diego Petreche Marcelo Eduardo Giacaglia Marco Antonio Rossi Rovilson Mafalda Rubens Augusto Amaro Junior Sérgio Leal Ferreira Thaís Regina Ueno Yamada Wilson Florio

Universidade Federal da Bahia, Brazil University of São Paulo, Brazil University of São Paulo, Brazil University of São Paulo, Brazil University of São Paulo, Brazil University of São Paulo, Brazil University of São Paulo, Brazil University of São Paulo, Brazil São Paulo State University, Brazil Federal University of ABC, Brazil University of São Paulo, Brazil University of São Paulo, Brazil São Paulo State University, Brazil Mackenzie University, Brazil

Staff Daniela dos Santos da Mata Gomes Lucas Soares Pereira Lucas Isaac Lima Trindade Thiago Machado Orlandi do Couto Daﬁco

University of São Paulo, Brazil University of São Paulo, Brazil University of São Paulo, Brazil University of São Paulo, Brazil

Contents

The 30th Anniversary of the ISGG On ISGG’s 30th Agenda: Legacy and Challenges . . . . . . . . . . . . . . . . . . Luigi Cocchiarella

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Theoretical Graphics and Geometry On the Diagonals of Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hellmuth Stachel

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Exploring the Steiner-Soddy Porism . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ronaldo Garcia, Liliana Gabriela Gheorghe, and Dan Reznik

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Circumparabolas in Chapple’s Porism . . . . . . . . . . . . . . . . . . . . . . . . . . Boris Odehnal and Dan Reznik

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Permutation Cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boris Odehnal

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Beyond the Nine-Point Conic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boris Odehnal

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Complex Solution of Engineering Problems by Graphic Methods . . . . . Aleksandr Yurievich Brailov

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Integer Sequences from Circle Divisions by Rational Billiard Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel Jaud

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Topographic Surfaces as Topological Sets . . . . . . . . . . . . . . . . . . . . . . . 107 Dilarom F. Kuchkarova and Dilnoza A. Achilova Generalizing Continuous Flexible Kokotsakis Belts of the Isogonal Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Georg Nawratil

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Contents

The Intersection Curve of an Ellipsoid with a Torus Sharing the Same Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Ana Maria Reis D’Azevedo Breda, Alexandre Emanuel Batista da Silva Trocado, and José Manuel Dos Santos Dos Santos Four-Dimensional Visual Exploration of the Complex Number Plane . . . 138 Jakub Řada and Michal Zamboj Regularity Conditions for Voronoi Diagrams in Hyperbolic Space . . . . 150 Alžbeta Mackovová and Pavel Chalmovianský Poncelet and the (Arquimedean) Twins . . . . . . . . . . . . . . . . . . . . . . . . . 163 Liliana Gabriela Gheorghe A Note on Local Intersection Multiplicity of Two Plane Curves . . . . . . 175 Adriana Bosáková and Pavel Chalmovianský Locally Flat and Rigidly Foldable Discretizations of Conic Crease Patterns with Reﬂecting Rule Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Erik D. Demaine, Klara Mundilova, and Tomohiro Tachi Transformations Between Developments and Perspectives of Three and Four Dimensional Cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Takafumi Otsuka and Akihiro Matsuura Perfect Circles, Amicable Triangles and Some of Their Properties – Angles Equality and About Two New Constants in the Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Michael Sejfried Applied Geometry and Graphics Geometrical Construction of Shape by a Weak-Visual Person in a Physical Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Lorena Olmos Pineda and Jorge Gil Tejeda Algebraic Surfaces and Their Geometric Bases . . . . . . . . . . . . . . . . . . . 228 Michael Manevich and Elizabeth Itskovich The Wrought Iron Beauty of Poncelet Loci . . . . . . . . . . . . . . . . . . . . . . 238 Dan Reznik Rectiﬁcation of an Edgy Photograph . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Boris Odehnal and Johannes Porsch Point Cloud Upsampling via Quadric Fitting . . . . . . . . . . . . . . . . . . . . . 263 Marcel Makovník and Pavel Chalmovianský Viewing Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Cornelie Leopold and Romy Link

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Geometric Aspects of Modeling Real Conditions of Solar Irradiation of Energy Efﬁcient Architectural Objects . . . . . . . . . . . . . . . . . . . . . . . . 288 Olga Krivenko, Alexey Pidgornyi, Vitaliy Zaprivoda, Viacheslav Martynov, and Andrey Zaprivoda MPS Simulation for the Japanese Sport Wellness Blowgun . . . . . . . . . . 298 Sande Gao, Naoki Ueno, and Loulin Huang Conservative Dynamical Systems in Oscillating Origami Tessellations . . . 308 Rinki Imada and Tomohiro Tachi Branching and Merging of Kumihimo Braiding Based on the Geodesics of Regular Tetrahedron . . . . . . . . . . . . . . . . . . . . . . . 322 Seri Nishimoto, Fuki Ono, Masaaki Miki, Kiichiro Domyo, and Tomohiro Tachi Study on Images of Historic Japanese Streetscapes Using Automatic Form Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Naomi Ando, Nobuhiro Yamahata, and Xiangjun Xu Town Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 Enrica Pieragostini and Salvatore Santuccio The Geometry of the Ramps in Vilanova Artigas Architecture . . . . . . . 358 Ana Tagliari and Wilson Florio 3D Reconstruction Using Sketch Based on Duchon Energy . . . . . . . . . . 370 Yexi Yin, Shujin Lin, and Fan Zhou Graphic Methodology in Structural Analysis of Historical Constructions. Application to Structural Behavior of the Domes’ Backﬁll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Francisco Javier Suárez, Thomas E. Boothby, and Jose A. González Using Telescopic Mapping for Inﬁnity Representation with an Example of Ventilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 Viktor Mileikovskyi and Tetiana Tkachenko Low-Cost 3D Scanning Applied to Packaging Design . . . . . . . . . . . . . . . 406 Joe Wallace Cordeiro, Marcelo Gitirana Gomes Ferreira, and Gilson Braviano From Natural Tree Forks to Grid Shells: Towards a Self-forming Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 Anton Donchev Kerezov, Mikio Koshihara, and Tomohiro Tachi Bio-digital Land Art Installation Inspired by Dandelion Leaf . . . . . . . . 431 Biljana Jovic, Dragica Obratov Petkovic, and Olga Gajanic

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Contents

Recognition of Archimedean Spiral Voronoi Diagrams from Linear Parastichies Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 Chanikan Sawatdithep and Supanut Chaidee Media Architecture as Innovative Method of Urban Environment Organizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Olga Semenyuk, Assem Issina, Rakhima Chekaeva, Zhazira Bissenova, Timur Yensebayev, Askar Kalikhin, Bayan Ozganbayeva, Madi Zhussupov, Zhansaya Ashimova, and Aida Slyamkhanova Le Corbusier’s Modulor and ‘le jeu des panneaux’: A Parametric Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Wilson Florio Grid as Memory in City and Architecture . . . . . . . . . . . . . . . . . . . . . . . 477 Yuji Katagiri, Taizo Iwashita, Hirotoshi Takeuchi, Takahiro Ohmura, Ikko Yokoyama, and Tatsuo Iwaoka Stencils Are like Pencils. On the Ambiguous Visuality of Laser-Cutting Templates from Model Making – Substance Versus Cutout as Constructive Vagueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 Niels-Christian Fritsche Gems Geometry: From Raw Structure to Precious Stone . . . . . . . . . . . 497 Nicola Pisacane, Pasquale Argenziano, and Alessandra Avella A Multi-scale Investigation of Visual Interactions in the Built Environment via the Generation of Parametric Procedures . . . . . . . . . . 509 Matteo Cavaglià Generating Spatial Conﬁgurations of Floating Settlement Branch Structures for Urban Atoll Islands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Jovana Stanković, Branislava Stoiljković, Sonja Krasić, and Nastasija Kocić Planning the Inﬁll Patterns and the Resulting Density Percentage Error in Additive Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Yasaman Farahnak Majd, Marcos de Sales Guerra Tsuzuki, and Ahmad Barari Equirectangular Pictures and Surrounding Visual Experience. Spherical Immersive Photographic Projections at: Boito Architetto Archivio Digitale, Historical Exhibition at Politecnico di Milano . . . . . . 541 Federico Alberto Brunetti Textile Drawing. A Geometric Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 554 Stefano Chiarenza

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Engineering Computer Graphics Soccer Player Pose Recognition in Games . . . . . . . . . . . . . . . . . . . . . . . 565 Rodrigo G. Reis, Diego P. Trachtinguerts, André K. Sato, Rogério Y. Takimoto, Fábio S. G. Tsuzuki, and Marcos de Sales Guerra Tsuzuki Parametric Design and Development of Wood Roof Based on Revit . . . 575 Ziru Wang and Chao Yu Computation and Amendment Method of Surface Deformation Based on Welding Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Pengfei Zheng, Jingjing Lou, Yunhan Li, Xiyuan Wan, Qingdong Luo, and Linsheng Xie Construction of Diffeomorphisms with Prescribed Jacobian Determinant and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 Zicong Zhou and Guojun Liao The Cuneiform Brick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 Francesco Di Paola, Calogero Vinci, and Fabrizio Tantillo PerForm – Designing Adaptable Furniture . . . . . . . . . . . . . . . . . . . . . . 624 Eva Hagen and Benedikt Blumenröder Automatic Lung Segmentation with Seed Generation and ROIFT Algorithm for the Creation of Anatomical Atlas . . . . . . . . . . . . . . . . . . 636 Jungeui Choi, Edson Kenji Ueda, Guilherme Cortez Duran, Paulo A. V. Miranda, and Marcos de Sales Guerra Tsuzuki A Pre-processing Tool for Particle-Based Fluid Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 Cezar Augusto Bellezi, Liang-Yee Cheng, and Lucas Soares Pereira Criteria and Procedures for the Geometric Parametrization of Existing Buildings: The Case Study of the Roof of the Frontón Recoletos in Madrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 Andrea Colombo and Andrea Giordano Design of Surfaces in Cylindrical Coordinates Using GeoGebra AR . . . 673 Alejandro Isaías Flores-Osorio and Dennis Alberto Espejo-Peña Real-Time Renderings of Multidimensional Massive DataCubes on Jupyter Notebook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 Antoine Lestrade, Mathias Marty, Artan Sadiku, Christophe Muller, Joep Neijt, Yann Voumard, and Stéphane Gobron Graphic Sciences and Documentary Heritages. A Shared Experience in Trentino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 Elena Bernardini and Giovanna A. Massari

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Contents

A Speciﬁc Segmentation Approach to Measure Deforestation from Satellite Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709 Raffael Paranhos and Aura Conci Implicit Curves: From Discrete Extraction to Applied Formalism . . . . . 717 Mathias Marty, Antoine Lestrade, Artan Sadiku, Christophe Muller, Joep Neijt, Yann Voumard, and Stéphane Gobron Rendering 360° Images in a 360° Theater . . . . . . . . . . . . . . . . . . . . . . . 729 Cecilia Maria Bolognesi, Simone Balin, and Elio Sasso The Use of Generative Adversarial Network as Graphical Support for Historical Urban Renovation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738 Angelo Lorusso, Barbara Messina, and Domenico Santaniello An Analysis and Consolidation of DfMA Based Construction Guidelines and Its Validation Through a Korean Case Study . . . . . . . . 749 Saddiq Ur Rehman, Soyeong Ryu, and Inhan Kim Topology Optimization of Capacitive MEMS Accelerometers for Seismic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 Hossein R. Najafabadi, Tiago G. Goto, Thiago C. Martins, Ahmad Barari, and Marcos de Sales Guerra Tsuzuki Reverse Engineering as Tool to Help Surgeons . . . . . . . . . . . . . . . . . . . 773 Renata A. Santos, Amanda M. de Souza, Marcel J. S. Tamaoki, Nicola A. Netto, and Elsa Vásquez-Alvarez Graphics Education Design and Development Process of an App with the Concepts of Descriptive Geometry for Students to Interact, Learn and Self-evaluate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787 Dina Rochman Images of Venice in Valeriano Pastor’s Project for Cannaregio Ovest, 1978 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800 Starlight Vattano The Exploration of the Mind Map Applied in the Teaching of Engineering Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 Dan Xu Graphic Computational Thinking in Descriptive Geometry . . . . . . . . . . 823 Haiyan Yu, Hongbo Shan, Hongming Cai, Yuanjun He, and Wenjun Zhang Transfer of Geometric Drawing, Descriptive Geometry and Technical Drawing Classes to a Remote Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 832 Maria Bernardete Barison

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Exploration and Reﬂection on the Teaching Mode of Cross-College Credit for Drawing Geometry and Engineering Graphics Courses . . . . 843 Xiaohao Li and Junqi Pan Emotional and Cognitive Maps for Urban Design Education: A Human-Centered Design Learning Approach . . . . . . . . . . . . . . . . . . . 849 Barbara E. A. Piga, Gabriele Stancato, and Giulio Faccenda Evaluation of Descriptive Geometry Dynamic Models Developed in Geogebra® for Online Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 Juliane Silva de Almeida and Márcio Schneider de Castro Modeling Bounded Surfaces Using Cylindrical Coordinates Using GeoGebra AR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870 Dennis Alberto Espejo-Peña and Alejandro Isaías Flores-Osorio An Introductory BIM Course for Engineering Students . . . . . . . . . . . . 880 Eduardo Toledo Santos and Sérgio Leal Ferreira A Project-Based Learning (PBL) Approach in an Engineering Design Graphics Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891 Liang-Yee Cheng, Sérgio Leal Ferreira, and Eduardo Toledo Santos Detecting and Correcting Errors in Mental Cutting Test Intersections Computed with Blender . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904 Róbert Tóth, Bálint Tóth, Marianna Zichar, Attila Fazekas, and Miklós Hoffmann Virtual Models of Survey and Use of Religious Architecture . . . . . . . . . 917 Luigi Corniello and Gennaro Pio Lento Educational Applications to Support the Teaching and Learning of Mental Cutting Test Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928 Róbert Tóth, Bálint Tóth, Marianna Zichar, Attila Fazekas, and Miklós Hoffmann Training of Bachelors in Descriptive Geometry and Graphics in the Republic of Kazakhstan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939 Serik Kuzembayev, Marat Alzhanov, Gulmarzhan Tuleuova, and Lyailya Elibaeva The Teaching of Technology-Mediated Architectural Design . . . . . . . . . 946 Ivan Silvio de Lima Xavier Visual Programming for Teaching Geometry in Architectural Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958 Pablo C. Herrera, Michael Hurtado, and Pedro Arteaga-Juárez

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Geometry and Graphics in History Shape and Geometric Tracing of the Arches of the Late Eighteenth-Century Neuilly Bridge Over the Seine in Paris . . . . . . . . . . 973 Ornella Zerlenga and Vincenzo Cirillo Study of the Basic Geometric Dimensions of the Mausoleum of Hodja Ahmed Yassawi in Central Asia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986 Elmira Kemelbekova, Auyez Baidabekov, and Mariyam Yeziyeva Nehir Fundamental Research on Descriptive Statics . . . . . . . . . . . . . . . . . . . . 994 Dajun Lin and Pengfei Zheng Methods of Measuring Inaccessible Architecture from the Treatise “Radio Latino” by L. Orsini and Commented by E. Danti . . . . . . . . . . 1004 Margherita Cicala The Meaning of Geometry in Land Art: The Architecture of the Woodpecker in Milano Marittima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016 Cristiana Bartolomei, Cecilia Mazzoli, and Caterina Morganti “Divers” by Hoyningen-Huene. Analysis of the Visual Fortune of a Successful Image Through the Principles of Art History by Heinrich Wölfﬂin and the Geometries of a Photographer . . . . . . . . . . . . . . . . . . . 1026 Matteo Giuseppe Romanato The Restoration of Mathematical Cabinets Between Rapid Prototyping and Augmented Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040 Michela Ceracchi An Interpretation of the Truncated Triangular Trapezohedron and the Sphere Depicted in “Melencolia I” by Albrecht Dürer . . . . . . . . . . . 1052 Hirotaka Suzuki Poster Reform of Engineering Course Practice Link Based on PBL Teaching Method—Taking Engineering Graphics Course as an Example . . . . . . . 1061 Xiaohao Li and Lin Mei A Study on Design Conformance Evaluation Model for DfMA Modular Design Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 Jiyoung Kim, Ahjin Lee, and Inhan Kim Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069

The 30th Anniversary of the ISGG

On ISGG’s 30th Agenda: Legacy and Challenges Luigi Cocchiarella(B) Department DASTU, Politecnico di Milano, 20133 Milan, Italy [email protected]

Abstract. According to Robert E. Schofield, looking back to the golden age of Scientific Societies we discover that, from the middle of XVII to the XIX century, rather than academic institutions they were considered as the proper alma mater by scientists [1]. Over time, the general reform of the university has gradually reversed this state of things, with few exceptions. This paper proposes some brief reflections on being a Scientific Society (of Geometry and Graphics) nowadays (in its 30th year), including a glance at the present COVID-19 pandemic impact. Keywords: Scientific Society · Scientific dissemination · International Society for Geometry and Graphics (ISGG)

1 Introduction

Fig. 1. The ISGG Community from 1995 to 2021: darker areas are those with a higher number of Individual members, dots represent Institutional Members and Associated National Organizations. Qualitative diagram based on ISGG register; historical data processed by Milena Stavric, current ISGG Treasurer; graphic editing by Matteo Cavagliá; image by author.

Looking at the world map in Fig. 1, one can see the long progression since 1992, when our Society was initially established in Melbourne. Its subsequent expansion largely © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 3–15, 2023. https://doi.org/10.1007/978-3-031-13588-0_1

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depends on its DNA, international right from the start. Indeed, the birth of ISGG was incubated over a long period from a series of Conferences having its early start in June 1978, with the International Conference on Descriptive Geometry, held in Vancouver, Canada. Based on a network of peer scientists, researchers, and teachers, mostly employed in various Universities worldwide, although open to independent scholars, the ISGG ‘DNA’ is definitely inspired by a modern open approach to the sharing of knowledge. In the article cited in the abstract, Robert E. Schofield does not fail to note that, in the past, mainly independent researches, even in opposition to academic institutions, were encouraged by Scientific Societies, which were often connected to a leading scientist or a distinguished scholar. Here he mentions a few outstanding cases, such as Galileo and the Accademia dei Lincei, Newton and the Royal Society, Huygens, Lavoisier and the Académie royale des Sciences, Leibniz and the Berlin Academy, Joule and the Manchester Literary and Philosophical Society. He concludes that, at that time, the History of Science was closely linked to the histories of Scientific Societies, which gave birth to the properly said scientific literature by regularly publishing their Proceedings. In our case, instead of encountering opposition, a tacit bond of collaborative friendship characterizes the relationships between Scientific Societies and Universities, offering teachers and scholars the opportunity to meet, discuss, and exchange ideas. Another point of difference is the reciprocal non-interference, so that, even if Scientific Societies are largely composed and chaired by academics, this has neither direct influence on their academic careers, neither has their academic status direct influence on their role in the Societies. But, each condition has effects on the scientific reputation of the participants in the other. Which is a strength from both a scientific and an ethical point of view. In a recent article, Cesare Giuseppe Cerri [2], warns about the possible confusion between scientific and trade union aspects in the activity of a Scientific Society, a risk that seems to mainly concern the so called disciplinary Societies, and particularly at a local national level, especially when their research fields coincide with ministerial disciplinary divisions for Universities. On the purpose, citing Imre Lakatos, he recommends Societies to give priority to the core of their theoretical assumptions, instead of to their academic power, which should be crystal clear from the bylaws, that is, from their federation pacts with/among the members. A modern Scientific Society also benefits of pros (and cons) of the present open evaluation system of the scientific literature, involving experts not necessarily belonging to the same Society, potentially offering different points of view on the submitted contributions, which, together with the blind reviewing process, more and more diffused, also aims to prevent any inappropriate approvals, rejections, or censorships (!). However, Cerri highlights some relevant issues. First, the sensitive increase in the number of Scientific Societies and scientific publications has generated an excessive demand for reviewers, and the consequent possible risks in terms of finding the right competences required as well as of having insufficient time to spend on in-depth evaluations of the contributions. Second, with difference to the past, scientific items are not only evaluated before the publication, but their impact is also permanently monitored after the publication, during their life cycle in the real scientific world. This further step needs to be

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carefully considered, since rankings are also sensitive to publishers’ reputation, and the index-based evaluation strategies may tend to reward quantity rather than quality. On the long wave of globalization, the growth of international Scientific Societies is still ongoing, and the topics briefly reported above synthetically define the context of their activity and some of the relevant issues to be faced.

2 ISGG Genealogy: A Society from a Conference, from a Society

Fig. 2. The Conference Proceedings published from 1978 on (2022 in progress): synoptic view of the covers; also published on the ICGG2018 webpage (icgg2018.polimi.it). Image by author.

History teaches that there are good reasons to relate Conferences to Scientific Societies, since the former concern a relevant part of the activities supported and carried out by the latter, and the latter take advantage of the scientific and human network provided by the former. In this regard, the birth of ISGG displays an exemplary virtuous circle. As Frank M. Croft states: “It all started in 1978 by a group of graphics professors associated with the American Society for Engineering Education in Vancouver, British Columbia to celebrate the 50th anniversary of the Engineering Design Graphics Division” [3], who organized a Conference, the International Conference on Descriptive Geometry, which is considered as the early point of origin of both the International Conference on Geometry and Graphics (ICGG), and the International Society for Geometry and Graphics (ISGG), as well as, consequently, the Journal for Geometry and Graphics (JGG). In this sense ISGG can be seen to have been generated by a Conference generated by a Society. Member of the Planning Committee of the 1978 Conference and author of the Preface to the Proceedings was Steve M. Slaby, at the time Associate Professor at the Princeton University, later on recognized as the father of the ISGG [4]. In line with the initial wish to inaugurate a series of meetings, a new Conference was held in Beijing, China, in 1984, titled International Conference on Engineering and Computer Graphics, where “The impact of the Conference in Vancouver was shown through a number of repeat presenters at this conference”, together with the first attendance of those who were to

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become the “founders” of ISGG, namely Hellmuth Stachel from Austria, Kenjiro Suzuki and Emiko Tsutsumi from Japan. Croft says that “The 1984 Conference in China served as a catalyst for future conferences and secured the sustainability of holding conferences on regular bases” [3]. It took another four years until, in 1988, a third Conference was organized in Vienna, the Third International Conference on Engineering Graphics and Descriptive Geometry, co-chaired by Hellmuth Stachel and Steve M. Slaby. It was there that the decision was made to hold the Conference as a series of meetings organized every two years. As the ordinal number – third – appearing in the title for the first time meant to show. On the occasion of the 40th Anniversary of the Conference, celebrated during the ICGG 2018 in Milano, Hellmuth Stachel recognized Steve M. Slaby as “the driving force behind the first international conference” and “the first to ‘kick the ball’ towards an international community”, mentioning that, at the beginning of January 1989, few months after the Vienna’s Conference, he proposed to arrange an ad-hoc committee to discuss the details at the next Conference. According to Stachel, this was the conclusion of the “Brainstorming” and the start of the “Decision” process [4]. Then, during the 4th International Conference on Engineering Computer Graphics and Descriptive Geometry, held in Miami in 1990, “a ‘Steering Committee’ was established with the declared goal to organize the activities of the ‘International Society for Geometry and Graphics’. This was the first mention of the new society”, whose mission statement was: “Foster international collaboration and stimulate the scientific research and teaching methodology in the fields of geometry and graphics”. Two years later, at the 5th International Conference on Engineering Computer Graphics and Descriptive Geometry, held in Melbourne in 1992, “the delegates voted to establish the International Society for Geometry and Graphics”. For the sake of an effective governance, it was planned to ideally subdivide the globe into three 3 regions: Asia/Australia/Oceania, Europe/Africa/Near East, North & South America, each under the responsibility of a Vice-President, cooperating with the ISGG President and supported by the ISGG Treasurer, and this is still the case today. After the elections organized during 1993/1994, the first Board officially took office at the 6 th International Conference on Engineering Computer Graphics and Descriptive Geometry in Tokyo, in 1994. Finally, in 2007 the Journal for Geometry and Graphics also appeared, as the official review of the ISGG, together with the ISGG logo [4]. This is how, in about nineteen years, thanks to the foresight and determination of some leading intellectuals, and the generous joint effort of participants worldwide, a Scientific Society in its 50th anniversary (ASEE) generated a Conference (ICDG 1978), which established a series of Conferences (ICECG, ICEGDG, ICECGDG, and since 2000 ICCG), which generated a new Society (ISGG 1990–1994), which generated a Journal (JGG 1997). On this journey, the torch of Geometry and Graphics touched several places in the world: Vancouver in 1978, Beijing in 1984, Vienna in 1988, Miami in 1990, Melbourne in 1992, Tokyo in 1994. Since then it has never stopped travelling around the world, also touching Cracow in 1996, Austin in 1998, Johannesburg in 2000, Kiev in 2002, Guangzhou in 2004, Salvador de Bahia in 2006, Dresden in 2008, Kyoto

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in 2010, Montreal in 2012, Innsbruck in 2014, Beijing in 2016, Milan in 2018, São Paulo in 2020/2021 and 2022 (Fig. 2). As we see the two stories, that of the Conferences and that of the Society, developed together and cannot be separated. The ASEE worked as the initially unaware incubator of a series of Conferences that ended by generating a new Scientific Society. The history of the first five Conferences is an exemplary model of international scientific cooperation, as well as of lucid determination, and – let’s say – academic style: note that, after preparing the way, the initial promoters kindly took a step back at the moment of the first election of the ISGG President. However, ISGG has given them the honor they deserved: Steve M. Slaby was in fact proclaimed Honorary President, and the Steve M. Slaby Award was established in his name. More recently, in 2020, Hellmuth Stachel has also been proclaimed Honorary President of ISGG, and Honorary Editor of the Journal for Geometry and Graphics (JGG), which he founded and directed as the Managing Editor since 1997. Concerning ISGG Board, the first President elected was Walter Rodriguez, who provided the first version of the bylaws [4], then a new President has been elected every four years: Kenjiro Suzuki, Gunter Weiss, Emiko Tsutsumi, Ted Branoff, Otto Roeschel, Yasushi Yamaguchi. More info can be found on our ISGG webpage [5]. The author of this paper, attending his first Conference at Austin in 1998, is the last in this list, to be in Office for the years 2021–2024. A great honor and responsibility, that also prompted him to look at the origins of ISGG, which has been a memorable experience. Considering that the Presidency Offices are nowadays composed on fourteen members, that is the President, Treasurer, three Vice-Presidents, nine Directors (three for each ISGG world region), it is clear that the story of ISGG is also a story of people, whose essential part is the whole ISGG community of members, that is, the ISGG, fueling our Scientific Society for over forty years now.

3 “Who We Are and What We Want”

Fig. 3. ISGG in a nutshell: mission, people, activity, and honor. Diagram by author.

Tracing the history of modern scientific societies in Europe between XVI and XIX centuries, Douglas McKie wrote: “The first scientific society was founded in 1560 in

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Italy, the cradle of what historians, for want of a better title, call the Renaissance.” [6]. He also observed that this largely coincided with the birth of modern Science, with the official establishment of History as a discipline, as well as with the expansion of the limits of the world known, thanks to voyages and great navigators, and to the diffusion of knowledge as an increasing effect related to the recent invention of printing by movable types. In the bigger world arising in the XVI century, specialization became more and more urgent in Science, and the rise of scientific associations of scientists, was part of a general process of separation of Science from Humanities. By this way, inspired by the Galilean method, validation of researches and dissemination of outcomes became the main components of the mission of a Scientific Society, as Cerri highlights, which led to a new ethics based on the non-pursuit of personal interests by participants, accepted in the best interest of Science [2]. The importance of the historical context seems equally relevant to the birth of the ISGG. Looking at the Proceedings of ICDG 1978, one can see that the first Conference had a double motivation. Indeed, the feeling of joy for the Celebration of the 50th anniversary of the Engineering Design Graphics Division offered, at the same time, the occasion to firmly express a critical and constructive position in front of the belittled role attributed to Geometry, not only inside the Universities. On this score, in the Preface Steve M. Slaby explicitly rebutted that, the Proceedings of what he called the “First” International Conference, would “represent the concrete evidence of the fact that Descriptive Geometry, the ‘Queen’ of the Engineering Disciplines is alive and thriving in the intellectual and applied arenas”, aiming at being “capable of contributing to enliven and preserve the genuine geometric spirit, a spirit which is missed sometimes in the mathematics of our time.” [7]. Not less considerable was the overall cultural context, which began to be dominated by the digital. In this regard, in the Introduction to Conference, Amogene Devaney, another member of the Planning Committee, wrote: “We are now in a new era of space description because of the revolutionary use of computer. However, though computers can enable us to do hitherto virtually impossible arithmetic calculations, it is still necessary for us to develop the mental concepts and visualize the space relations underlying engineering problems.” [7]. Based on these assumptions, together with the EDGD anniversary, that Conference celebrated Descriptive Geometry in fact, comparing and contrasting traditional and new approaches, including workshops where participants could solve problems on the applications of descriptive geometry and space structures, even by elementary and advanced computer graphics procedures. Of course, no words would suffice to eulogize the scientific and pedagogic value of this idea. Actually, it is really hard to imagine the impact of computer graphics at the time, and the consequent revolutionary paradigm shift that occurred, on the research routines, first of all in the field of Mathematics, but not only in that field. Among the most alert and sensitive to the wide foreseeable implications, Lionel March and J. Philip Steadman, referring to the profound sense of the Gaspard Monge’s legacy affirmed in their paper: “In our view there is no academic future in descriptive geometry as mere drawing or even as computer graphics per se: it must once again, albeit in new guise, establish itself at the intellectual center of engineering and architecture activity.” [7]. The paper was titled From Descriptive Geometry To Configurational Engineering, endorsing that

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computational approach to space representation and design which is still at the base of AI systems in scientific and everyday life applications all over the world today. As it was in 1978, even today we have to deal with new items, some unimaginable except in the science fiction until a few years ago, and with the hyperbolic acceleration of Technology and Science. However, differently from XVI century, the complexity level of the present challenges requires the joint contribution of disciplinary and interdisciplinary approaches. After centuries of specialization, it seems once again time to integrate knowledge. So, the absolute centrality of Geometry and Graphics, exploded in almost all human activities shows at the same time a great opportunity and a great challenge, for research, for education, and profession. And, of course, for the Scientific Societies, including ISGG. “Who We Are and What We Want”, a title inspired by the former homepage of the ISGG website appears, as probably never before, the permanent question to consider. Geometry as a way of thinking, and Graphics as a way of visualizing thoughts and things, supported by digital approaches and tools, may work as the efficient components of a meta-language facilitating the exchange of knowledge and the collaboration across the traditional ‘official’ disciplinary boundaries. To defeat the field from naïve misinterpretations, quite paradoxically, this will not require lower, but higher specializations, flanked by a wider cultural background. Which is now having and will in the future have a big impact on education, regardless of the severe setbacks suffered during the present pandemic spread. After the impressive power of calculation contributed to bring theoretical and applied research closer and closer, the present challenge seems to be in the direction of a new alliance among Science, Technique, Art, and Humanities, aiming at integrating rigor and creativity. On closer inspection, in recent years the list of our Conference Topics has also been expanded, including historical and epistemological researches, together with new theoretical and application fields, and the Conference Proceedings have been indexed as SCOPUS records to increase their visibility and, hopefully, their impact [8]. Summing up, still in the long wave of the digital turn, the increasing attractive tension among the disciplinary fields orbiting around Geometry and Graphics as the shared background, as well the growing hybridization between theory and praxis, soft and hard sciences, as the new collaborative strategies supported by the network, seem to show fertile conditions for a paradigm shift, into which we are day by day diving into research, education, industry and profession. To this purpose, it could be interesting to read again what Thomas Samuel Kuhn said about the scientific revolutions [9]. Back to present time, in an inspiring and well documented book titled The Game, Alessandro Baricco has recently outlined the relevant steps in the combined development of the Digital and the Internet, and its effects on the taxonomy of knowledge and styles of research. As well as, on our everyday lives. Starting from the early birth of the video games as tools for leisure addressed to limited groups of people, based on a detailed historical analysis he finally directs the attention to the present diffusion of the Apps. Which, in many respects, he considers as the digital progeny of the early game engines, today pervasively used worldwide, and which already result from a collaboration between soft and hard sciences. So that any activity, including research and education, by ignoring it, may expected to end up being cut out of “the Game” going on worldwide nowadays [10]. Far from taking it in a pessimistic way, we see that new channels for recording, using

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and diffusing knowledge are now available, which require our attention. As it happened after Gutenberg, it is up to us to take advantage of them, in accordance with and for the benefits of “Who We Are and What We Want”.

4 So (Possibly) What?

Fig. 4. The ISGG HUB, and the Internet as its virtual house. Diagram by author.

The evolution of the ongoing processes mentioned above will quite naturally sprout and bloom through our activities as researchers and teachers which, we hope, will continue to be presented and discussed at our Conferences and related international events. However, in this new and permanently evolving context a question about the identity and role of a contemporary Scientific Society is to be posed. The proper collocation of a Scientific Society seems to be a sort of limbo, suspended between classic Academies and modern Universities, somehow looking at industrial and professional contexts. In fact, research and education are not a direct part of the mission of a Scientific Society anymore. But this can be seen as a plus, since Scientific Societies can consequently apply all their energies to playing a relevant role as permanent HUBs for researchers and educators, as well as for industry and eventually the professions. Thus, they can work as ‘bridges’ allowing to share scientific results and expertise, and to promote opportunities for international cooperation, taking advantages of the network of researcher and educators working in more than one University and in more than one country (Fig. 4). Moreover, scientific communities are today quite different compared with the early germinal one at the Plato’s Academy (or School), which was constituted in Athens in the IV century BC. Quite paradoxically, present Societies are only virtually ‘geo-referenced’ where the President resides, but their members do not use to meet there. However, even if their physicality looks quite evanescent, many participants from all over the world join them as Members and use do periodically meet in occasion of their official Conferences. Covid-19 has recently obliged us to make even this event virtual, so that ICGG 2020/2021 took place as an online meeting, which by the way, was turned into an opportunity to test a new way, successfully carried out thanks to the extraordinary efforts of the Brazilian Local Committee, and the Executive Conference Chair Liang-Yee Cheng.

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All in all, after asking “Who We Are” and “What We Want”, in these new condition a crucial query is about how to proceed. The diagram in Fig. 3 would show an updated picture of the current relevant items and activities of the International Society for Geometry and Graphics regarded as a HUB, which is a good base from which to start for some reflections in view of the future expectations that (we hope) await us, and that relevance and attractiveness of the ISGG scientific core – Geometry & Graphics – claims for. As a long term Society, one of the key points for a flourishing survival will concern how to permanently attract and sensitize people to ISGG from generation to generation. Maybe due to the present pandemic spread – not completely defeated yet –, we are recently noting a declining trend in the number of the ISGG Members, as well as in the requests for becoming new ISGG Members. In addition to a – let’s say – already known sort of identification of the Society with the Conference, so that we have participants in ICGGs who are not, or do not wish to become, members of ISGG, which is perfectly normal. It may be worthwhile to remember that ISGG has Individual Members, and Institutional Members, the former are people, the latter are mostly national Scientific Associations from all over the world. This would suggest two complementary options: to persevere in the habit of promoting ISGG every two years during ICGGs; and encourage a more systematic promotion of ISGG even independently from the ICGGs. As regards the first, fortunately our ICGG conferences are well attended and ISGG is always presented there. However, a dedicated ISGG Desk could also be ‘officially’ arranged at the ICGG events, where information about Membership and promotional materials about ISGG mission and activities can be distributed and explained to the participants. As well as targeted links to specific promotional contents concerning ISGG, which could be made available on the Conference website, or attached to the Calls, during the interval – about one year – of the preparation of the ICGGs. Concerning the second, an ISGG Newsletter is sent to all the ISGG Members about twice in a year, to inform them about relevant items and upcoming ISGG events. However, in the present era dominated by the Internet – even more so during COVID-19 – the ISGG webpage (isgg.net) is for sure the most effective and promising source of information to refer to, and to rely on, even in the future, that is, as the place where ISGG Members may feel to find their virtual house. Which is expected to flourish even more over time, including relevant public information about ISGG mission, patronages, activities and expertise, accessible and retrievable by anyone; scientific ISGG archives and documents reserved to ISGG members; and a private domain where an ISGG Member can access her/his own Membership information. Little details that could further contribute to nourish the sense of belonging to our Scientific Community. This seems to be a trend for Scientific Societies, as parts of competitive networks, which might also be assessed in the future, similarly to the publications. But, as far as we are concerned, as a non-profit Community our main goal remains that of realizing that HUB-effect we mentioned above. Now, apart from strategies and organizational aspects, the question is: why should an individual or an institution decide to apply to remain or become a Member of ISGG? What may ISGG offer to them? The answer to this question is definitely to be sought in the ISGG bylaws [11], which are to be considered as our “constitution”. It is written there, in the Sect. 1.2:

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“The objective of the Society is to foster international collaboration and stimulate scientific research and teaching methodology in the fields of geometry and graphics. To pursue its objectives the Society shall seek to: 1. foster geometry and graphics technology transfer; 2. enhance the quality of graphics and geometry education of designers, engineers, and geometricians, through international co-operation; 3. co-operate with international agencies and corporation; 4. sponsor international conferences, seminars and symposiums in graphics and geometry; 5. establish sponsored research and funding mechanisms; 6. seek membership from graphics and geometry organizations and individuals”. These are the high goals of our Society, which require great energy and a large number of people actively engaged for achievement, especially in the present time of fluid and fast developments, which demand a new impulse. As a first step in this direction, a forum with the representatives of the ISGG Institutional Members has been planned during the ICGG 2022, in order to open a discussion on the new challenges in the fields of Geometry and Graphics from our point of view as an International Society. And to possibly collaborate in establishing an active network which could serve as a reference point, in order to share info about research opportunities and to foster international cooperation, for the benefit of the younger generations of scholars as well. Maybe a series of designated Committees should be also arranged in the future to help ISGG to match the objectives enounced in the list above. As we said before, it is hard to imagine anything different from a webpage to offer a virtual platform to this network. Looking again at the map presented in Fig. 1, we note an unequal distribution of the ISGG Members around the world. It is for our Community a constant reminder of what is still to be done like a warning, recalling the great effort made by the founding fathers of the ISGG, since it was initially conceived in the intangible form of an idea. As a part of our mission addressed to foster international collaboration in the fields of Geometry and Graphic, engaging developing countries is also one of the expected challenges to pursue. About the relevance of this action, Leo Tan Wee Hin and R. Subramaniam stated: “invigorating the scientific society movement in these countries can be a useful way to catalyse the promotion of a popular science culture among their people.” [12]. To this purpose, one can think of some areas of Central Africa, with which is still difficult to communicate, but several other regions of the world could also be mentioned. To the contrary the case of regions where the number of participants, from initially high, decreased over time, as in North American regions, that is, where ISGG had its origins (!), and again, other regions could be mentioned.

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Fig. 5. The ISGG Board: composition and the assigned world regions: a task force designed for promoting a global network in the field of Geometry and Graphics. Diagram by author.

Staying on the distribution of the ISGG Members in the world, the far-sighted wisdom of the founding fathers is to be again remarked. Indeed, as mentioned before, the articulation of the ISGG Board of Regents, together with Honorary President, Former Presidents, current President, and the Treasurer, includes three Vice-Presidency Boards, one for each of the three world regions of ISGG, and each Vice-President is supported by three appointed Directors. A task force well distributed around the globe, with the arduous task of keeping the Society alive and in good health, managing as probably never before to encourage and coordinate local Members of our Society – Individual and Institutional Members – to expand our ramifications, as Fig. 5 aims to symbolize. In order to nurture interest in ISGG worldwide, and to engage new members – Individual and Institutional Members – through an everyday and capillary dissemination of info about ISGG mission and activities in their own countries, universities, institutions, agencies and corporations, local associations and independent scholars [13].

5 Conclusion On the occasion of the 30th anniversary of the decision to establish the International Society for Geometry and Graphics (Melbourne 1992), in honour of this legacy, some reflections have been proposed in this paper on the challenges facing ISGG in the present global context, including the global pandemic event of Covid-19. With a retrospective glance at the origin of our Society, and to its scientific mission, dreamt and carried out since the founding fathers conceived the idea of a scientific community aiming to promote Geometry and Graphics worldwide [14]. It led to the need to reformulate the answers to the basic questions “Who we are”, “What we want”, and must of all “What to do”, further leading to the final question about what our Scientific Society should offer to its members nowadays?, asking which the author felt as a part of his task as the current

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President in Office of ISGG. Far from offering solutions, which are to be discussed by, and shared with, the whole ISGG Community, it seems that some emerging points for an updated agenda can be agreed. First, confirming and reaffirming the idea that ISGG should more and more serve as a HUB, to share information about research opportunities and expertise, to offer patronages and scientific support to conferences and events, to appropriately foster a “think tank” in the vast field of Geometry and Graphics. Secondly, the Webpage is expected to be more and more considered as the “virtual house” of the Members where the opportunity to freely access valuable public information about ISGG mission and current activities and to access by login reserved scientific ISGG archives and document as well as one’s own personal Membership information is offered. This would increase the sense of belonging to a living and active Scientific Community. Finally, but ‘not least in the list’, a permanent, active engagement of the Members of the Board of Regents in a capillary local promotion of the ISGG in all the world regions is recommended. This engagement should include developing countries, asking all the ISGG Members of good will around the world for help. Committees should be arranged with specific tasks according to the ISGG mission, as formulated in the bylaws, to attract New Members, and especially the new generations (!), from universities, institutions, agencies and corporations, local associations, independent scholars. This not so much as to guarantee the ISGG to thrive in splendour, but at least to survive. Acknowledgments. As the current ISGG President, in Office for the Term 2021–2024, the author would like to acknowledge all the Members of the Board of Regents for their active collaboration and wise support, including Honorary and Former Presidents, Treasurer, Vice-Presidents and Directors, and to express his gratitude to all the ISGG Members who nominated and elected him. Thanks to Mrs. Sarah Pye and Prof. Dario Coronelli for their kind help in revising the manuscript.

References 1. Schofield, R.E.: Histories of scientific societies: needs and opportunities for research. History Sci. 2(1), 70–83 (1963) 2. Cerri, C.G.: Cambia il modo di fare scienza: da accademie a società scientifiche. DentalAcademy.it, dentaljournal.it/cambia-scienza-accademie-societa-scientifiche. Accessed 27 Jan 2022 3. Croft, F.M.: The history of the international conference on geometry and graphics – one persons’s reflection. J. Geom. Graph. 20(2), 253–262 (2016) 4. Stachel, H.: Giving birth to the international society for geometry and graphics – a documentation of the years 1988–1994. In: Cocchiarella, L. (ed.) Proceedings of THE 18th International Conference on Geometry and Graphics – 40th Anniversary – Milan, Italy, 3–7 August, 2018, AISC 809, pp. 3–6. Springer, Cham (2019) 5. ISGG webpage. https://isgg.net. Accessed 02 Mar 2022 6. McKie, D.: The rise of scientific societies and periodicals. Phys. Educ. 1, 213–222 (1966) 7. Hilliard, G.K., Vanderwall, W.J. (eds.) Proceedings – INTERNATIONAL CONFERENCE ON DESCRIPTIVE GEOMETRY – Vancouver, Canada, 14–18 June, 1978. ASEE EDGD, Vancouver (1978) 8. Proceedings of ICGG 2018, 2020, 2022, edited by Springer 9. Kuhn, T., S.: The Structure of Scientific Revolutions. The University of Chicago, Chicago and London (1962)

On ISGG’s 30th Agenda: Legacy and Challenges

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10. Baricco, A.: The Game. Einaudi, Torino (2018) 11. See ISGG bylaws at. https://isgg.net/board/isgg-bylaws/. Accessed 2022/02/05 12. Tan Wee Hin, L., Subramaniam, R.: Scientific academies and scientific societies as agents for promoting science culture in developing countries. Int. J. Technol. Manage. 46(1/2), 132–145 (2009) 13. Latour, B.: Pandora’s Hope: An Essay on the Reality of Science Studies. Harvard University Press, Cambridge Mass (1999) 14. Gardner, H., Csikszentmihalyi, M., Damon, W.: Good Work: When Excellence and Ethics Meet. Basic Books, New York (2001)

Theoretical Graphics and Geometry

On the Diagonals of Billiards Hellmuth Stachel(B) Vienna University of Technology, Vienna, Austria [email protected] https://www.geometrie.tuwien.ac.at/stachel

Abstract. A billiard is the trajectory of a mass point in a domain with ideal physical reﬂections in the boundary e. If e is an ellipse, then the billiard’s sides are tangents of a confocal conic called caustic c. The variation of billiards in e with caustic c is called billiard motion. We recall and extend a classical result of Poncelet on the diagonals of billiards which envelope motion-invariant conics. Each billiard in e with caustic c is the ﬂat pose of a Henrici framework. Its spatial poses deﬁne focal billiards in an ellipsoid with a ﬁxed focal conic c . We prove that for even j the j-th diagonals are located on a motion-invariant one-sheeted hyperboloid.

Keywords: Ellipse

1

· Billiard · Caustic · Poncelet grid · Focal billiard

Introduction

A billiard is the trajectory of a mass point in a domain with ideal physical reﬂections in the boundary. Already for two centuries, billiards in ellipses and their projectively equivalent counterparts have attracted the attention of mathematicians, beginning with J.-V. Poncelet [7] and C.G.J. Jacobi [5]. Computer animations carried out by Dan Reznik [8] stimulated a new vivid interest on these well studied objects. They oﬀer an arena where problems can be attacked with analytic and algebraic methods (see, e.g., [2,13]). The sides of any billiard in an ellipse e are tangent to a confocal ellipse or hyperbola c called caustic (Fig. 1). Accordingly, we speak brieﬂy of elliptic or hyperbolic billiards in e. It was Poncelet who proved in the projective setting [7] that if one billiard in e with caustic c closes after N reﬂections, then it closes for each choice of the initial vertex P1 ∈ e. The variation of P1 along e deﬁnes a socalled billiard motion, though it neither preserves angles or side lengths nor is a projective motion. However, the total length of periodic billiards remains constant, and D. Reznik [8] identiﬁed about 50 other invariants, e.g., the sum of cosines of the exterior angles θi (Fig. 1), which was ﬁrst proved in [1]. As shown in [10], the billiards in e with caustic c can be isometrically transformed into spatial billiards in the ellipsoid E through e with the focal conic c . These billiards, which in the periodic case share the total length, are called focal billiards in E since their side lines are generators of confocal one-sheeted hyperboloids H1 and therefore focal lines of E (see [6, p. 284]). c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 19–33, 2023. https://doi.org/10.1007/978-3-031-13588-0_2

20

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/2 θθθ222222/2 /2 P P P2222222

P P P P333333 P P P3333333

P2 e

Q Q Q111111 Q

Q Q22222222222 Q Q Q

c

Q Q Q2222222

Q Q Q333333

Q4444444444 Q Q Q

P P P444444 P P P P444444

Q Q Q111111 Q Q Q555555 Q Q

O

Q3333333333 Q Q Q

P P P1111111

Q Q Q555555 Q Q Q444444 Q

P1

P P P555555

P P P555555 P

Fig. 1. Periodic billiard P1 P2 . . . P5 in e with caustic c and its conjugate P1 P2 . . . P5 .

The goal of this paper is to present new invariants of billiard motions for planar billiards as well as for spatial focal billiards. These invariants are related to the diagonals. In 1822, Poncelet proved in the projective setting that the envelopes of diagonals are conics [7]. A few years later, in 1828 Jacobi showed in [5, p. 388] that in the concyclic case, where the circumscribed and inscribed conics are circles, the diagonals envelope circles, too. It must be noted that the computation of billiards is not as easy as one might expect. The vertices of billiards can either be determined iteratively or, due to Jacobi’s brilliant disclosure, be explicitely represented only in terms of Jacobian elliptic functions (see, e.g., [12]). Structure of the Article. We begin with planar billiards. In Sect. 2 we extend Poncelet’s result by presenting formulas for the envelopes of the j-th diagonals of elliptic billiards, and we determine the contact points. Similar results for hyperbolic billiards follow in Sect. 3, while Sect. 4 focuses on a general equivalence in the projective setting: We prove for polygons with circumconic and inconic that there exists a polarity which sends the vertices to the ﬁrst diagonals. Finally in Sect. 5, we show for the spatial case and even j, that the j-th diagonals of focal billiards are generators of one-sheeted hyperboloids. For the sake of completeness, we repeat at the beginning a few theorems and proofs from the author’s paper [11].

2

Diagonals of Elliptic Billiards

The extended sides of a billiard P1 P2 . . . intersect at points which deﬁne the associated Poncelet grid and are located on confocal ellipses and hyperbolas. We follow the notation in [9] and deﬁne1 1

Note that XY denotes the segment bounded by the points X and Y , while [X, Y ] denotes the connecting line.

On the Diagonals of Billiards

(j) Si

:=

[Pi−k−1 , Pi−k ] ∩ [Pi+k , Pi+k+1 ] for j = 2k, [Pi−k , Pi−k+1 ] ∩ [Pi+k , Pi+k+1 ] for j = 2k − 1

21

(1)

(j)

where i, j = 1, 2, . . . (Fig. 2). For ﬁxed j, the points Si are located on a confocal ellipse e(j) , which remains invariant under the billiard motion [9, Theorem 3.6]. For example, the principal semiaxes of e(1) and e(2) are ae (a2e b2e − d2 ke )2 + 4d2 b2e ke2 ac (a2e b2e − d2 ke ) , a = ae|1 = e|2 a2c b2c − ke2 (b2c a2e − 3b2e ke )(a2e b2e − d2 ke ) − 4d2 b2e ke2 with (ac , bc ) and (ae , be ) as respective semiaxes of the confocal ellipses c and e, and moreover (2) ke = a2e − a2c and d2 = a2c − b2c = a2e − b2e . (1)

(3)

For ﬁxed i, the points Si , Si , . . . belong to the confocal hyperbola through (2) (4) Qi (but not necessarily to the same branch), while Si , Si , . . . are located on the confocal hyperbola through Pi (Fig. 2). As introduced in [9], for each elliptic billiard P1 P2 . . . in e with the contact points Qi ∈ c exists a conjugate billiard P1 P2 . . . in e with contact points Qi ∈ c . It can be deﬁned in the following way (see Fig. 1). There is an aﬃne transformation ac bc x, y with e → c, (3) α : (x, y) → ae be which sends Pi to Qi and Pi to Qi−1 .2 Thus, the relation between the original billiard P1 P2 . . . and its conjugate P1 P2 . . . is symmetric. Below we use the (j) symbol Si for the vertices of the Poncelet grid associated with the conjugate (1) (3) billiard. Then for ﬁxed i, the points Si , Si , . . . belong to the confocal hyper(2) (4) bola through Qi and Pi+1 , while the points Si , Si , . . . are located on the confocal hyperbola through Pi and Qi (Fig. 2). Theorem 1. Let P1 P2 P3 . . . be an elliptic billiard in the ellipse e with the caustic c. Then for ﬁxed j = 1, 2, . . . , the envelope of the diagonals [Pi , Pi+j+1 ] is a coaxial ellipse he|j , provided that in the particular case of N -periodic billiards with even N holds j ≤ [ N 2−3 ]. The ellipse he|j has the semiaxes aj =

ae ac be bc and bj = , ae|j be|j

(4)

where (ae|j , be|j ) are the semiaxes of the ellipse e(j) (Fig. 2). The ellipses he|2 , he|3 , . . . belong to the pencil spanned by c and e. Proof. We focus on the j-th diagonal Pi Pi+j+1 (with j vertices between Pi and Pi+j+1 ). For the case of N -periodic billiards with even N we assume 0 < j ≤ [ N 2−3 ] in order to exclude main diagonals passing through the center O. 2

An aﬃne transformation which keeps the coordinate axes ﬁxed is called aﬃne scaling.

22

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(1) (1) (1) (1) (1) S S S1(1) S 1 1 1 11

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(1) (1) (1) S S S888(1) 88 8

(1) (1) (1) (1)

(1) (1) (1) S S S777(1) S 7 77

Fig. 2. Periodic elliptic billiard P1 P2 . . . P8 inscribed in e with caustic c along with the conjugate billiard P1 P2 . . . P8 (dashed) and the envelopes he|1 of the ﬁrst diagonals (green) and he|2 of the second diagonals (orange). The polarity in the ellipse pe|1 (dotted) sends the vertices Pi to the adjacent ﬁrst diagonals [Pi−1 , Pi+1 ].

The aﬃne scaling α, as deﬁned in (3), sends Pi Pi+j+1 to Qi−1 Qi+j , where and Qi+j are contact points of the conjugate billiard P1 P2 . . . with the caustic c . The pole of [Qi−1 , Qi+j ] w.r.t. c is the point of intersection between , Pi ] and [Pi+j , Pi+j+1 ]. This point belongs to the ellipse e(j) included in the [Pi−1 associated Poncelet grid. According to the notation explained in (1), we obtain (j) the point Si+k for j = 2k as well as for j = 2k + 1 (Fig. 2). For further details see [11], where it is also proved that the standard equation of he|j is an aﬃne combination of the standard equations of c and e. Qi−1

According to (4), the envelope he|j of the j-th diagonals can also be determined as the image of e under an aﬃne scaling with e(j) → c . The following lemma holds for elliptic and hyperbolic billiards. Lemma 1. Let P1 P2 P3 . . . be a billiard in the ellipse e with Q1 , Q2 , Q3 , . . . as (j) contact points with the caustic c and with Si ∈ e(j) for j = 1, 2, 3, . . . as points of the associated Poncelet grid according to (1).

On the Diagonals of Billiards

23

If there is an aﬃne scaling (j)

β : e(j) → c with Si

(j)

→ Qi or Si

→ Qi ,

(5)

then β sends e to the envelope he|j of the j-th diagonals, and the diagonal [Pi , Pi+j+1 ] contacts he|j at the point of intersection between the j-th diagonals [Qi−1 , Qi+j ] and [Qi , Qi+j+1 ] of the polygon Q1 Q2 Q3 . . . . Proof. We distinguish two cases. 1. j is odd, say j = 2k − 1: Then, according to the properties of the Poncelet (j) grid, β sends Si to Qi . We proceed in two steps: We determine the β-image of Pi ∈ e and we show that the j-diagonal [Pi−k , Pi+k ] is the β-image of a tangent to e . This conﬁrms the claim. (i) Due to (1), the extensions of the two sides Pi−1 Pi and Pi Pi+1 through (j) (j) (j) (j) the vertex Pi ∈ e are the lines [Si−k−1 , Si+k−1 ] and [Si−k , Si+k ]. Hence, the intersection Ti of their β-images [Qi−k−1 , Qi+k−1 ] and [Qi−k , Qi+k ] is the β-image of Pi ∈ e (Fig. 2). (ii) The vertex Pi+k is the intersection of the tangents to c at Qi+k−1 and Qi+k . Therefore, the β-preimage Ri+k of Pi+k is the intersection of the (j) (j) (j) tangents to e(j) at Si+k−1 and Si+k . The four tangents drawn from Si+k−1 (j)

and Si+k to c form a quadrilateral where Pi and Pi+2k are two opposite vertices. Now we refer to [9, Theorem 3.5] which says that this quadrilateral is concyclic; the tangents to c at Pi and Pi+2k and those to e(j) (j) (j) at Si+k−1 and Si+k are concurrent angle bisectors. This means that the tangent to e at Pi passes through Ri+k . After replacing k by −k, the same reasoning yields that the tangent to e at Pi also passes through the β-preimage Ri−k of Pi−k (see also [9, Fig. 6]). Thus we conﬁrmed that β sends e to a conic that contacts the j-th diagonal [Pi−k , Pi+k ] at the image Ti of Pi . (j) 2. j even, say j = 2k. In this case holds β : Si → Qi . In view of the j-th diagonal [Pi−k , Pi+k+1 ] we note that (j)

(j)

(j)

(j)

Pi = [Pi−1 , Pi ] ∩ [Pi , Pi+1 ] = [Si−k−1 , Si+k ] ∩ [Si−k , Si+k+1 ]

with the β-image Ti = [Qi−k−1 , Qi+k ]∩[Qi−k , Qi+k+1 ]. In order to show that β sends the tangent to e at Pi to the j-th diagonal [Pi−k , Pi+k+1 ], we proceed similar to the previous case. The only diﬀerence is that the quadrilateral (j) circumscribed to c consists now of the c-tangents passing through Si+k+1 (j)

(j)

(j)

or Si+k at the endpoint Pi+k+1 and through Si−k−1 or Si−k at the other endpoint Pi−k . In the elliptic case the conics e(j) are ellipses confocal with e and c . Therefore, (j) (j) there exists an aﬃne scaling β : e(j) → c with Si → Qi for odd j and Si → Qi for even j (note, e.g., [3, p. 40]). Thus, by virtue of Lemma 1 follows (see Fig. 3)

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P P P333333 cc c cc Q Q222222 Q

TTT222222 h h he|2 e|2 e|2 h e|2 e|2 TT111111 TT TT TT888888

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Fig. 3. Envelope he|2 of the second diagonals of an elliptic billiard (top) and envelope he|3 of the third diagonals of a hyperbolic billiard (bottom) along with the contact points Ti .

Theorem 2. Referring to the previous notation, the envelope he|j of the j-th diagonals of the elliptic billiard P1 P2 P3 . . . is the image of e under an aﬃne scaling β with e(j) → c . The envelope he|j equals the locus of the points of intersection Ti+[ j ] = [Qi−1 , Qi+j ] ∩ [Qi , Qi+j+1 ] between consecutive j-th diagonals 2

of the polygon Q1 Q2 Q3 . . . inscribed in the caustic c (Fig. 3). The following theorem is a consequence of one of D. Reznik’s experiments. Theorem 3. Let P1 P2 . . . be an elliptic billiard in the ellipse e . Then for any and j ∈ {1, 2, . . . }, the diagonal line [Pi , Pi+j+1 ] is polar for even j to Pi+(j/2) for odd j to Pi+[(j+1)/2] w.r.t. a coaxial ellipse pe|j with semiaxes ac bc ap|j = ae and bp|j = be . ae|j

be|j

A proof can be found in [11, p. 147]; an alternative proof follows in Sect. 4.

3

Diagonals of Hyperbolic Billiards

Hyperbolic billiards in ellipses, i.e., billiards with a hyperbola as caustic c diﬀer in various ways from their elliptic counterparts. We recall a few of them.

On the Diagonals of Billiards (2) (2) (2) (2) (2)

(2) (2) (2) SS S66666(2) 6 S 6 6

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(2) (2) (2) S S S8(2) 8 8 S 8 88 (1) (1) (1) (1) (1) (1) (1) (1) e(1) ee ee

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(2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) S (2) (2) S (2) (2) (2) SS S S4444(2) (2) (2) S12 SS 4 12 12 S10 12 10 4 10 10 12 12 10 S S S 10 10 2 2 (5) (5) 2 (5) (5) 2 (5) 22 (5) (5) eee(5) (3) (3) (3) (3) (3) SS S111(3) (3) (3) 1 (3) (3) 1 (3) 1 SS S777(3) 7 Q Q Q111111 7 7

(3) (3) (3) (3) (3) ee e(3) (1) (1) (1) (1) e

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25

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Q Q 12 12 Q Q12 12 12 12

P P 12 12 P12 12 12 12 P P P P2222222 P 10 10 P P10 10 P 10 10 Q Q 11 11 11 Q Q 11 11 11 (3) (3) (3) (3) (3) (3) SS S11 11 11 11 11 11 (5) (5) (5)

(5) (5) (5) SS S777(5) 7 7 7 (2) (2)

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(2) (2) (2) (2) (2) (2) SS (2) S11 11 11 (2) (2) 11 11 11 S1(2) SS 1 1 S 1 11

Fig. 4. Periodic billiard P1 P2 . . . P12 with τ = 1 in the ellipse e with the hyperbola c as caustic, together with the hyperbolas e(1) , e(3) , the secondary axis e(5) , and the ellipse e(2) .

During the billiard motion the vertices of a hyperbolic billiard vary only either on an upper or a lower subarc of the ellipse e (note Figs. 4 or 5). The turning number τ of a periodic billiard counts how often the vertices run to and (1) (3) fro along these arcs. According to [9, Theorem 3.12], the points Si , Si , . . . of the Poncelet grid are located on confocal ellipses through the contact point Qi (2) (4) of [Pi , Pi+1 ] with c , while the points Si , Si , . . . are located on the confocal hyperbola through Pi , but not necessarily on the same branch. For even j, the conic e(j) is a confocal ellipse (if ﬁnite), while for odd j we obtain confocal hyperbolas e(j) or an axis of symmetry. As stated in [9, Corollary 4.3], periodic N -sided billiards with N ≡ 0 (mod 4) are symmetric w.r.t. the secondary axis. For N ≡ 2 (mod 4) and odd turning number τ (Fig. 5), the hyperbolic billiards are centrally symmetric, for even τ symmetric w.r.t. the principal axis of e and c . This results in an exceptional behavior of the N 2−2 -th diagonals: If N ≡ 0 (mod 4), then these diagonals are parallel to the principal axis, while for N ≡ 2 (mod 4) and odd turning number they are diameters of e and otherwise orthogonal to the principal axis. In Theorem 4 we exclude these cases. According to [9, Lemma 3.14], also for hyperbolic billiards there exist one or two conjugate billiards (note Fig. 5). However, there is no kind of symmetry between the contact points Qi ∈ c and the vertices Pi ∈ c since there is no

26

H. Stachel

aﬃne transformation α between the ellipse e and the hyperbola c . This is the reason, why the proofs in Sect. 3 cannot be transferred one-to-one from elliptic to hyperbolic billiards. Theorem 4. Let P1 P2 P3 . . . be a billiard in e with the hyperbola c as caustic. Then, apart from the exceptions listed above, the j-th diagonals envelope for odd j a coaxial ellipse he|j , for even j a coaxial hyperbola. These envelopes belong to the pencil spanned by e and c and have the semiaxes aj =

ae b b a and bj = e c , ac e|j be|j

where (ae|j , be|j ) are the semiaxes of e(j) (Fig. 2). The construction of contact points according to Lemma 1 is still valid for hyperbolic billiards. Proof. 1. j even, say j = 2k, and the conic e(j) is an ellipse, provided that we (j) exclude the particular case with points Si at inﬁnity: There exists an aﬃne scaling ae be (j) γ : (x, y) → ± x, ± y with e(j) → e and Si → Pi (6) ae|j be|j (j)

(j)

for a particular choice of the signs. This aﬃne scaling sends [Si , Si+j+1 ] to the diagonal line [Pi , Pi+j+1 ]. The preimage is the extension of the side Pi+k Pi+k+1 and contacts the caustic c at the point Qi+k . Consequently, the j-th diagonal [Pi , Pi+j+1 ] contacts the γ-image of the caustic at the γ-image γ(Qi+k ) of Qi+k . Thus, we obtain the hyperbola with semiaxes ac ae /ae|j and bc be /be|j as the envelope he|j .3 How to determine the contact point γ(Qi+k ) of [Pi , Pi+j+1 ] with the envelope he|j ? Let πc denote the mapping of lines to their poles w.r.t. the caustic c . Then the coordinate representations of these mappings show that γ ◦ πc = πc ◦ γ −1 and therefore −1 γ(Qi+k ) = γ ◦πc ([Pi+k , Pi+k+1 ]) = πc ◦ γ ([Pi+k , Pi+k+1 ]) (j) (j) = πc [Si+k , Si+k+1 ] = [Qi−1 , Qi+j ] ∩ [Qi , Qi+j+1 ].

In the last equation we use the rule that the pole of the connection of two points is the intersection of the two respective polar lines. 2. j is odd, say j = 2k − 1, and the conic e(j) is a hyperbola. There is an aﬃne transformation (j) β : e(j) → c with Si → Qi , and the claim follows directly from Lemma 1 (note Fig. 3, bottom). The aﬃne combination in [11, p. 146] reveals that also in the case of hyperbolic billiards the conics c, e and he|j belong to a pencil—in alignment with the arguments used below in the proof of Theorem 5. 3

Note that the axial scaling γ can also be used for proving Theorem 1.

On the Diagonals of Billiards

27

(2) (2) (2) (2) (2) (2) (2) (2) (2) (2)

eee

(3) (3) (3) (3) (3) eee(3)

(2) (2) (2) (2)

(2) (2) (2) S S S1(2) 1 1 S 1 11

ccc

h h e|2 e|2 he|2 e|2 e|2 e|2

(2) (2) (2) (2) SS S77777(2) SS S999(2) 9 7 9 9

(1) (1) (1) (1) (1) eee(1)

(2) (2) (2) (2) (2) SS S55(2) (2) (2) 5 (2) 5 (2) 5 (2) (2) 5 5 S S S11 11 11 S 11 11 11

P P777777 P9999999 P P P

(2) (2) (2) (2) (2) ee e e(2) (2) (2) (2) (2) (2) (2) (2) (2) (2) S (2) (2) S S333(2) 3 S S 3 S S13 3 13 13 13 13 13

eee

P P P333333 13 13 P P P13 13P 13 13

P P P P5555555P 11 11 P P11 11 11 11 P P P1111111

h h e|3 e|3 he|3 e|3 e|3 e|3 (4) (4) (4) (4) (4) ee e e(4)

P P888888 P P P4444444P P12 12 P P P 12 12 12 12 12

P P P666666 P 10 10 P P10 10 10 10

P 14 14 P P P14 14 P P222222 P 14 14

(2) (2) (2) (2)

(2) (2) (2) SS S888(2) 8 8 8

(2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) SS S10 S 10 10 S66666(2) 10 S 6 10 10 10 6 6 (2) (2) (2) (2)

(2) (2) (2) (2)

(2) (2) (2) (2) (2) (2) (2) (2) (2) (2) SS S14 S 14 14 14 14 S S S2222 14 14

(2) (2) (2) SS S4444(2) (2) (2) (2) 4 (2) (2) 4 (2) (2) 4 SS S12 12 12 12 12 12

22

Fig. 5. Twofold covered periodic hyperbolic billiard P1 P2 . . . P14 with τ = 3 and its mirror (dashed) along with the conjugate billiard, the second diagonals (green) enveloping the hyperbola he|2 , and the third diagonals (orange) enveloping the ellipse he|3 .

4

An Excursion to Projective Billiards

There is a certain converse of Theorem 3 (Fig. 2). We present a projective version which is valid in all pappian projective Fano planes. Theorem 5. Let P1 P2 P3 . . . with Pi = Pi+1 for i = 1, 2, 3 . . . be a polygon inscribed in a conic e . Then the extended sides [Pi , Pi+1 ] are tangent to a conic c if and only if there exists a polarity w.r.t. a conic p which sends the vertices P2 , P3 , . . . to the respectively adjacent ﬁrst diagonals [P1 , P3 ], [P2 , P4 ], . . . . Remark 1. The statement in Theorem 5 is trivial for N -periodic billiards with N ≤ 5, since there always exists a second-degree curve with ﬁve given tangents or ﬁve given pairs (pole, polar). Proof. We use homogeneous coordinates (x0 : x1 : x2 ) with e : x0 x2 − x21 = 0 and an inhomogeneous parameter t on e such that for the billiard’s vertices holds Pi = (t2i :

ti : 1). The vertices are the poles of the ﬁrst diagonals [Pi−1 , Pi+1 ] w.r.t. p : pik xi xk = 0 with pik = pki if and only if for all i ∈ {1, 2, 3, . . . } consecutive vertices Pi and Pi+1 are conjugate w.r.t. p . This is equivalent to p00 t2i t2i+1 + p11 ti ti+1 + p22 + p01 ti ti+1 (ti + ti+1 ) + p02 (t2i + t2i+1 ) + p12 (ti + ti+1 ) = 0. (7)

28

H. Stachel

On the other hand, the line [Pi , Pi+1 ] with homogeneous line coordinates (u0 : u1 :

u2 ) = (1 : −(ti + ti+1 ) : ti ti+1 ) is tangent to c given by the tangential equation cik ui uk = 0 with cik = cki iﬀ c00 +c11 (ti +ti+1 )2+c22 t2i t2i+1 −2c01 (ti +ti+1 )+2c02 ti ti+1 −2c12 ti ti+1 (ti +ti+1 ) = 0. (8)

The conditions in (7) and (8) are equivalent iﬀ p00 : p01 : p02 : p11 : p12 : p22 = c22 : −2c12 : c11 : 2(c11 + c02 ) : −2c01 : c00 . (9) Clearly, for a given conic c with coeﬃcient matrix (cik ) the second-degree curve p with the coeﬃcient matrix (pik ) is uniquely deﬁned, and vice versa. The geometric meaning guarantees that c and p are irreducible, i.e., det cik = 0 and det pik = 0, which proves the claim. Theorem 5 provides a new approach to Poncelet’s classical result. Corollary 6. If a polygon has a circumconic e and an inconic c, then the ﬁrst diagonals envelope the conic he|1 which is polar to e w.r.t. p . The conic he|1 belongs to the pencil spanned by e and c . Proof. The envelope he|1 is polar to e w.r.t. the conic p , since the polarity in p sends points of e to tangents of he|1 and tangents of p to points of he|1 . If we specify the vertex Pi ∈ e at a point of intersection with c, then the neighbouring vertices Pi−1 , Pi+1 ∈ e coincide and the ﬁrst diagonal [Pi−1 , Pi+1 ] becomes a tangent of e . Hence, the p-pole of this tangent, namely the point Pi ∈ e , must be located on he|1 . If all points of intersection between e and c have the multiplicity 1 , then in the complex extension of the real projective plane and in all projective planes over an algebraically closed ﬁeld K, the conics e, c and he|1 share four points and belong to a pencil. In all other projective planes we proceed in the following way. Let E, C and P denote the respective symmetric coeﬃcient matrices of the conics e, c and p as used above in the proof of Theorem 5. The points Kx of e satisfy xT Ex = 0, and their polars w.r.t. p have the line coordinates u = Px, or conversely, x = P−1 u . Thus, the envelope he|1 has the tangential equation uT P−1 EP−1 u = 0 . The points Ky of he|1 satisfy yT PE−1 Py = 0. It remains to prove that there exist λ, μ ∈ K such that −1 We substitute C = cik ,

PE−1 P = λE + μC.

⎛

⎞ ⎛ ⎞ ⎞ ⎛ 0 0 1 −2c12 c11 0 0 2 c22 1 −1 E = ⎝ 0 −2 0 ⎠ , E = ⎝ 0 −1 0 ⎠ , P = ⎝ −2c12 2(c11 + c02 ) −2c01 ⎠ , 2 c11 −2c01 c00 1 0 0 2 0 0

and obtain as their product PE−1 P the matrix ⎛

⎞ 2c12 c22 − 2c212 −2c01 c22 + 2c02 c12 c00 c22 − 2c01 c12 + c211 2 2 ⎝ −2c01 c22 + 2c02 c12 8c01 c12 − 4c02 c11 − 2c02 − 2c11 −2c00 c12 + 2c01 c02 ⎠ , c00 c22 − 2c01 c12 + c211 −2c00 c12 + 2c01 c02 2c00 c11 − 2c201

On the Diagonals of Billiards

29

and this equals λ E+μ C with λ = c00 c22 −4c01 c12 +2c02 c11 +c211 , μ = 2 det cik , which conﬁrms the claim. Remark 2. Theorem 5 holds not only for the ﬁrst diagonals of any elliptic billiard in an ellipse e, but also for the j-th diagonals with j = 3, 5 . . . . This follows iteratively, when we replace c by one of the envelopes of diagonals he|1 , he|3 , . . . . In the billiard case, there is also another extension: We can replace e with one of the conics e(1) , e(2) , . . . of the Poncelet grid and apply an aﬃne scaling γ with e(j) → e .

5

Diagonals of Focal Billiards

We follow the notation in [9] and [10]. Let E be the ellipsoid with the semiaxes ae > be > ce . Then, its focal ellipse c has the semiaxes ac , bc with ac2 = a2e − c2e and bc2 = b2e − c2e . The quadrics which are confocal with E satisfy the equations x2 y2 z2 + 2 + = 1 with + k bc + k k

ac2

k ∈ R \ {−ac2 , −bc2 , 0}

(10)

as a parameter called elliptic coordinate. The family of confocal central quadrics contains ⎧ 0 < k = k0 < ∞ triaxial ellipsoids, ⎨ −bc2 < k = k1 < 0 one-sheeted hyperboloids, for (11) ⎩ −ac2 < k = k2 < −bc2 two-sheeted hyperboloids, c in the plane the focal ellipse c as the limit for k = 0, and the focal hyperbola 2 y = 0 as the limit for k = −bc with the semiaxes ac = ac2 − bc2 = d and bc = bc . We recall that the family of confocal quadrics sends through each point P = (x, y, z) with xyz = 0 three mutually orthogonal surfaces, one of each type (see, e.g., [6, p. 279]). The parameters (k0 , k1 , k2 ) of these quadrics are the elliptic coordinates of P and satisfy x2 =

(ac2 + k0 )(ac2 + k1 )(ac2 + k2 ) 2 (b 2 + k0 )(bc2 + k1 )(bc2 + k2 ) 2 k0 k1 k2 , y = c , z = 2 2 . 2 2 2 (ac − bc )ac bc2 (bc2 − ac2 ) ac b c

Conversely, eight points in space, symmetrically placed w.r.t. the coordinate frame, share the elliptic coordinates (k0 , k1 , k2 ). The given ellipsoid E has the elliptic coordinate k0 = c2e > 0 . As described in [10], each elliptic billiard with circumellipse e and caustic c in the plane z = 0 can be isometrically transformed into a focal billiard on the ellipsoid E through e with focal ellipse c . The sides of this spatial billiard lie on generators of a confocal hyperboloid H1 , which intersects E along a line of curvature e , the common trajectory of the vertices during a billiard motion (Fig. 6).

30

H. Stachel (2)

S15

(2)

S19

(2)

S1 (2)

S11

e

13 P P P13 13 13 13 13

e(2) (2)

S7

(2) (2) (2) (2) (2) (2) S S S17 17 17 17 17 17

P P P999999 P (2) (2) (2) (2) (2) (2) S S S21 21 21 21 21 21 P P P111111 P

(2) S3

P P P17 17 17 P 17 17 17

(2) (2) (2) (2)

e ee e

P P 15 15 P15 P 15 15 15

(2) (2) (2) S S13 S S 13 13 13 13 13 P 11 P 11 P11 11 11 11

H1 P7

P16 P P 16 16 16 16 16 16

19 P P 19 P19 19 19 19

(2)

S9

E (2)

S14

P P 20 20 P20 20 20 20

(2)

S18 P P 14 14 14 P14 14 14 14

P P 10 10 P10 10 P 10 10 10

(2) S16 (2) S12 (2)

S8

(2)

P666666 P P

S22 e(2)

H1 (2)

S4

Fig. 6. N -periodic focal billiard (red) with N = 22 and turning number τ = 5 in the ellipsoid E along with the line e(2) (blue) of the associated spatial Poncelet grid and the (2) (2) (2) inscribed focal billiard (orange) consisting of two 11-gons Si Si+3 Si+6 . . . with τ = 5 each on the confocal one-sheeted hyperboloid H1 .

Theorem 7. Let P1 P2 . . . be a focal billiard on the ellipsoid E with vertices on the line of curvature e being the intersection between E and the confocal onesheeted hyperboloid H1 with semiaxes ah1 , bh1 and ch1 . Then for even j = 2k, the diagonals [Pi , Pi+j+1 ] are generators of a coaxial one-sheeted hyperboloid Dj which belongs to the pencil of quadrics spanned by E and H1 . The hyperboloid Dj has the semiaxes ad|j =

ae ah1 be bh1 ce ch1 ce ch1 , bd|j = , cd|j = = , ae|j be|j ce|j a2e|j − ac2

where ae|j , be|j , ce|j are the semiaxes of the confocal ellipsoid through the ellipse e(j) of the planar Poncelet grid. Only for N -periodic focal billiards and j = N2 −1, the diagonals belong to a quadratic cone. In the plane z = 0 of the focal ellipse c , the trace points of the j-th diagonals form a polygon where the extended sides coincide with the j-th diagonals of the polygon formed by the trace points of the original focal billiard. The same holds for the plane y = 0 of the focal hyperbola c .

On the Diagonals of Billiards

31

Remark 3. It is worth to be noted that at focal billiards any two consecutive j-th diagonals [Pi , Pi+j+1 ] and [Pi+1 , Pi+j+2 ] are intersecting when j is even. Proof. The j-th diagonals form a spatial polygon with vertices Pi , Pi+j+1 , Pi+2(j+1) , Pi+3(j+1) , . . . on e , provided that the billiard is not (2j + 2)-periodic. In a similar way, the associated spatial Poncelet grid on the hyperboloid H1 (j) (note Fig. 6 or [10, Figs. 6 and 7]) contains the quartic e(j) , and the points Si , (j) (j) Si+j+1 , Si+2(j+1) , . . . on e(j) are vertices of a spatial polygon with sides along the rulings of H1 , provided that j is even.4 There is an axial scaling of the form be ce ae (j) x, ± y, ± z with e(j) → e and Si → Pi . δ : (x, y, z) → ± ae|j be|j ce|j Hence, the j-th diagonals belong to the δ-image of H1 , which is a one-sheeted hyperboloid Dj with the stated semiaxes and the gorge ellipse in the plane z = 0 . The axial scaling ac bc x, y, 0 α : (x, y, z) → ah1 bh1 sends H1 to the exterior of the focal ellipse c in z = 0 , the j-th diagonals of the focal billiard to j-th diagonals of e , and the gorge ellipse of H1 to c . The restriction of α to z = 0 is bijective and maps the top views of the j-th diagonals to the j-th diagonals of e . Thus, the envelope to the top views of the spatial diagonals, i.e., the gorge ellipse of Dj , is bijectively related to the envelope he|j , and we can transfer the construction of contact points of he|j , as given in Lemma 1, to that of trace points of the spatial diagonals. Finally, it should be noted that several of the presented theorems can also be veriﬁed using the canonical parametrization of the billiards in terms of the Jacobian elliptic functions to the modulus d/ac (see, e.g., [4] and [12, Theorem 4.3]). As an example, we demonstrate this for Theorem 7. According to [10, Theorem 13], the two components of e = E ∩ H1 can be parametrized as ae ah1 be bh1 ce ch1 u) = − sn u ˜, cn u ˜, ± dn u ˜ (12) e1,2 (˜ ac bc bc where the transition from any vertex Pi of the billiard to the next one Pi+1 corresponds to the parameter’s shift by a constant 2Δ˜ u combined with a change ˜, then for even of the sign of the z-coordinate. Hence, if Pi has the parameter u ˜j := u ˜ + 2(j + 1)Δ˜ u , i.e., Pi = e1 (˜ u) j the vertex Pi+j+1 has the parameter u uj ). and Pi+j+1 = e2 (˜ 4

For odd j the points Pi and Pi+j+1 belong to the same component of e(j) , and the (j) (j) line [Si , Si+j+1 ] is no generator of H1 .

32

H. Stachel

In order to verify that [Pi , Pi+j+1 ] ⊂ Dj , it is necessary and suﬃcient to show that for even j the points Pi and Pi+j+1 are conjugate w.r.t. Dj . From [12, Cor. 4.5] follows u ˜ −˜ u

ae|j =

a dn j2 u] ac dn [(j + 1)Δ˜ = c u˜j −˜ , u cn [(j + 1)Δ˜ u] cn 2

be|j =

bc bc = , u ˜ −˜ u cn [(j + 1)Δ˜ u] cn j2

hence ke|j = a2e|j − ac2 = ac2

dn2

u ˜j −˜ u ˜ −˜ u u − cn2 j2 2 u u ˜ −˜ cn2 j2

=

u ˜j −˜ u 2 u u ˜ −˜ cn2 j2

bc2 sn2

.

Thus, it remains to show that a2e|j a2e a2h1 b2e|j b2e b2h1 ke|j c2e c2h1 sn u ˜ sn u ˜ + cn u ˜ cn u ˜ + dn u ˜ dn u ˜j = 1, j j 2 2 a2e ah1 ac2 b2e bh1 bc2 c2e c2h1 bc2 hence dn2

u ˜j − u ˜ u ˜j − u ˜ u ˜j − u ˜ sn u ˜ sn u ˜j + cn u dn u ˜ dn u ˜j = cn2 . ˜ cn u ˜j + sn2 2 2 2

This is an identity due to the addition theorems and half-angle theorems of elliptic functions [4].

6

Conclusion

We extended Poncelet’s classical results and discussed the diagonals of elliptic and hyperbolic billiards P1 P2 . . . in an ellipse e as well as that of focal billiards in an ellipsoid E. We proved that the envelopes of the j-th diagonals in the plane belong to a pencil of conics, and we disclosed a remarkable relation between the j-th diagonals of the original billiard P1 P2 . . . and that of the contact points Q1 Q2 . . . . In space, the j-th diagonals form ruled quadrics contained in a pencil through E, but only for even j . Periodic billiards yield periodic polygons of j-th diagonals. All obtained results can immediately be projectively generalized to statements on planar polygons with circumconic and inconic as well as to spatial counterparts.

References 1. Akopyan, A., Schwartz, R., Tabachnikov, S.: Billiards in ellipses revisited. Eur. J. Math. (2020). https://doi.org/10.1007/s40879-020-00426-9 2. Bialy, M., Fierobe, C., Glutsyuk, A., Levi, M., Plakhov, A., Tabachnikov, S.: Open problems on billiards and geometric optics. Arnold Math. J. (2022). https://doi. org/10.1007/s40598-022-00198-y 3. Glaeser, G., Stachel, H., Odehnal, B.: The Universe of Conics. From the ancient Greeks to 21st Century Developments. Springer, Heidelberg (2016). https://doi. org/10.1007/978-3-662-45450-3

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4. Hoppe, R.: Elliptische Integrale und Funktionen nach Jacobi (2004–2015). http:// www.dfcgen.de/wpapers/elliptic.pdf. Accessed May 2021 5. Jacobi, C.G.J.: Ueber die Anwendung der elliptischen Transcendenten auf ein bekanntes Problem der Elementargeometrie. Crelle’s J. 3(4), 376–389 (1828) 6. Odehnal, B., Stachel, H., Glaeser, G.: The Universe of Quadrics. Springer, Heidelberg (2020). https://doi.org/10.1007/978-3-662-61053-4 7. Poncelet, J.-V.: Trait´e des propri´ete´es projective des ﬁgures. Tome premier, Bachelier, Libraire, Paris 1822. deuxi`eme edition, Gauthier-Villiars, Imprimeur-Libraire, Paris (1865) 8. Reznik, D., Garcia, R., Koiller, J.: Fifty new invariants of n-periodics in the elliptic billiard. Arnold Math. J. 7/2 (2021). https://doi.org/10.1007/s40598-021-00174-y 9. Stachel, H.: The geometry of billiards in ellipses and their poncelet grids. J. Geom. 112, 40 (2021). https://doi.org/10.1007/s00022-021-00606-2 10. Stachel, H.: Isometric billiards in ellipses and focal billiards in ellipsoids. J. Geom. Graph. 25(1), 97–118 (2021) 11. Stachel, H.: Billiard motions in ellipses - invariants of projective nature. In: Velichov´ a, D., L´ aviˇcka, M., Szarkov´ a, D. (eds.): Proceedings of the Slovak-Czech Conference on Geometry and Graphics 2021, Koˇcovce/Slovakia, pp. 143–148 (2021). http://www.ssgg.sk/scg/archiv/scg2021 online.pdf 12. Stachel, H.: On the motion of billiards in ellipses. Europ. J. Math. (2022). https:// doi.org/10.1007/s40879-021-00524-2 13. Tabachnikov, S.: Geometry and Billiards. American Mathematical Society, Providence/Rhode Island (2005)

Exploring the Steiner-Soddy Porism Ronaldo Garcia1 , Liliana Gabriela Gheorghe2 , and Dan Reznik3(B) 1

2 3

Federal University of Goi´ as, Goiˆ ania, Brazil [email protected] Federal University of Pernambuco, Recife, Brazil [email protected] Data Science Consulting, Rio de Janeiro, Brazil [email protected]

Abstract. We explore properties and loci of a Poncelet family of polygons – called here Steiner-Soddy – whose vertices are centers of circles in the Steiner porism, including conserved quantities, loci, and its relationship to other Poncelet families.

Keywords: Triangle Invariant

1

· Inversive · Pedal polygon · Porism · Poncelet ·

Introduction

A Steiner chain is a set of pairwise-tangent circles, all of whom are tangent to a pair of disjoint circles, called the “Soddy” circles, see Fig. 1 (left). The chain is poristic since it is the inversive image of a set of identical, mutually-tangent circles, centered at the vertices of a regular polygon, see Fig. 1 (right). In this article, we explore the family of “Steiner-Soddy” polygons whose vertices are the centers of circles in a Steiner chain. Main Results. Let P denote any polygon in the Steiner-Soddy family. – Any polygon P is conic-inscribed and circumscribe a circle, and is therefore in a Poncelet poristic family [7]. – The outer conic of the P has foci on the centers of the inner and outer Soddy circles of the Steiner chain. – When the center of the caustic is on the circumference of the inner Soddy circle, the family of polygons P becomes parabola-inscribed and the outer Soddy circle degenerates to a line. – P conserves the sum of powers of half-angle tangents, up to power N − 1. – The locus of the perimeter centroid1 is a conic. When N = 3 (P are poristic triangles), we further obtain: 1

The weighted average of the midpoints of sides, where the weights are sidelengths.

c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 34–46, 2023. https://doi.org/10.1007/978-3-031-13588-0_3

Exploring the Steiner-Soddy Porism

35

– The sum of tangents of half angles is the same as the sum of the cotangents of its intouch triangle and are both conserved. – The aforementioned sum is less than, equal, or greater than 2, iﬀ the poristic family is ellipse, parabola, or hyperbola inscribed, respectively. – When the family is parabola-inscribed, the locus of the orthocenter is a line. – In the spirit of [15], we tour loci of some triangle centers over the P, showing some to be stationary, circles, lines, conics, and non-conics. Related Work. In [20] it is proved that the moments of curvatures, up to N − 1 are invariant over Steiner’s porism. Loci of vertex, area, and perimeter centroids of a generic Poncelet family are studied in [19], where it is shown that the ﬁrst two are always conics. In [12] a condition is derived which guarantees that a certain center of a triangle exists iﬀ the sum of half-tangents is less than two; by [9] this is equivalent to whether the two Soddy circles are nested or not. The Poncelet family which is the polar image of bicentrics was studied in [3], and that of harmonic polygons, in [17]. Seminal studies of loci of triangle centers over families of Poncelet triangles include [15,21,24,25]. A theory for the type of locus swept by triangle centers over the confocal Poncelet family is in [13].

Fig. 1. Left: A regular Steiner chain of N pairwise tangent circles centered at the vertices of a regular polygon; an inner and an outer “Soddy” circles can be drawn tangent to all circles of the chain. Right: A Steiner chain is the inversive image of a regular chain, with respect to an inversion circle (shaded red). A Steiner-Soddy polygon P (orange) is any polygon whose vertices are the centers of the circles in a Steiner chain. The vertices of P lie on a ﬁxed conic, whose foci are the centers of the inversive images of the two regular Soddy circles, and their sides tangent a ﬁxed circle (dashed) centered at I: a Poncelet porism is in place. The harmonic polygon, e.g. the pedal polygon of P with respect to I (magenta), is also Ponceletian.

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Article Organization. In Sect. 2 we prove new properties of the Steiner-Soddy family for an arbitrary N . In Sect. 3 we specialize to the N = 3 case and prove sharper facts. Loci of triangle centers in the N = 3 are toured in Sect. 4. We ﬁnish with a discussion in Sect. 5 comparing invariants of various Poncelet families. Explicit formulas and equations for some of the objects mentioned herein are collected in the Appendix.

2

The Steiner-Soddy Porism

The set of all polygons P with vertices at the centers of circles in a Steiner porism is a “Steiner-Soddy” family. The points of contact between consecutive circles in the chain are concyclic, since these lie on the inversive image of a regular polygons’ incircle; call this circle C and its center I (see Fig. 1). Denote by H the polygon whose vertices are the aforementioned contact points. H is known to be harmonic, [5]. The tangents at circles in a Steiner chain, at their contact points, meet at I [23, pp. 120, 244–245], H is the pedal polygon of P with respect to I, and the sides of P are tangent to C. The Steiner porism has two ﬁxed “Soddy” circles S and S which are tangent be their inversive pre-images in the to all N circles in chain. Let Sreg and Sreg regular chain in Fig. 1 (left). Proposition 1. P is inscribed in a conic E which is an (i) ellipse, (ii) hyperbola, or (iii) parabola if the inversion center is (i) interior to Sreg or exterior to Sreg , , or (iii) on either circle. (ii) in the annulus between) Sreg and Sreg Furthermore, the foci of E are the centers of the inner and outer Soddy circles S and S . In the case of a parabola, the center of S is at infinity and S becomes a line parallel to the directrix. Proof. The locus of the center of a circle internally tangent to a pair of circles is an ellipse (resp. hyperbola) if the circles are nested (resp. disjoint). Said locus is a parabola, when one circle has inﬁnite radius. Since P is inscribed to a conic E and circumscribed to a circle C: Corollary 1. The family P is Ponceletian. Remark 1. In [11] it is shown that the pedal family H is also Ponceletian. Its envelope (or caustic) is the Brocard inellipse B; see Appendix for explicit expressions. Let θi denote the internal angle of P at its i-th vertex Pi . N Theorem 1. The P family conserves i=1 tan(θi /2). Proof. Referring to Fig. 1, let Pi (resp. Pi ) be the centers (resp. the contact points between) consecutive circles in the Steiner chain. Let r (resp. ri ) denote the radius of C (resp. of a circle in chain centered at Pi ). Since |Pi Pi | =

Exploring the Steiner-Soddy Porism

37

|Pi Pi+1 | = ri and |OPi | = |OPi+1 | = r, the line Pi O bisects Pi Pi Pi+1 = θi . Since H is the I-pedal of P , C is tangent to Pi Pi+1 at Pi , so P P O = 90◦ ; i i OPi hence, tan(P . Thus: i Pi O) = Pi Pi

r θi tan = 2 ri

hence

N i=1

k

tan (θi /2) =

N k r i=1

ri

, for any k.

(1)

The claim is obtained by invoking a result from [20, Theorem 1], namely that over Steiner’s porism, the sum of the k th powers of curvatures of circles in chain, up to N − 1, is invariant. When E is a hyperbola, the sign of tan(θi /2) must be ﬂipped for the two angles whose neighboring vertices lie on diﬀerent branches of the hyperbola. Remark 2. Richard Schwartz [18] has suggested an elegant interpretation of conservations of the above type: All homogeneous polynomials in 2-variables of degree less than N have the same average on the unit circle as they do on a regular N -gon inscribed in the circle.

Fig. 2. Left: If the inversion center is on Sreg , S degenerates to a line (vertical cyan), and the Soddy polygon P (orange) is inscribed into a parabola (gray), whose directrix is parallel to S ; in this case, I, the center of the caustic C (dashed grey), is on S (cyan circle). Right: When S and S are external, P is hyperbola-inscribed. The family oscillates between two states: (i) all but one vertex lie on a single branch (shown), and (ii) all vertices lie on a single branch (not shown).

Proposition 2. Over a hyperbola-inscribed Steiner-Soddy family, the vertices of P oscillate between two states: (i) all on a single branch of the hyperbola or (ii) N − 1 contiguous vertices on one branch and one on the other branch. This corresponds to whether I is interior (resp. exterior) to H, the I-pedal polygon.

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Proof. The vertices of P are the centers of the circles in the Steiner chain. Since S and S are tangent to all the circles in the chain, when they are external (as in Fig. 2), the centers of the circles in the chain are located on a hyperbola with foci at the Soddy’s circle’s centers. The hyperbola’s branch closer to the smaller Soddy circle, contains the centers of circles that tangent externally S and S , while the distal branch, contains the centers of the chain that tangent internally both. The vertices of P will be on the same branch iﬀ all the circles in the chain tangent externally the Soddy circles. By contrast, if there exist some circle in the chain, Γ , centered on the distal branch, then Γ will tangent externally both S and S , as well as two of its neighbours in the chain. But this implies that Γ will contain all the circles in the chain: otherwise, at least one of the two Soddy cannot be tangent to one of those aforementioned circle. There cannot be another circle in the chain, say Γ also centered on the distal branch; otherwise, Γ would necessarily contain all the circles in the chain, including Γ , which is absurd. To prove the claim on I, note that if Γ is internally tangent to two of its neighbours in the chain, the intersection of the two tangents drawn at the tangency points of the latter two with Γ will be outside Γ , hence outside the pedal polygon itself. But this point is precisely I, the center of the caustic circle. A Relation with the Homothetic Family. By homothetic family we mean a Poncelet family associated to a pair of concentric and homothetic conics. Let their foci be called “inner” and “outer” ones. In [17] it was shown that the harmonic family is the polar image of the homothetic family with respect to an inversion circle centered at one of the outer foci. Referring to Fig. 3:

Fig. 3. The Steiner-Soddy family, orange (resp. harmonic family, magenta), is the polar image of the “homothetic” family, with respect to a circle (not shown) centered at an inner (resp. outer) focus.

Exploring the Steiner-Soddy Porism

39

Proposition 3. The Steiner-Soddy family is the polar image of the homothetic family with respect to an inversion circle centered at one of its inner foci. Centroid Loci. In [19] it was proved that the locus of C0 , the vertex centroid and of C2 , the area centroid, over any Poncelet family, are conics; by contrast, the locus of C1 is not always a conic. Arseniy Akopyan reminded us that if a polygon circumscribes a circle whose center is in I, then C1 , C2 , I are collinear and |IC1 |/IC2 | = 3/2 [1]. Thus, referring to Fig. 4: Corollary 2. Over the Steiner-Soddy porism of any number N of sides, the locus of the perimeter centroid C1 is a conic. Due to symmetry, the loci of C0 and C1 are coaxial with the outer conic of the porism.

Fig. 4. Over the Soddy-Steiner porism (orange), the loci of the vertex, area, and perimeter centroids C0 , C1 , C2 are conics (dashed red, green, blue) coaxial with the outer ellipse (gray). The points I, C2 , C1 are collinear and |IC1 |/|IC2 | = 3/2

3

The Special Case of N = 3

Consider a Steiner-Soddy family of triangles as in Fig. 5 (left). Above, we give a direct proof of Theorem 1, based on Descartes’ theorem. Proof. Let 1/ρ = 1/r1 + 1/r2 + 1/r3 denote the sum of the curvatures of the circles in the chain. By Descartes’ theorem [22]:

1 1 1 1 + + + r1 r2 r3 r4 1 1 1 1 + + + r1 r2 r3 r5

2 =2 2

1 1 1 1 + 2 + 2 + 2 r12 r2 r3 r4

1 1 1 1 =2 2 + 2 + 2 + 2 r1 r2 r3 r5

40

R. Garcia et al.

where r4 and r5 denotes the radii of the inner and outer Soddy circles, and r1 , r2 , r3 those of the circles in the chain. Subtracting and factoring, we obtain:

1 1 − r4 r5

2 1 1 + + ρ r4 r5

=2

1 1 − r4 r5

1 1 + r4 r5

hence 2/ρ = (1/r4 + 1/r5 ). By Fig. 5 (left): tan

A B C r + tan + tan = 2 2 2 ρ

(2)

Since r4 , r5 , and r are ﬁxed, ρ is necessarily constant, hence so is the sum of half tangents, ending the proof. Remark 3. The outer Soddy circle is a line iﬀ r5 = +∞; by Eq. (2) this led to B C tan A 2 +tan 2 +tan 2 = 2. The latter condition was derived in [12], by a diﬀerent approach. Proposition 4. Let ABC a triangle in a Steiner-Soddy family and let A B C be its pedal with respect to I, the center of the former’s i-circle. Over the porism, the left and right hand side of each equation are equal and conserved: tan tan2

B C A + tan + tan = cot(A ) + cot(B ) + cot(C ) 2 2 2

A B C + tan2 + tan2 = cot2 (A ) + cot2 (B ) + cot2 (C ) 2 2 2

Proof. As in Fig. 5, let the circles in the Steiner chain be centered at A, B, C. I is (both) the i-center of ABC, as well as the circumcenter of A B C . By the properties of tangents at a circle, AI ⊥ B C ; by hypothesis, IC ⊥ AB. B are either equal, or supplementary. Since both are and IC Therefore, IAC A . Thus, = IC acute, they have the same measure; similarly, IBC cot(C ) = cot(

+B A π−C C ) = cot( ) = tan 2 2 2

and Equation (1) ﬁnishes the proof. Proposition 5. In the N = 3 case, the outer Soddy circle degenerates to a line when any one of the following equivalent conditions is fulfilled: (i) tan A 2 + tan B2 + tan C2 = 2; (ii) the outer conic is a parabola; (iii) the circumcenter of the I-pedal family (N = 3 Brocard porism) is at a co-vertex of its (Brocard) √ √ inellipse, whose aspect ratio is 5/2 = φ − 1/2 ≈ 1.118, where φ = (1 + 5)/2 is the golden ratio. Proof. Consider the 1d-family of triangles P1 = (R cos t, R sin t), P2 = (R cos(t+ 2π 2π 2π 2π 3 ), R sin(t + 3 )) and P3 = (R cos(t − 3 ), R sin(t − 3 )) which are the centers of a Steiner chain of 3-circles interscribed in two Soddy circles centered √ at (0, 0) √ √ (2± 3)R and of radius rext = (2+ 3)R/2 and rint = (2− 3)R/2. Let I = (± , 0) 2

Exploring the Steiner-Soddy Porism

41

and consider the inversion with respect to a circle of radius λ and centered at I. √ (2+ 3)R Taking I = (− , 0) and computing the parabola stated in (ii) it follows, 2 from straightforward calculations, that: √ √ √ 2 + 3 9 3λ2 + 2 R2 − 16 λ2 2 3 + 3 Ry 2 . − x= 4 λ2 4R Computing the Brocard porism as shown in Fig. 6 it follows that: ⎛

2+

⎞2 √ 2 √ √ 2 2 3 − 3 λ4 3 4 3λ − 3 R2 − 6 λ2 ⎠ + y2 − =0 6R 9 R2

2+

⎞2 √ √ √ 3 15 R2 − 18 λ2 + 8 3λ2 4 λ4 4 3 − 7 4 y2 ⎠ + + =0 30 R 5 75 R2

⎝x − ⎛ ⎝x +

Direct analysis, leads to the result stated in item (iii).

Fig. 5. Left: A Steiner-Soddy chain consist in three circles (blue) centered at ABC’s vertices and mutually tangent at A ,B , C ,, along with two Soddy circles (light blue). The angles at ABC’s vertices are bisected and same color marks equal angles. Right: Let ABC be a parabola-inscribed Poncelet family of triangles whose caustic is a circle centered at the parabola axis. Over the family, the locus of the orthocenter X4 is a line (purple) parallel to the directrix (dashed red).

4

Loci in the N = 3 Case

In this Section we will refer to triangle centers using the Xk notation as in [14]. Referring to Fig. 5 (right): Proposition 6. For any parabola-inscribed Poncelet family whose caustic is an axis-centered circle, the locus of the orthocenter X4 is a line parallel to the directrix.

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The following proof was kindly contributed by Alexey Zaslavsky [26]: Proof. Consider the unit parabola y = x2 , and let the center and radius of the incircle be given by (0, y0 ) and r. Then points (r, y0 − r) and (−r, y0 − r) lie on the parabola, thus y0 = r2 +r. Now let A(a, a2 ), B(b, b2 ), C(c, c2 ) be the vertices of the triangle. Then the distances from the incenter (0, r2 + r) to lines AB and AC equal r and we obtain b + c and bc as functions of a. Thus, the y−coordinate of the orthocenter is −(1 + ab + ac + bc) = r2 + 2r − 1 is independent of a. Remark 4. Consider the parabola 4cy = x2 and the circle centered at (0, y0 ). For the pair to admit it can be shown that r = 4y0 c/ 16c2 + x20 ,

Poncelet triangles, where x0 = 2 −2c2 + y0 c + 2 c3 (c + y0 ). In this case, the locus of X4 is the line y = −4c + x20 /(4c) = (−6c2 + y0 c + 2 c3 (c + y0 ))/c. For example, assume the parabola is 4cy = x2 . Then the locus of X4 is the line y = −7c/4. √ Remark 5. For y0 = (7±2 7)c the locus X4 coincides with the directrix y = −c. Corollary 3. Over a parabola-inscribed Soddy-Steiner family, the locus of its orthocenter X4 is a line parallel to the directrix.

Fig. 6. Over the Steiner-Soddy family (larger red triangle), the loci of the centroid X2 (green), circumcenter X3 (blue) and orthocenter X4 (purple) are conics. The smaller (red) triangle is Brocard-poristic, [10]. live.

It is known that if the intouch triangle is acute, its circumcenter X3 (resp. symmedian X6 ) coincides with the incenter X1 (resp. Gergonne point X7 ) of the reference.

Exploring the Steiner-Soddy Porism

43

Let S3 denote the N = 3 Steiner-Soddy family. While the intouch of S3 is acute (its harmonic O-pedal), X1 and X7 are stationary. Otherwise, while S3 is everted (it is inscribed in a hyperbola), X1 and X7 follow arcs curves of degree at least four. Here are some experimental observations about the loci of Xk , some of which are explicitly derived in Sect. 5: 1. 2. 3. 4. 5.

Xk , k =2, 3, 4, 5 are conics, as predicted in [13], see Fig. 6. live Xk , k =6, 8, 9, 10 are also conics, though not in general [13]. live Xk , k =13, 14, 15, 16, 80 are circles. live Xk , k =20, 77, 170 are segments along the major axis of E. live X105 is a circle centered at the major axis of E and tangent to it at two points. live

Remark 6. Let R be the circumcircle of the regular polygon in the inversive preimage √ of the Steiner-Soddy family, and let (x0 , 0) be the inversion center. With √ x0 = ( 10 ± 6)R/4, the Steiner Steiner-Soddy family is hyperbola-inscribed, and the locus of X4 is identical to said hyperbola. See it live.

5

Comparing Poncelet Conservations

Conservations of various Poncelet families studied so far as well as their polar images with respect to certain signiﬁcant points is shown in Table 1. As we study more of these canonical families, one of our goals (still elusive) is to identify a functional pattern in the conserved quantities. Table 1. Poncelet families, polar transformations, and invariants Family

Invariant Polar center/conic cos θi [2, 4] Outer ell confocal Inner/outer focus cos θi [17] Outer circle bicentric Incircle inward cos θi [2] incircle Outer ell Incircle inward cos θi [2, 6] Outer circle circum cot θi [8] homothetic outer focus Inner focus cot θi [11] Outer circle harmonic Focus

New family

New invariant cos(2θi ) [16] Tangential cos θi Bicentric sin(θi /2) [3] Bic-tang cos θi MacBeath sin θi Circum sin θi Circum cos θi Incircle cot θi Harmonic tan(θi /2) Steiner-Soddy tan(θi /2) Steiner-Soddy cot θi Harmonic

Acknowledgements. We thank Alexey Zaslavsky and Arseniy Akopyan for contributing some proofs and facts, and Richard Schwartz for useful discussions. The ﬁrst author is fellow of CNPq and coordinator of Project PRONEX/CNPq/FAPEG 2017 10 26 7000 508.

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Appendix: Explicit Formulas Referring to Fig. 1, let the inversion circle be centered at inv = (x0 , 0) and have radius λ. Let α be shorthand for π/N . Let E the Poncelet conic the Steiner-Soddy family is inscribed into; let C denote the family’s circular caustic. Proposition 7. The x coordinates of foci f1 and f2 and vertex V of E are given by: f1,2 =

x0 R4 cos4 α − R2 cos2 α(λ2 − 2x20 ) ± 2R2 sin αλ2 + 2R2 (λ2 − 2x20 ) − x20 (λ2 − x20 ) R4 cos4 α + 2R2 cos2 αx20 − 4R2 x20 + x40

cos(2α)R2 x0 + R(Rx0 + 2λ2 − 4x20 ) − 2x0 (λ2 − x20 ) Vx = R2 cos(2α) + R2 − 4Rx0 + 2x20

Proposition 8. The parameters of the caustic C = (I, r) are given by: I = x0 +

2x0 λ2 , 0 , 2 2(R2 cos2 α − x0 )

λ2 R cos α |R2 cos2 α − x20 |

r=

Proposition 9. Let the Brocard inellipse (caustic of the pedal family with respect to I) be centered at O with semi-axes a , b . These are given by:

O =

x0 +

R2 (R2

λ2 R(x20 − R2 cos2 α) cos2 α , − 4x20 ) cos4 α + 2R2 x20 cos2 α + x40

a =

λ2 x0 (R2 cos2 α cos(2α) − x20 ) ,0 − 4x20 ) cos4 α + 2R2 x0 2 cos2 α + x40

R2 (R2

λ2 R cos2 α b = R2 (R2 − 4x20 ) cos4 α + 2R2 x20 cos2 α + x40

Proposition 10. The loci of X2 , X3 and X4 are the conics given by: 2

2

4

2

2

4

2

2

2 3 2

6

4

2

X2 :(R − 4x0 )(R − 56R x0 + 16x0 )x + (R − 4x0 ) y − 2x0 (R − 60R x0 4

2

+ 12R λ + 240R 2

2

4

4 x0 2

− 192R 2

2

2 2 x0 λ

2

2

−

6 64x0

4

+

4 2 64x0 λ )x

6

4

2 6 x0 (R

+

2

2

4

2

4

2

− 60R x0 + 24R λ 4

+ 240R x0 − 384R x0 λ + 144R λ − 64x0 + 128x0 λ − 64x0 λ ) = 0 4

X3 :(R − 56R

2

2 x0

2

+

2 10 4x0 )(R

− 2x0 (R − 4

4 2 16x0 )(R

4

2 4 2 4x0 ) x

−

− 68R

2

8

2 x0

2

+ (R − 8

2 2 4 4x0 ) (R

2

+ 28R λ + 736R

2

8

2

+

4 (1008x0

6

2

2 2 2080x0 λ

6

4 x0

2

2

4 2 2

2 2 x0 λ

− 2944R x0

+ 40R x0 + 16x0 ) y

− 928R

10

6

8

4

2

2

− 768R x0 λ + 4352R x0 − 2560R x0 λ − 1024x0 + 1024x0 λ )x + x0 (R +

2 (−72x0

+

4 4 256x0 (63x0 2

2

+ 56λ )R

2 4

10

2 2 4x0 λ

+

2

−

4

4

− 18λ )R −

4

2

4

+ 784λ )R −

6 2 2048x0 (x0

2

4

8

2

−λ

2

2 )(9x0

2 4 256x0 (23x0 2

2

− 2λ )R +

2 2 2

6

12 2

2

4

− 23x0 λ + 16λ )R

8 2 4096x0 (x0

8

6

2 2

−λ ) )=0

2

6

2

X4 :(R − 4x0 ) x + (R − 56R x0 + 16x0 )(R − 4x0 ) y + x0 (−2R + 32R x0 + 40R λ 4

4

4

2

2

2

6

2

4

2

8

6

2

− 192R x0 + 96R x0 λ + 512R x0 − 1152R x0 λ − 512x0 + 512x0 λ )x +

2 8 x0 (R

− 896R

− 16R

2

2 4 x0 λ

6

+

2 x0

6

2

− 40R λ + 96R

8 256x0

−

6 2 512x0 λ

+

4

4 x0

− 96R

4 4 256x0 λ )

4

2 2 x0 λ

4

4

2

6

2

4

2

=0

8

4

4

8

2

2 2

2

4

4 2

2

2 3 2

X6 :(R − 4x0 )(R + 544R x0 + 256x0 )(R + 4x0 ) x + (R + 16x0 ) (R − 4x0 ) y 2

− 2x0 (R +

+

2

+

2 2 10 2 4x0 ) (R (8λ

− 48λ

+

− 2176R

4 4 34x0 )R

+

2

− 4R λ − 4R

2

2 x0

4 x0

8

4

+ 1536R λ 2 2 x0 (R

2 2 10 4x0 ) (R

4

6 x0

2

8

2 x0

2

6 x0

− 512R λ

+

2 8 4x0 )R

4 2 256x0 (λ

−

6

2

2

8 x0

− 160R λ + 256R 4

−

2 x0

2 2 x0 )R

+ 544R 2

6

4 x0

8

10

+ 1024λ x0 − 1024x0 )x

2

+ (16λ − 320λ

2 2 x0 )(3λ

2

+ 400R λ − 256R x0 + 1152R x0 λ

Proposition 11. The locus of X6 is the ellipse given by: 2

2 x0

−

+

4 6 544x0 )R

6 2 1024x0 (λ

−

2

4

− 64x0 (15λ 2 2 x0 ) )

6

=0

Exploring the Steiner-Soddy Porism

45

Proposition 12. When the center of inversion is internal to the Soddy circle Sreg or exterior to Sreg , the locus X10 is the ellipse given by: 4

2

2

4

2

2 2

2

2

2 4 2

X10 : (R − 56R x0 + 16x0 )(R − 4x0 ) x + (R − 4x0 ) y 2

2

6

2 32(2x0

2

4

2

4

2

2

4

2

2

2

6

4

2

− 2x0 (R − 4x0 )(R − 60R x0 + 16R λ + 240R x0 − 176R x0 λ − 64x0 + 64x0 λ )x 8

+ (R − +

4 2 256x0 (x0

6

− λ )R + 2 2

−λ )

2 )x0

4 (480x0

−

2 2 480x0 λ

4

4

+ 256λ )R −

2 2 512x0 (x0

2

−λ

2 )(2x0

2

− λ )R

2

=0

Proposition 13. When the center of inversion is internal to the Soddy circle, Sreg the locus X15 is the circle given by:

X15 :

x− =

x0 (−256x60 + 144R2 x40 − 24(R4 + 6R2 λ2 )x20 + R6 + 12R4 λ2 ) R6 − 24R4 x20 + 144R2 x40 − 256x60

2 +y

2

36864R2 x80 λ4 (R6 − 24R4 x20 + 144R2 x40 − 256x60 )2

− + Proposition 14. The locus of X20 is a segment [X20 , X20 ] contained in axis x of length L20 . These points are given by: +

X20 =

x0 ζ + 4x0 R5 + 304R3 x0 λ2 − 64x30 (x20 − λ2 )(R + x0 ) (R2 + 8Rx0 + 4x20 )(R − 2x0 )3 (R + 2x0 )

X20 =

x0 (ζ − 4x0 R5 − 304R3 x0 λ2 + 64x30 (x20 − λ2 )(R − x0 )) (R2 − 8Rx0 + 4x20 )(R + 2x0 )3 (R − 2x0 )

L20 =

18432x40 R5 λ2 , (R2 − 4x20 )3 (R4 − 56R2 x20 + 16x40 )

−

where ζ = R6 + (76λ2 − 28x20 )R4 + 16x20 (12λ2 + 7x20 )R2 .

References 1. Akopyan, A.: Private communication (2022) 2. Akopyan, A., Schwartz, R., Tabachnikov, S.: Billiards in ellipses revisited. Eur. J. Math. (2020). https://doi.org/10.1007/s40879-020-00426-9 3. Bellio, F., Garcia, R., Reznik, D.: Parabola-inscribed Poncelet polygons derived from the bicentric family. J. Croatian Soc. for Geom. Gr. (KoG) (2021, to appear) 4. Bialy, M., Tabachnikov, S.: Dan Reznik’s identities and more. Eur. J. Math. (2020). https://doi.org/10.1007/s40879-020-00428-7 5. Casey, J.: A Sequel to the First Six Books of the Elements of Euclid, 5th edn. Hodges, Figgis & Co., Dublin (1888) 6. Chavez-Caliz, A.: More about areas and centers of Poncelet polygons. ArnoldMath J. (2020). https://doi.org/10.1007/s40598-020-00154-8 7. Dragovi´c, V., Radnovi´c, M.: Poncelet Porisms and Beyond: Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics. Frontiers in Mathematics. Springer, Basel (2011). https://doi.org/10.1007/978-3-0348-0015-0 8. Galkin, S., Garcia, R., Reznik, D.: On aﬃne images of regular polygons (2022). in preparation 9. Garcia, R., Gheorghe, L., Moses, P., Reznik, D.: Triads of conics associated with a triangle. J. Croatian Soc. Geom. Gr. (KoG) (2022, to appear). arXiv:2112.15232 10. Garcia, R., Reznik, D.: A matryoshka of Brocard porisms. Eur. J. Math. (2022). https://doi.org/10.1007/s40879-022-00529-5

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11. Garcia, R., Reznik, D., Roitman, P.: New properties and invariants of harmonic polygons (2021). arXiv:2112.02545 12. Hajja, M., Yﬀ, P.: The isoperimetric point and the point of equal detour in a triangle. J. Geom. 87(7), 76–82 (2007) 13. Helman, M., Laurain, D., Reznik, D., Garcia, R.: Poncelet triangles: a theory for locus ellipticity. Beitr. Algebra Geom. (2022). https://doi.org/10.1007/s13366-02100620-0 14. Kimberling, C.: Encyclopedia of triangle centers (2019). bit.ly/3mOOver 15. Odehnal, B.: Poristic loci of triangle centers. J. Geom. Graph. 15(1), 45–67 (2011) 16. Reznik, D., Garcia, R., Koiller, J.: Fifty new invariants of n-periodics in the elliptic billiard. Arnold Math. J. 7, 341–355 (2021) 17. Roitman, P., Garcia, R., Reznik, D.: New invariants of Poncelet-Jacobi bicentric polygons. Arnold Math. J. 7(4), 619–637 (2022). https://doi.org/10.1007/s40598021-00188-6 18. Schwartz, R.: Private communication (2021) 19. Schwartz, R., Tabachnikov, S.: Centers of mass of Poncelet polygons, 200 years after. Math. Intell. 38(2), 29–34 (2016) 20. Schwartz, R.E., Tabachnikov, S.: Descartes circle theorem, Steiner porism, and spherical designs. Am. Math. Monthly 127(3), 238–248 (2020) 21. Skutin, A.: On rotation of a isogonal point. J. Classical Geom. 2, 66–67 (2013) 22. Weisstein, E.W.: CRC concise encyclopedia of mathematics, 2nd edn. Chapman and Hall/CRC, Boca Raton (2002) 23. Wells, D.: The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, London (1991) 24. Zaslavsky, A., Chelnokov, G.: The Poncelet theorem in Euclidean and algebraic geometry. Math. Educ. 4(19), 49–64 (2001). in Russian 25. Zaslavsky, A., Kosov, D., Muzafarov, M.: Trajectories of remarkable points of the Poncelet triangle. Kvant 2, 22–25 (2003) 26. Zaslavsky, A.: Private communication (2021)

Circumparabolas in Chapple’s Porism Boris Odehnal1(B) 1 2

and Dan Reznik2

University of Applied Arts Vienna, Vienna, Austria [email protected] Data Science Consulting Ltd., Rio de Janeiro, Brazil

Abstract. We study the two-parameter manifold of parabolas circumscribed to triangles in a Poncelet porism between two circles (Chapple’s porism). It turns out that the focal points of the parabolas in a certain one-parameter subfamily trace a straight line. The vertices of these parabolas move on rational cubic curves whose acnodes trace an ellipse centered at the poristic stationary triangle center which is the midpoint of the common incenter and the common circumcenter. The axes of the circumparabolas envelop a Steiner hypocycloid over the course of a porism. Varying the pivot point of the circumparabola, the one-parameter family of Steiner cycloids envelops two ellipses, one with the fixed common incenter and circumcenter as foci, the other one carrying the cusps of all cycloids. Keywords: Circumparabola · Triangle · Porism · Point orbit Envelope · Focus · Axis · Vertex · Quintic · Septic

1

·

Introduction

Overview of Known Results. The one-parameter family of triangles interscribed between a common circumcircle u and a common incircle i is known as Chapple’s porism, see [5], referred below as the “poristic family”. Within the past ten years, porisms in more general forms (including Chapple’s) have been studied focused on various aspects: (i) poristic traces of triangle centers in [8–11,21,22], (ii) derivation of invariants by means of numerical experiments in [15,17,23], (iii) ellipticity of poristic traces of triangle centers and bicentric pairs [16], (iv) historical point of view [4]. Experiments in [24] in particular have motivated a detailed study of circumparabolas of the poristic family. The results presented here try to verify the numerical and experimental results by means of algebraic techniques. We shall keep technical details aside and try to formulate proofs short and traceable. Therefore, we will sometimes not lay down all equations in detail, especially if they are of enormous length and high complexity. Techniques. Since we are dealing with the contents of Euclidean geometry, we use Cartesian coordinates (x, y) for points. Whenever favorable, we will switch to c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 47–58, 2023. https://doi.org/10.1007/978-3-031-13588-0_4

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−1 homogeneous coordinates x0 : x1 : x2 by setting x = x1 x−1 0 and y = x2 x0 with x0 = 0. Thereby, we will perform the projective closure of the Euclidean plane (with the ideal line ω : x0 = 0) and we shall also allow homogeneous coordinates to be complex. This allows us to describe a pair of complex conjugate points

I = 0 : 1 : i,

J = I = 0 : 1 : −i

on the ideal line ω, called the absolute points of Euclidean geometry (see [14]). Each Euclidean circle contains both I and J, and each conic through I and J is a Euclidean circle. Further, a line in the projectively-closed complex extension of the Euclidean plane is called isotropic if it contains either I or J. We choose a Cartesian coordinate frame such that the equations of the circumcircle u and the incircle i read u : (x − d)2 + y 2 = R2 , i : x2 + y 2 = r2 .

(1)

It is natural to parametrize u by means of trigonometric functions as u = (R cos τ + d, R sin τ ),

with τ ∈ R.

(2)

In what follows, rational parametrizations of point orbits are essential (if available). Whenever computations are carried out by a computer algebra system (CAS), rational parametrizations are preferred. Therefore, trigonometric functions will be replaced with their rational equivalents cos τ =

1 − T2 , 1 + T2

sin τ =

2T . 1 + T2

(3)

Present Contributions. In Sect. 2, we ﬁrst parametrize the poristic family. Then, we determine the ideal points of the circumparabolas with the help of the isogonal conjugation. Subsequently, we derive the equations of the circumparabolas of all triangles in the poristic family. Section 3 is devoted to the traces of the parabolas’ foci, the envelopes of their axes which turn out to be Steiner’s hypocycloids. The latter envelop an ellipse, while the cusps trace another ellipse. Finally, we look at the traces of the vertices and other points.

2

Parametrization of the Porism and Circumparabolas

Vertices of the Triangles. We assume that the circumcircle u and the incircle i of a poristic triangle family are given by their Eq. (1), where the inradius r, the circumradius R, and the central distance d (distance between the incenter X1 and the circumcenter X3 ) are related via the Euler triangle formula d2 = R2 − 2Rr

(4)

which guarantees a porism (cf. [5,18,19]). Here and thereafter, triangle centers are labeled according to C. Kimberling’s encyclopedia, cf. [19,20]. Clearly,

Circumparabolas in Chapple’s Porism

49

r, R ∈ R+ , and no restriction is imposed if we assume d ∈ R+ (the coordinate frame can always be chosen such that d > 0). First, we parametrize the one-parameter family of triangles Δ = P1 P2 P3 interscribed between u and i, i.e., the poristic family. Let the triangle vertex P1 be given by (2). The points P2 and P3 are found as the intersections of the tangents from P1 to the incircle i with the circumcircle. This yields σδ · P2,3 = 2R(2dR cos τ + R2 + d2 )2 · ± W(R2 − d2 ) sin τ − 4dR3 cos2 τ − (R4 + 6d2 R2 + d4 ) cos τ − 4d3 R, sin τ (d4 − R4 − 4d2 R2 − 4dR3 cos τ ) ∓ W((R2 + d2 ) cos τ + 2dR) where W :=

√

(5)

8dR3 cos τ + 3R4 + 6R2 d2 − d4 and σ := R + d,

δ = R − d.

Ideal Points of the Circumparabolas, Isogonal Conjugation. Each triangle Δ = P1 P2 P3 in the Euclidean plane can serve as the fundamental triangle of a special quadratic Cremona transformation, called the isogonal transformation, cf. [14,19]. Any point Q (not on the sidelines of Δ) is mapped to its isogonal conjugate ι(Q) by intersecting the reﬂections of the Cevians [Q, Pi ] in Δ’s (interior) angle bisectors [X1 , Pi ]. It is easily shown that if Q = Pi is chosen on Δ’s circumcircle u, ι(Q) is an ideal point (point at inﬁnity). Further, ι is quadratic, i.e., it maps lines (not incident with any Pi ) to conics passing through all fundamental points Pi . Therefore, a tangent of the circumcircle is mapped to a parabola circumscribed to the fundamental triangle. Some circumparabolas of a certain triangle in the poristic family are shown in Fig. 1. Equations of the Circumparabolas Related to the Poristic Family. Having deﬁned the isogonal transformation, we can now determine the equations of the one-parameter family of circumparabolas for each triangle in the poristic family. In fact, we are about to determine a two-parameter family of parabolas: The ﬁrst parameter T determines one particular triangle in the poristic family. The second parameter U determines one particular parabola circumscribed to Δ. We may assume that the tangents to the circumcircle touch the circumcircle at a point Q ∈ u which can be given by means of rational coordinate functions as Q(U ) = u(U ) (cf. (2)) with some real parameter U . Since X1 = (0, 0) (center of i from (1)), we ﬁnd the directions of the axes of all circumparabolas by reﬂecting [P1 , Q] in [P1 , X1 ]. This yields the direction vector a of the circumparabolas, or if we use homogeneous coordinates, the ideal points A(U, T ) = ι(Q) of all circumparabolas as A(U, T ) = 0 : T 3 δ 2 − δ(δ + 2σ)T 2 U − σ(2δ + σ)T + σ 2 U : : −T 3 U δ 2 − δ(δ + 2σ)T 2 + σ(2δ + σ)T U + σ 2 .

(6)

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Fig. 1. Circumparabolas pi of a triangle P1 P2 P3 as isogonal images of the tangents ti .

The points P1 , P2 , P3 , and A determine a unique parabola since the tangent at A is known: It is the line at inﬁnity. A homogeneous equation of the circumparabolas will have the form p : a00 x20 + 2a01 x0 x1 + 2a02 x0 x2 + a11 + 2a12 x1 x2 + a22 x22 = 0

(7)

with coeﬃcients aij ∈ R[δ, σ, U, T ]. In order to make p a parabola, the coeﬃcients aij have to satisfy (8) a11 a22 − a212 = 0, since this is the condition on p to touch the ideal line ω : x0 = 0. Inserting the rational equivalents of P1 , P2 and P3 from (5), and (6) into (7) and using (8), we ﬁnd a00 : a01 : a02 : a11 : a12 : a22 = δσ(σ + δU 2 )(1 + T 2 )(δ 2 T 2 + σ 2 )2 : : (σ 2 − δ 2 U 2 )(1 + T 2 )(δ 2 T 2 + σ 2 )2 : 2δσU (1 + T 2 )(δ 2 T 2 + σ 2 )2 : : −(δ + σ)(δ 2 T 3 U + δ(δ + 2σ)T 2 − σ(2δ + σ)T U − σ 2 )2 : : −2(δ + σ)(δ 2 T 3 U + δ(δ + 2σ)T 2 − σ(2δ + σ)T U − σ 2 )· ·(δ 2 T 3 − δ(δ + 2σ)T 2 U − σ(2δ + σ)T + σ 2 U ) : −(δ + σ)(δ 2 T 3 − δ(δ + 2σ)T 2 U − σ(2δ + σ)T + σ 2 U )2 .

(9)

Note that the one-parameter family of circumparabolas of any triangle Δ (in the poristic family) is itself a conic in the Veronese model V22 ∈ P5 : Inserting

Circumparabolas in Chapple’s Porism

51

the homogeneous coordinates of the three vertices P1 , P2 , P3 for x0 , x1 , x2 in (7) yields three hyperplanes in P5 which are to be intersected with the quadratic cone determined by (8).

3

Properties of the Circumparabolas

Focal Traces. According to von Staudt, a point F is a focus of a planar algebraic curve if the tangents from F to c are a pair of complex conjugate isotropic lines, cf. [1,6,14]. Now, it is rather elementary to determine the tangents tI and tJ from I and J to p that diﬀer from the line at inﬁnity. Then, the one and only (real) focus of p is the point F = tI ∩ tJ with homogeneous coordinates f0 : f1 : f2 = 4(δ + σ)(U −T )(1 + U 2 )· · δ 3 (δ + 2σ)T 2 U 2 − δ 2 σ 2 T 2 + 2δσ(δ + σ)2 T U − δ 2 σ 2 U 2 + (2δ + σ)σ 3 : : δ 4 (δ 2 −4σ 2 )T 3 U 4 −2δ 3 σ(2δ 2 +7δσ+4σ 2 )T 3 U 2 −δ 2 σ 2 (4δ 2 +4δσ−σ 2 )T 3 − −δ 3 (δ+2σ)(δ 2 −4δσ−4σ 2 )T 2 U 5 + 2δ 2 σ(δ+2σ)(4δ 2 +7δσ+2σ 2 )T 2 U 3 + +δσ 2 (δ+2σ)(4δ 2 −σ 2 )T 2 U + δ 2 σ(σ+2δ)(4σ 2 −δ 2 )T U 4 + +2δσ 2 (σ+2δ)(2δ 2 +7δσ+4σ 2 )T U 2 + σ 3 (σ+2δ)(4δ 2 +4δσ−σ 2 )T + (10) +δ 2 σ 2 (δ 2 −4δσ−4σ 2 )U 5 − 2δσ 3 (4δ 2 +7δσ+2σ 2 )U 3 − σ 4 (4δ 2 −σ 2 )U : : δ 4 (δ + 2σ)2 T 3 U 5 + 2δ 3 σ 2 (δ + 2σ)T 3 U 3 + δ 2 σ 2 (4δ 2 + 8δσ + σ 2 )T 3 U + +δ 3 (δ + 2σ)(δ 2 + 8δσ + 4σ 2 )T 2 U 4 + 2δ 3 σ(δ + 2σ)(σ + 2δ)T 2 U 2 + +δσ 2 (δ+2σ)(σ+2δ)2 T 2 − δ 2 σ(σ+2δ)(δ+2σ)2 T U 5 − −2δσ 3 (δ+2σ)(σ+2δ)T U 3 − σ 3 (σ+2δ)(4δ 2 +8δσ+σ 2 )T U − −δ 2 σ 2 (δ 2 + 8δσ + 4σ 2 )U 4 − 2δ 2 σ 3 (σ + 2δ)U 2 − σ 4 (σ + 2δ)2 . where U is as deﬁned in Eq. (6). Now we are able to show: Theorem 1. Over Chapple’s porism, and a fixed U , the foci of circumparabolas of triangles in the poristic family trace a straight line given by F : 4(δ + σ)(1 + U 2 )(δ 2 U 2 − σ 2 )x − 8δσ(δ + σ)U (1 + U 2 )y+ (11) +δ 2 (δ 2 −4δσ − 4σ 2 )U 4 −2δσ(2δ 2 +3δσ+2σ 2 )U 2 −σ 2 (4δ 2 +4δσ−σ 2 ) = 0. Proof. With (10), we have a parametrization of the trace of the foci of the circumparabolas of all triangles in the poristic family. If we eliminate T from (10), we obtain an equation of the trace of the focus of the circumparabolas. This yields (11) which is the equation of a straight line. Lines F from (11) envelop a sextic curve. This can be easily shown by elimd F. inating parameter U from F and its derivative dU The parametrization of the foci allows us to verify a result established in [13]: Theorem 2. The locus of the focus of all circumparabolas of three points is a circular quintic.

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Fig. 2. Over poristic triangles (variable T ), if U (and thus, Q) is fixed, the focus of circumparabolas (for fixed U ) move along a straight line while the axes pass through a fixed point F .

Proof. In order to ﬁnd the equation of the locus F of the foci of all circumparabolas, we have to eliminate U from the aﬃne parametrization (f1 f0−1 , f2 f0−1 ). (without loss of generality, the computation is simpliﬁed if we ﬁx T , or even set it equal to 0). The rather long equation for F can be found (for a special type of coordinatization) in [13]. Fig. 3 shows the focus curve F for one particular triangle. Indeed, this result is not related to porisms. However, each triangle in the poristic family deﬁnes its own quintic. Parabolas’ Axes. With the direction (6) of the axis and the focus (10), the axes a of all parabolas are well-determined, yielding a : 2(δ+σ)(1+U 2 ) (δ 2 T 3 U +δ(δ+2σ)T 2 −σ(2δ+σ)T U −σ 2 )x+ +(δ 2 T 3 −δ(δ+2σ)T 2 U −σ(2δ+σ)T +U σ 2 )y + (12) +δ 4 T 3 U 3 −δ 2 σ(2δ+σ)T 3 U +δ 2 (δ+2σ)2 T 2 U 2 −δσ 2 (δ+2σ)T 2 + −δ 2 σ(2δ+σ)T U 3 +σ 2 (2δ+σ)2 T U −(δ 2 σ 2 +2δσ 3 )U 2 +σ 4 = 0. This allows us to compute the set of vertices of circumparabolas over triangles in the poristic family. Thereby, we can verify another well-known result (see [12]): Theorem 3. The locus of all vertices of circumparabolas of a triangle is a circular septic curve.

Circumparabolas in Chapple’s Porism

53

Fig. 3. The vertices of all circumparabolas lie on a septic curve V (red), while their foci lie on a quintic curve F (violet).

The Envelope of Axes over the Poristic Family. The equations of the axes (12) depend on two parameters: (i) The parameter T describing P1 ∈ u and the other two vertices of poristic triangles. (ii) The parameter U determining Q ∈ u, and therefore, a parametrization of the family of circumparabolas. This allows us to consider the axes (12) as two independent one-parameter families of lines, each of which envelops a certain curve. Referring to Fig. 2, we ﬁrst show that: Theorem 4. For a fixed pivot Q ∈ u, i.e., a fixed ideal point ι(Q), the axes of circumparabolas of triangles in the poristic family pass through a fixed point F . The set of all points F (while Q traverses u) is an ellipse ei with the semi-axes ai = centered at X1358

R 2 + d2 δ2 + σ2 = , 4(δ + σ) 4R = d2 , 0 .

bi =

R 2 − d2 δσ = 2(δ + σ) 4R

(13)

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Proof. Eliminating the poristic parameter T from the two equations a (given in d a yields the poristic envelope. Over the real numbers, the (12)) and aT := dT resultant of a and aT factors into 3 polynomials, one of degree 1 L : 2(δ + σ)U (1 + U 2 ) U x + y + U (U 2 δ 2 − 2δσ − σ 2 ) = 0 (14) and two of degree 2. The two quadratic polynomials describe two pairs of complex conjugate lines (one of which is an isotropic pair) emanating from the real point 2 1 σ − U 2δ2 . (15) F = 2δσU 2(δ + σ)(1 + U 2 ) This point is incident with the line L, and over the course of all pivot points Q traces the circumcircle u, while the point F itself moves on the ellipse ei : 4δ 2 σ 2 (δ+σ)2 x2 +2δ 2 σ 2 (δ−σ)(δ+σ)2 x+(δ+σ)2 (δ 2 +σ 2 )2 y 2 −δ 4 σ 4 = 0 (16) which has semi-axes (13) and is centered at X1358 .

The ellipse ei has for real foci the incenter X1 = (0, 0) and circumcenter X3 = (d, 0) common to all triangles in the porism. It is therefore centered at X1385 . (For details on relations between said triangle centers see [20].) It can be also be shown: Theorem 5. The axes of all circumparabolas of a fixed triangle P1 P2 P3 envelop a Steiner cycloid. Proof. Computing the envelope of the axes for a ﬁxed triangle can be done by d a. This results in eliminating parameter T from a (given in (12)) and aU = dU a quartic factor Q and a linear factor M. Q = 0 describes a Steiner cycloid (rational, bicyclic, quartic curve with three cusps of the ﬁrst kind, cf. [2,3,6,7, 25]) (it is rather technical to show that Q = 0 indeed describes a one-parameter family of Steiner cycloids). The linear factor M : 2T (δ + σ)(δ 2 T 2 − 2δσ − σ 2 )x − 2(δ + σ)(δ(δ + 2σ)T 2 − σ 2 )y+ +δ 2 T (δ 2 T 2 − 2δσ − σ 2 ) = 0 is the equation of a line which is tangent to the cycloid Q = 0 for all T ∈ R. Figure 4 shows the Steiner cycloid enveloped by the axes of all circumparabolas of a certain triangle in the poristic family. Envelopes of Steiner Cycloids and a Further Porism Theorem 6. The envelope of all Steiner cycloids over the poristic family consists of two ellipses ei , ec : ei (given in (16)) is internally tangent to all cycloids, while ec carries the cusps of the cycloids. Like ei , ec is also centered at X1385 and has the following semi-axes: ac =

3R2 − d2 δ 2 + 4δσ + σ 2 = , 4(δ + σ) 4R

bc =

3R2 + d2 δ 2 + δσ + σ 2 = . 2(δ + σ) 4R

(17)

Circumparabolas in Chapple’s Porism

55

Fig. 4. While the circumparabola p traverses the family of all circumparabolas of a fixed triangle in the poristic family, its axes envelop a Steiner cycloid s.

Proof. The quartic polynomial Q given in the proof of Theorem 5 depends on the porism parameter T . Therefore, it describes a one-parameter family of Steiner d Q = 0. Eliminating T from the cycloids whose envelope is given by Q = 0 and dT latter two equations, we obtain an implicit equation for the envelope factoring into two ellipses with the semi-axes (13) and (17). Figure 5 shows some Steiner cycloids together with the trace ec of the cusps which is obviously traced thrice in the course of one poristic round. Figure 5 also shows the ellipse ei enveloped and touched by the Steiner cycloids. Since the cycloids’ cusps on the outer ellipse ec are singular points (of multiplicity two) considered on the cycloids, they contribute to the envelope of the cycloids. Theorem 6 does not only describe the envelope of the Steiner cycloids, but it also allows us to formulate another kind of porism: Consider two nested and concentric ellipses ei and ec such that the major axis of ei lies on the minor axis of ec . If it is possible to draw a Steiner cycloid with cusps on ec such that the cycloid touches the inner ellipse ei three times for one particular starting point C (cusp) on ec , then the same is possible for any choice of C ∈ ec . Of course, ellipses ec and ei must satisfy certain conditions (maybe not as simple as those in [14, Thm. 9.5.4, p. 432]) in order to guarantee the existence of such a porism to exist.

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Fig. 5. Over poristic triangles, the Steiner cycloids as envelopes of the axes of the circumparabolas envelop two ellipses ei and ec . The triangles’ incenter X1 and circumcenter are the real foci of ei .

Fig. 6. The poristic trace of the vertices of the circumparabolas is a rational cubic C.

Vertices and Other Points. With the equations of the parabolas (7), (9), and the equations of their axes (12), we can ﬁnd the vertices. The intersection of a parabola with its axis yields only one proper point, the vertex V . The homogeneous coordinates v0 : v1 : v2 of V are bi-variate polynomials vi (T, U ) of bi-degree (9, 7) and reads:

Circumparabolas in Chapple’s Porism

57

v0 : v1 : v2 = 4(δ+σ)(T −U )(1+T 2 )(1+U 2 )2 (δ 2 T 2 +σ 2 )2 · · δ 3 (δ + 2σ)T 2 U 2 − δ 2 σ 2 T 2 + 2δσ(δ + σ)2 T U − δ 2 σ 2 U 2 + σ 3 (2δ + σ) : : δ 8 (4σ 2 − δ 2 )T 9 U 6 + . . . + δ 7 (δ + 2σ)(δ 2 − 4δσ − 4σ 2 )U 7 T 8 + . . . : : −δ 8 (δ + 2σ)2 T 9 U 7 + . . . + δ 7 (δ + 2σ)(δ 2 − 4δσ − 4σ 2 )T 8 U 6 + . . . . (18) The parametrization (18) allows us to verify the following: Theorem 7. Over poristic triangles Δ = P1 P2 P3 , the vertices of all circumparabolas move on a cubic curve. Proof. We consider (v1 v0−1 , v2 v0−1 ) as a parametrization of a curve depending on the poristic parameter T . The elimination (by means of resultant) of T from x = v1 v0−1 and y = v2 v0−1 yields a product of a cubic polynomial 2 2 2 2 2 2 2 4(δ + σ)2 (1 + U ) (x + y ) (σ − δ U )x + 2δσU y − −(1+U 2 )(δ+σ) 4δ 2 (δ 2 −δσ−σ 2 )U 4 −4δσ(δ 2 +3δσ+σ 2 )U 2 − −4σ 2 (δ 2 +δσ−σ 2 ) x2 + 12δσ(σ 2 − U 2 δ 2 )U xy+ + (δ 2−4δσ−4σ 2 )δ 2 U 4 −2δσ(2δ 2−3δσ+2σ 2 )U 2 −σ 2 (4δ 2+4δσ−σ 2 ) y 2 − (19) − (δ 2 −4δσ−4σ 2 )δ 2 U 4 −2δσ(2δ 2 +3δσ+2σ 2 )U 2 −σ 2 (4δ 2 +4δσ−σ 2 ) · · δ 2 U 2 x − 2δσU y − σ 2 x + δσ(δU 2 + σ)(δ 2 U 2 + σ 2 )2 = 0 and degree one polynomial −4(δ+σ)(1+U 2 ) δ 4 (δ+2σ)2 U 10 +δ 2 (δ+2σ)(4δ 3 +10δ 2 σ+δσ 2 −2σ 3 )U 8 + +δ(δ 5 + 6δ 4 σ + 4δ 3 σ 2 − 20δ 2 σ 3 − 21δσ 4 − 4σ 5 )U 6 + +σ(4δ 5 + 21δ 4 σ + 20δ 3 σ 2 − 4δ 2 σ 3 − 6δσ 4 − σ 5 )U 4 + +σ 2 (σ + 2δ)(2δ 3 − δ 2 σ − 10δσ 2 − 4σ 3 )U 2 − σ 4 (σ + 2δ)2 x+ +4(δ+σ)U (1+U 2 ) δ 3 (δ+2σ)(δ 2 +2δσ+4σ 2 )U 8 −δ 2 (δ 4 +4δ 3 σ−9δ 2 σ 2 − −32δσ 3 −8σ 4 )U 6 +δσ(4δ 4 + 39δ 3 σ + 88δ 2 σ 2 + 39δσ 3 + 4σ 4 )U 4 +

+σ 2 (8δ 4 +32δ 3 σ+9δ 2 σ 2 −4δσ 3 −σ 4 )U 2 +σ 3 (σ+2δ)(4δ 2 +2δσ+σ 2 ) y+ +4δσ 4 (δ + σ)(σ + 2δ)2 + 4δ 4 σ(δ+σ)(δ+2σ)2 U 12 − −δ 3 (δ+2σ)(3δ 4 −6δ 3 σ−56δ 2 σ 2 −60δσ 3 −16σ 4 )U 10 + 2 +4δ σ(6δ 5 + 45δ 4 σ + 110δ 3 σ 2 + 106δ 2 σ 3 + 43δσ 4 + 5σ 5 )U 8 + +2δσ(2δ 6 + 34δ 5 σ + 136δ 4 σ 2 + 199δ 3 σ 3 + 136δ 2 σ 4 + 34δσ 5 + 2σ 6 )U 6 + +4δσ 2 (5δ 5 + 43δ 4 σ + 106δ 3 σ 2 + 110δ 2 σ 3 + 45δσ 4 + 6σ 5 )U 4 + +σ 3 (σ + 2δ)(16δ 4 + 60δ 3 σ + 56δ 2 σ 2 + 6δσ 3 − 3σ 4 )U 2 = 0.

(20)

The cubic polynomial is annihilated by the vertices coordinates (18) and describes a one-parameter family of rational circular cubic curves. Their singularities are the points given in (15) which are isolated double points and located on the ellipse ei with the equation (13). In a similar way, we can show that there are some further points on the parabolas’ axes that trace cubic curves. Even the envelopes of the parabolas’ directrices are rational cubic curves. Theorem 7 is illustrated in Fig. 6.

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References 1. Berger, M.: Geometry, pp. 1–2. Springer, Berlin (1987) 2. Brieskorn, E., Kn¨ orrer, H.: Planar Algebraic Curves. Birkh¨ auser, Basel (1986) 3. Burau, W.: Algebraische Kurven und Fl¨ achen. I - Algebraische Kurven der Ebene. De Gruyter, Berlin (1962) 4. del Centina, A.: Poncelet’s Porism: a long story of renewed discoveries. I. Arch. Hist. Exact Sci. 70(1), 1–122 (2016) 5. Chapple, W.: An essay on the properties of triangles inscribed in and circumscribed about two given circles. Misc. Curiosa Math. 4, 117–124 (1746) 6. Coolidge, J.L.: A Treatise on Algebraic Plane Curves. Dover Publications, New York (1959) 7. Fladt, K.: Analytische Geometrie spezieller ebener Kurven. Akademische Verlagsgesellschaft, Frankfurt am Main (1962) 8. Garcia, R.A., Odehnal, B., Reznik, D.: Loci of poncelet triangles in the general closure case. J. Geom. 113(17), 1–17 (2021) 9. Garcia, R.A., Odehnal, B., Reznik, D.: Poncelet porisms in hyperbolic pencils of circles. J. Geom. Graph. 25(2), 205–225 (2021) 10. Garcia, R.A., Reznik, D.: Loci of the Brocard points over selected triangle families. Intl. J. Geom. 11(2), 35–45 (2022) 11. Garcia, R.A., Reznik, D., Koiller, J.: New properties of triangular orbits in elliptic billiards. Am. Math. Mon. 128(10), 898–910 (2021) 12. Gibert, B.: Higher Degree Related Curves. https://bernard-gibert.pagespersoorange.fr/curves/q077.html 13. Gibert, B.: Degree Related Curves. https://bernard-gibert.pagesperso-orange.fr/ curves/q079.html 14. Glaeser, G., Stachel, H., Odehnal, B., The Universe of Conics. From the Ancient Greeks to 21st Century Developments. Springer, Heidelberg (2016). https://doi. org/10.1007/978-3-662-45450-3 15. Helman, M., Garcia, R.A., Reznik, D.: Intriguing invariants of centers of ellipseinscribed triangles. J. Geom. 112, 2 (2021). Paper No. 28, 22 p 16. Helman, M., Laurain, D., Garcia, R., Reznik, D.: Invariant center power and elliptic loci of poncelet triangles. J. Dyn. Control Syst. (2021). https://doi.org/10.1007/ s10883-021-09580-z 17. Jaud, D., Reznik, D., Garcia, R.: Poncelet plectra: harmonious curves in cosine space. Beitr. Algebra Geom. (2021). https://doi.org/10.1007/s13366-021-00596-x 18. Kerawala, S.M.: Poncelet porism in two circles. Bull. Calcutta Math. Soc. 39, 85–105 (1947) 19. Kimberling, C.: Triangle Centers and Central Triangles. Congressus Numerantium, vol. 129. Utilitas Mathematica Publishing, Winnipeg (1998) 20. Kimberling, C.: Encyclopedia of Triangle Centers. http://faculty.evansville.edu/ ck6/encyclopedia 21. Odehnal, B.: Poristic loci of triangle centers. J. Geom. Graphics 15(1), 45–67 (2011) 22. Reznik, D., Garcia, R.A., Koiller, J.: The ballet of triangle centers on the elliptic billiard. J. Geom. Graph. 24(1), 79–101 (2020) 23. Reznik, D., Garcia, R.A., Koiller, J.: Fifty new invariants of N-periodics in the elliptic billiard. Arnold Math. J. 7(3), 1–15 (2021) 24. Reznik, D.: Poncelet parabola pirouettes. Math. Intelligencer (to appear). arXiv:2110.06356 25. Wieleitner, H.: Spezielle ebene Kurven. G.J. G¨ oschen’sche Verlagshandlung, Leipzig (1908)

Permutation Cubics Boris Odehnal(B) University of Applied Arts Vienna, Vienna, Austria [email protected]

Abstract. The permutation of the trilinear coordinates of a point yield the six permutation points which are conconic. This idea leads to the following generalization: The six permutations of the trilinears of a point together with the six permutations of the trilinears of the image of that particular point under a certain quadratic Cremona transformation yield twelve points which lie on a single cubic curve. This note is devoted to the study of the thus defined cubic curves, especially those defined by some triangle center and its isogonal or isotomic conjugate.

Keywords: Triangle cubic transformation

1

· Permutation points · Quadratic cremona

Introduction

Along with a triangle in the Euclidean plane, there are many known cubic curves: the Neuberg cubic, the Thomson cubic (17-point cubic), the McCay cubic, the Darboux cubic, the Napoleon (or Feuerbach) cubic, the Orthocubic, and approximately 1200 more cubics, cf. the exhaustive collection on B. Gibert’s page [1]. These curves are deﬁned by means of algebraic or geometric properties (see [6]) or they are the images of lines and conics under certain algebraic transformations (cf. [1–3]). In [5], the homogeneous coordinates p : q : r of a point P in the plane of a triangle are permuted which yields in total six points, the so-called permutation points. It is of minor importance whether p : q : r are trilinear or barycentric coordinates (or homogeneous coordinates deﬁned by an arbitrarily chosen unit point). In the following, we will use homogeneous trilinear coordinates, i.e., the incenter X1 of the base triangle is the unit point of the underlying projective frame (cf. [2,3]). The labeling Xi for the ith triangle center follows the list given in C. Kimberling’s book [3] and the Encyclopedia of Triangle Centers [4]. The vertices A, B, C, and the incenter X1 of the base triangle Δ are described by the following homogeneous trilinear coordinates A = 1 : 0 : 0,

B = 0 : 1 : 0,

C = 0 : 0 : 1,

X1 = 1 : 1 : 1.

Any point P in the plane of the triangle is uniquely deﬁned by a homogeneous coordinate triple p : q : r = 0 : 0 : 0, and any such triple deﬁnes a unique c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 59–70, 2023. https://doi.org/10.1007/978-3-031-13588-0_5

60

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point. Sometimes, we shall prefer the vector notation of coordinates, and then, we simply write p = (p, q, r). At ﬁrst glance, it is surprising that the six permutation points are conconic, i.e., they lie on a single conic with the equation p2 · ξη = ξ 2 · pq, which denotes the cyclic see [5]. Here, we have introduced the symbol sum. This is to be understood literally: For example a = a + b + c, or ξ 2 = ξ 2 + η 2 + ζ 2 , i.e., the argument is replaced twice cyclically in alphabetic order (a → b → c → a, ξ → η → ζ → ξ, p → q → r → p, . . . ). However, the cyclic symmetry of the coeﬃcients of the conic’s equation displays the invariance of the conic with respect to the permutation of the coordinates. In the geometry of the triangle, two special quadratic Cremona transformations play an important role: the isogonal conjugation ι and the isotomic conjugation τ . These mappings are deﬁned for all points in the projectively extended plane of Δ, except on the side lines of Δ. In terms of homogeneous (trilinear) coordinates, these mappings are given by ι(ξ, η, ζ) → (ηζ, ζξ, ξη),

τ (ξ, η, ζ) → (b2 c2 ηζ, c2 a2 ζξ, a2 b2 ξη).

(1)

In both cases, the permutation yields 6 permutation points. Thus, an arbitrary point P deﬁnes its six permutation points and so does its isogonal conjugate ι(P ) (or its isotomic conjugate τ (P )) which makes 12 points in total. In the following Sect. 2, we shall focus on the isogonal conjugation and show that the latter 12 points indeed lie on a single cubic which we will call a permutation cubic. Within the family of permutation cubics, only one cubic is known and shows up in Gibert’s list [1] as K228 . All other permutation cubics are apparently new triangle cubics. In Sect. 3, we look for rational and degenerate cubics among the permutation cubics and study their conﬁguration of real inﬂection points. In addition, their dual curves and their Hessians are derived. We also consider permutation cubics with triangle centers for their pivot points which yields a relatively small number of cubics containing more than just a pair of isogonal conjugate centers. Finally, in Sect. 4, we replace the isogonal conjugation with the isotomic conjugation and study the thus deﬁned cubics. The latter cubics diﬀer slightly from those deﬁned with the help of the isogonal conjugate of the pivot point.

2

The Permutation Cubics

In general, the permutation points of a point P lie on one conic, while the permutation points of its isogonal conjugate ι(P ) lie on another conic. However, it is surprising that we have 12 points with the following property:

Permutation Cubics

61

Theorem 1. The 12 points obtained from P = p : q : r by permuting P ’s homogeneous coordinates as well as those of its isogonal image ι(P ) = qr : rp : pq are located on the self-isogonal cubic C = P ξηζ − Q(ξ 2 η + ξη 2 + ξ 2 ζ + ξζ 2 + η 2 ζ + ηζ 2 ) = 0, where P = pq(p + q) and Q = pqr.

(2)

Proof. The fact that the 12 points lie on the cubic (2) is best shown by inserting the respective coordinates. The cubic is self-isogonal: Assume X = ξ : η : ζ is a point on C, then its isogonal image ι(X) = ηζ : ζξ : ξη is also contained in C regardless of the point X. Figure 1 shows the distribution of the permutation points of a given point P as well as those of the isogonal image ι(P ). C p:q :r qr : pq : rp

q :p:r pq : qr : rp

C rp : pq : qr

r:q :p

rp : qr : pq r:p:q

pq : rp : qr qr : rp : pq

q :r:p p:r:q

Fig. 1. The permutation cubic C with pivot point p : q : r contains 12 a priori known points.

Because of the cyclic symmetry of the Eq. (2) of the permutation cubics, these cubics are invariant under all six permutations of the homogeneous coordinates. Since these permutations induce collineations in the triangle plane, we can say: Theorem 2. Each of the permutation cubics (2) is transformed into itself under each element of the discrete group of collineations induced by the permutation of coordinates. As can be easily read oﬀ from (2), each permutation cubic passes through all three vertices of the base triangle.

62

3 3.1

B. Odehnal

Properties of Permutation Cubics Rational and Singular Permutation Cubics

At ﬁrst, we shall determine those curves in the family (2) which degenerate. For that purpose, we derive conditions on p, q, r so that the gradient grad C = (∂ξ C, ∂η C, ∂ζ C) of the cubics’ Eq. (2) vanishes. We decide to eliminate the variables ζ and η from ∂ξ C = 0, ∂η C = 0, ∂ζ C = 0 and ﬁnd 0 = 81(pqr)15 ξ 16 (q + r)6 (p + r)6 (p + q)6 (r + q + p)2 (pq + pr + qr)2 2 (6pqr − pq(p + q)) (5pqr + 2 pq(p + q)) 49p2 q 2 r2 +5 p2 q(q(p2 + q 2 ) + 2p(pr + q 2 ) + 6pr(q + r)) .

(3)

(The choice of variables to be eliminated is not essential. Other combinations than the chosen one lead to products involving sixteenth powers of η or ζ. The other factors (homogeneous polynomials in p, q, and r) remain the same.) If either of p, q, r equals zero, we obtain the completely degenerate cubic ξηζ = 0, i.e., the three side lines of the base triangle. If either p = −q or q = −r or r = −p, we obtain the singular cubic (ξ + η)(η + ζ)(ζ + ξ) = 0, i.e., the side lines of the excentral triangle Δe . The factor p + q + r set equal to zero is the equation of the line L1 which is polar to the incenter X1 with regard to Δ. L1 is called the antiorthic axis of Δ, see [3]. The points on L1 determine the degenerate cubic (ξ + η + ζ) · (ξη + ηζ + ζξ) L1

= 0.

(4)

Steiner circumellipse e

It is the union of L1 and the Steiner circumellipse e. The points on L1 correspond to points on e and vice versa, since ι(L1 ) = e and ι2 = idP2 . Thereby, the meaning of the quadratic factor pq + qr + rp is also disclosed. The ﬁrst cubic factor yields the equation (5) C1 : 6 ξηζ − ξη(ξ + η) = 0 of the cubic determined by X1 , since P (1, 1, 1) = 6 and Q(1, 1, 1) = 1. In fact C1 has an acnode at X1 = 1 : 1 : 1 and is therefore rational. In terms of a homogeneous parameter u0 : u1 = 0 : 0, its points can be given as u0 (u1 − u0 )(2u0 + u1 ) : u1 (u0 + 2u1 )(u0 − u1 ) : (u0 + u1 )(u0 + 2u1 )(2u0 + u1 ). This cubic can be found as cubic K228 in [1] and it is an isogonal circum-conicopivotal cubic which contains the triangle centers with Kimberling indices 1, 1022, 1023, 23889 – 23894. For a speciﬁc triangle, the cubic C1 = K228 is shown in Fig. 2.

Permutation Cubics

63

The second cubic factor corresponds to an elliptic cubic E whose points do not lead to degenerate or rational permutation cubics. Finally, the sextic factor is the equation of an elliptic curve S of degree 6 with three tacnodes at the vertices of the base triangle Δ and ordinary double points at W1 = [A, B] ∩ L1 = 1 : −1 : 0, W2 = [B, C] ∩ L1 = 0 : 1 : −1, W3 = [C, A] ∩ L1 = 1 : 0 : −1. The ellipticity (genus = 1) of the latter curves makes clear that the points on these curves do not yield degenerate or rational permutation cubics. We summarize our results in: C

C1

A X23889

X23891 X1023

C1

X1

C1

B X23893 X23894 X1022

X23890

X23892

Fig. 2. The only rational cubic C1 = K228 with an acnode at X1 contains eight further triangle centers.

Theorem 3. The family of permutation cubics (2) contains one rational cubic, the cubic C1 = K228 determined by X1 . In the family of permutation cubics, there exist only 3 degenerate cubics: (i) the union of the Steiner circumellipse e and the polar L1 of X1 , (ii) the union of the side lines of the excentral triangle Δe , (iii) the union of the side lines of the base triangle Δ. Figure 3 shows the degenerate curves in the family of permutation curves. The elliptic cubic E and the elliptic sextic S are also shown. 3.2

The Dual Curves

We compute the dual curves of (2) by eliminating ξ, η, and ζ from the system of algebraic equations ∂ξ C = ρu0 , ∂η C = ρu1 , ∂ζ C = ρu2 , C = 0

64

B. Odehnal

which yields C : Q4 u60 +2Q3 (P +Q) u50 (u1 +u2 ) + Q2 (P 2 −4P Q−13Q2 ) u40 (u21 + u22 ) + 2Q2 (2P 2 +10P Q+15Q2 )u0 u1 u2 u30 + 2Q(P 3 +4P 2 Q+P Q2 −8Q3 )u0 u1 u2 u20 (u1 +u2 ) + (P 4 −6P 2 Q2 +36P Q3 +90Q4 )u2u u21 u23 = 0.

S

W2

S

(6)

C1

E

W3 S

C

A2

L1

A1

X1 A

C1

W1 B

e S

E

S

C1

Fig. 3. The locus C44 = e∪L1 and C1 of points in the triangle plane that define singular permutation cubics and the computational artifacts E and S.

For the curve C1 (pivot X1 ) with Eq. (5), we have p : q : r = 1 : 1 : 1 or P = 6 and Q = 1. This reduces the equation of the dual curves (6) to the self-isogonal quartic C1 : u20 (u20 + 12u0 (u1 + u2 ) − 26u21 + 244u1 u2 ) = 0. To be more precise: With P = 6 and Q = 1 the factor u0 +u1 +u2 (corresponding to the antiorthic axis) splits oﬀ from (6) with multiplicity 2. 3.3

Hessian Curves

The Hessian curves of cubics are again cubics and are obtained as the determinant of the Hessian matrix of the form C. Hence, we have HC : 8Q2 (P +2Q)(ξ 3 + η 3 + ζ 3 )+2(P 3 −12P Q2 −24Q3 )ξηζ + − 2P 2 Q(ξ 2 η + ξη 2 + ξ 2 ζ + ξζ 2 + η 2 ζ + ηζ 2 ) = 0.

(7)

Permutation Cubics

65

For arbitrary choices of P and Q, i.e., for an arbitrary choice of a pivot point with homogeneous coordinates p : q : r, none of the Hessian curves (7) will be a permutation curve. This could only be the case if the coeﬃcient of ξ 3 + η 3 + ζ 3 vanishes: If Q = 0 (or equivalently pqr = 0), the pivot point is located on a side of the bases triangle and the Hessian curve becomes the union of the side lines of Δ. In the case P + 2Q = 0, (7) simpliﬁes to −8Q3 (ξ + η)(η + ζ)(ζ + ξ) = 0 and describes the union of the side lines of the excentral triangle Δe . We are able to show the following: Theorem 4. All cubics in the family of permutation cubics share their three real points of inflection which lie on the line L1 and on the side lines of Δ. Proof. Possible candidates for points of inﬂection on a curve C are located on its Hessian curve HC. The common points of L1 and the side lines of Δ are W1 = 1 : −1 : 0,

W2 = 0 : 1 : −1,

W3 = 1 : 0 : −1

and it is easily checked that W1 , W2 , W3 are regular points on C (independent of the choice of p : q : r) and lie also on HC with Eq. (7). Therefore, they are inﬂection points of all curves C given by (2). For the regular curves in the family (2), the three inﬂection tangents are not concurring. Note that the conﬁguration of inﬂection points on the permutation cubics shows a similar behavior than that of the distance product cubics (cf. [6]). 3.4

Triangle Centers on Permutation Cubics

Each triangle center determines a permutation cubic. Since the cubics (2) are invariant under isogonal transformations, each triangle center Xi shares its permutation cubic Ci with the isogonal conjugate ι(Xi ). The cubic C1 determined by X1 is a special case: It is the only rational (nondegenerate) permutation cubic and is deﬁned by X1 (cf. Theorem 3). Since X1 is self-assigned under the isogonal conjugation, there is no corresponding point on C1 . Earlier, we have mentioned that C1 contains 9 triangle centers of which 8 can be arranged in 4 pairs of conjugate points. The conjugate pairs on C1 are: (X1022 , X1023 ), (X23889 , X23894 ), (X23890 , X23893 ), (X23891 , X23892 ). We shall not go through all permutation cubics deﬁned by the approximately 47400 known triangle centers (as to April 2022). There are only a few remarkable examples, which contain more than just one pair of triangle centers.

66

B. Odehnal

The centroid X2 and the Symmedian point X6 are each others isogonal conjugates. The conjugate pairs on C2 are (X2 , X6 ) and (X3570 , X3572 ). The center X3572 is the intersection of C2 ’s tangents at X2 and at X6 . The tangent of C2 at X3570 meets the cubic at X3572 (cf. Fig. 4). The circumcenter X3 and the orthocenter X4 form a conjugate pair on C3 . Further, this cubic contains the center X1981 whose isogonal conjugate is a yet unknown center with trilinear center function α = a(b − c)(a2 − b2 − c2 )(a3 c + a2 (b2 − 2c2 ) − ac(b2 − ac2 ) − b4 + b2 c2 ) ·(a3 b − a2 (2b2 − c2 ) + a(b3 − bc2 ) + b2 c2 − c4 ). On the cubic C9 determined by the Mittenpunkt X9 , we ﬁnd the following conjugate pairs: (X9 , X57 ) and (X1024 , X1025 ) (cf. Fig. 5). Similar to the case of C2 , we observe that the tangents to C9 at X9 and X57 intersect on the cubic at X1024 , while the tangents at X1024 and X1025 meet at a further yet unnamed center with a rather lengthy trilinear representation of algebraic degree 24 in a, b, c (the side lengths of Δ). C2 C

C2

C2

A

X6

X2

X3570

B

C2

X3572

Fig. 4. The permutation cubic C2 = C6 .

The center X101 lies on Δ’s circumcircle, and therefore, its isogonal conjugate X513 is a point on the line at inﬁnity. The tangents at the latter two centers to the cubic C101 intersect in X34906 ∈ C101 . The tangents at the points of the isogonal conjugate pair (X34905 , X34906 ) intersect on C101 .

Permutation Cubics

67

X1025

C9 C

X9 C9

A

X57

C9

X1024

C9

B

Fig. 5. The permutation cubic C9 = C57 .

(6)

(5)

263 7 104 174 95 257 20 80 8 43 650

224 43

654

262 4 250 9

223 3 260 0 223 0 21 894 ,(2 )

950 8

263 8515

26 30130 3

513

⎧ 14299 ⎪ ⎪ ⎪ ⎪ 2590 ⎪ ⎨ 14298 (6)= 2522 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩656 652 ⎧ 9404 ⎪ ⎪ ⎪ ⎨822 (5)= 40137 ⎪ ⎪ ⎪ ⎩657 30600

⎧ ⎧ ⎧ ⎧2183 ⎧ ⎧ 29361 9393 2229 ⎪ 2236 40109 ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2173 ⎪ ⎪ ⎪ ⎪ ⎪ 9356 2591 9511 2238 2245 ⎪ ⎪ ⎪ ⎪ ⎪ (4)= ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪3330 ⎪ ⎪ ⎪ ⎪ ⎪ 15586 2312 2228 2232 2244 ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎪ ⎪ ⎪ 910 ⎪ 1755 39690 ⎪ ⎪ 2235 2290 (8’)= 2348 ⎪ ⎪ ⎪ ⎨ (8)= ⎪ ⎪ ⎪ ⎪ 2313 2272 ⎪ ⎪ 899 ⎪ ⎪2610 672 ⎪ ⎪ ⎪ ⎪ (10)= ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ 3000 29357 2227 2642 ⎪ ⎪ 659 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (12)= ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ 38472 2579 2516 2511 2243 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ (14)= ⎪ ⎪ ⎪ ⎪ 2225 2182 2314 7655 2641 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (8 ⎪ ⎪ ⎪ ) (4 ⎪ ⎪ ⎪ 22108 896 ⎩3768 ⎪ ⎪ ) (1 ⎪ ⎪ 0) ⎪ ⎪ 1155 ⎪ ⎪ (1 2231 2240 ⎪ ⎪ ⎪ ⎪ 4) ⎪ ⎪ ⎪ ⎪ 44 ⎪ ⎩9360 ⎪ ⎪ 2234 ( 1635 ⎪ ⎪ 1 2) ⎪ ⎪ ⎪ ⎩2239 (8’) 2237

23 23515 03 22 43965 224 4 7

1575 2254

245 33 209 79 225 3

(2)=

Fig. 6. The distribution of triangle centers on the antiorthic axis L1 ⊂ C44 .

There are 224 triangle centers on the degenerate cubic C44 = e ∪ L1 given by (4). On the Steiner circumellipse, the 97 centers (cf. Fig. 7) with the Kimberling indices

68

B. Odehnal

88, 100, 162, 190, 651, 653, 655, 660, 662, 673, 771, 799, 823, 897, 1156, 1492, 1821, 2349, 2580, 2581, 3257, 4598, 4599, 4604, 4606, 4607, 8052, 20332, 23707, 27834, 29059, 32680, 34085, 34234, 36083 – 36102, 37128 – 37143, 37202 – 37223, 38340, 40110, 43069, 43192 can be found. The remaining 127 centers with the Kimberling numbers 44, 649, 650, 652, 654, 656, 657, 659, 661, 672, 770, 798, 822, 851, 896, 899, 910, 1155, 1491, 1575, 1635, 1755, 2173, 2182, 2183, 2225, 2227 – 2240, 2243 – 2247, 2252 – 2265, 2272, 2290, 2312 – 2315, 2348, 2483, 2484, 2503, 2509, 2511, 2515, 2516, 2522, 2526, 2578, 2579, 2590, 2591, 2600, 2610, 2624, 2630, 2631, 2635, 2637, 2641, 2642, 3000, 3013, 3287, 3330, 3768, 4394, 4724, 4782, 4784, 4790, 4813, 4893, 4979, 7655, 7659, 8043, 8061, 9356, 9360, 9393, 9404, 9508, 9511, 10495, 13401, 14298 – 14300, 15586, 17410, 17418, 17420, 18116, 20331, 20979, 21127, 21894, 22108, 22443, 23503, 24533, 29357, 29361, 30600, 38472, 39690, 40109, 40137, 40338

⎧ 823 ⎪ ⎪ ⎪ ⎪36093 ⎪ ⎪ ⎪ 38340 ⎪ ⎪ ⎪ ⎪ 36097 ⎪ ⎪ ⎪ ⎪ 29059 ⎪ ⎨ 36092 (12)= ⎪653 ⎪ ⎪ ⎪ 162 ⎪ ⎪ ⎪ ⎪ 36099 ⎪ ⎪ ⎪ ⎪ 36088 ⎪ ⎪ ⎪ ⎪ ⎩37141 36094

(12)

37128 360 91

37135

B

88, 7

20

37

2581 37214

2

89 7

325

7

4 13 37 60 6 73 6

23 4 9 37221 36101 23707

129 36085,37

6 115 7 460 0 3713

37219

37143

10 401 8 0 372

37202 37213 36100

6 08 2 36033 2 2 3713

X1

1821

3610

371 36 36 0 326 360 83 80 89 371 39 36 360090 36 84 08 7 65 5 3 34 721 08 7 5, 37 14 0

C

34234 37142

372 20

36098

43069

37 21 6 27 83 4

651

4604

e

37137 36096 4599

A 06 372

37138 1492

4598

662

1 3721 100 771 52 80 12 372 6 460210 2 37 15 22 2 37 37 0 19

37 79 3 21 9 37 722 8 2 3 37 09 37 1 372 31 371 203 04 33 37205

43192 2580 6095

are located on the line L1 (see Fig. 6). The ideal point X513 is the only real improper center on the singular permutation cubic C44 .

Fig. 7. The distribution of triangle centers on the Steiner ellipse e ⊂ C44 .

4

Isotomic Instead of Isogonal Conjugation

The cubics mentioned in Theorem 1 contain the permutation points of the isogonal image of the pivot point P = p : q : r. The isogonal transformation can be replaced with any other quadratic Cremona transformation.

Permutation Cubics

69

However, the isotomic conjugation is the second prominent quadratic Cremona transformation closely related to the geometry of a triangle. In this case, a point P = p : q : r is mapped to a point τ (P ) = b2 c2 qr : c2 a2 rp : a2 b2 pq. Again, there are 12 points obtained from P (its permutation points and the permutation points of its isotomic conjugate) which lie on cubics with the equations T : pqr a6 b6 p3 q 3 − a4 b4 c4 pqr p3 ξη(ξ + η) − a2 b2 c2 p2 q 2 r2 a2 b2 pq(a2 p + b2 q) − a2 b2 c2 pq(p + q) ξ 3 + ξηζ a2 b2 c2 pqr a2 b2 pq(p4 + q 4 ) + a6 b6 p4 q 4 (p + q) + a4 b4 p3 q 3 r(b2 (c2 − b2 )q 2 + b2 (c2 − a2 )p2 ) + a2 b2 p3 q 3 r2 (b4 (c4 − a4 )p + a4 (c4 − b4 )q)) = 0.

(8)

Compared to the cubics (2), the latter cubics do not pass through the vertices of the base triangle. With the help of a CAS it is a rather simple task to show that the following holds true: Theorem 5. The six permutation points of a point P = p : q : r and the six permutation points of its isotomic conjugate lie on a triangle cubic with equation (8). All cubics in the family share the inflection points which agree with the inflection points of the self-isogonal permutation cubics (2). The cubic with pivot X1 is rational and has an acnode at X1 . The cubic with pivot point X2 degenerates completely, i.e., its equation is the zero form. The cubics (8) whose pivot points are the triangle centers with Kimberling numbers 649, 650, 652, 654, 656, 657, 659, 661, 672, 770, 798, 799, 822, 851, 896, 899, 910, 1155, 1491, 1575, 1635, 1755, 2173, 2182, 2183, 2225, 2227 – 2240, 2243 – 2247, 2252 – 2254, 2265, 2272, 2290, 2312 – 2315, 2348, 2483, 2484, 2503, 2509, 2511, 2515, 2516, 2522, 2526, 2578, 2579, 2590, 2591, 2600, 2610, 2624, 2630, 2631, 2635, 2637, 2641, 2642, 3000, 3013, 3287, 3330, 3768, 4394, 4724, 4782, 4784, 4790, 4813, 4893, 4979, 7655, 7659, 8043, 8061, 9356, 9360, 9393, 9404, 9508, 9511, 10495, 13401, 14298 – 14300, 15586, 17410, 17418, 17420, 18116, 20331, 20979, 21127, 21894, 22108, 22443, 23503, 24533, 29357, 29361, 30600, 38472, 39690, 40109, 40137 split into the line L1 and a further conic. Especially, if i = 661 and i = 799, the degenerate cubics are equal to that given in (4). Note that the permutation cubics that carry the permutation points of the isotomic conjugate of the pivot point are not self-isotomic.

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B. Odehnal T3

T6 T4

T3

T5

T7 T9 T10 T11 T12

4 L1 T5

T6

3

T6

T11

T4

Fig. 8. Some permutation cubics (8) with low-indexed centers for their pivot points.

Figure 8 shows some of the cubics Ti deﬁned by triangle centers with low Kimberling indices i ∈ {1, . . . , 12}. Note that T7 = T8 since X8 = τ (X7 ).

References 1. Gibert, B.: Cubics in the Triangle Plane. https://bernard-gibert.pagesperso-orange. fr/index.html. Accessed 1 Apr 2022 2. Glaeser, G., Stachel, H., Odehnal, B.: The Universe of Concis. From the ancient Greeks to 21st century developments. Springer-Spektrum, Springer-Verlag, Heidelberg (2016). https://doi.org/10.1007/978-3-662-45450-3 3. Kimberling, C.: Triangle Centers and Central Triangles. (Congressus Numerantium, vol. 129) Utilitas Mathematica Publishing, Winnipeg (1998) 4. Kimberling, C.: Encyclopedia of Triangle Centers. http://faculty.evansville.edu/ ck6/encyclopedia. Accessed 1 Apr 2022 5. Kimberling, C.: Permutation ellipses. J. Geom. Graph. 24(2), 233–247 (2020) 6. Odehnal, B.: Distance product cubics. KoG 24, 29–40 (2020)

Beyond the Nine-Point Conic Boris Odehnal(B) University of Applied Arts Vienna, Vienna, Austria [email protected]

Abstract. The nine-point conic n contains the three diagonal points and the midpoints of the six sides of a complete quadrangle. We show that for any quadrilateral Q = P1 P2 P3 P4 and an arbitrarily chosen point P there exists a conic l passing through ten points: P , the three diagonal points of Q, and the six inverses of the poles of the lines [Pi , Pj ] with respect to any circumconic k of Q and the inversion center P . The circumconic k can be any conic from the pencil circumscribed to Q. The nine-point conic shows up as a special case of the conic l. The projective nature of the deﬁnition of the conic l has implications on a certain normal problem of asymptotic quadrilaterals in the hyperbolic plane. Keywords: Nine-point conic · Quadrangle · Projective inversion Pencil of conics · Hyperbolic plane · Asymptotic quadrangle

1

·

Introduction

Let four points P1 , P2 , P3 , P4 in the Euclidean plane form a quadrilateral Q, i.e., no three points are collinear. Assume further that D1 = [P1 , P2 ] ∩ [P3 , P4 ], D2 = [P1 , P3 ] ∩ [P2 , P4 ], D3 = [P1 , P4 ] ∩ [P2 , P3 ] are the diagonal points and M12 , M13 , . . . , M34 are midpoints of the edges P1 P2 , P1 P3 , . . . , P3 P4 . Now, it is well-known that the diagonal points and the six midpoints of Q’s edges lie on a single conic n which is frequently referred to as the nine-point conic. This ˆ cher (cf. [3]). result is usually ascribed to the American mathematician M. Bo ˆ cher later recognized that W.K. Clifford However, there is evidence that Bo and J.J. Sylvester may have been aware of the existence of such a conic a little bit earlier (in 1864). Whitworth’s monograph [15] on trilinear coordinates mentions the ninepoint conic in two exercises without explicitly calling it a nine-point conic. Perhaps, in [2], this particular conic was called nine-point conic for the ﬁrst time. Later on, various attempts by means of analytic and synthetic geometry towards the nine-point conic were made in [1,4,9,12,13] and some spatial analog was described in [7]. It is clear that the midpoints of the quadrilateral’s sides are the harmonic conjugates of their ideal points with respect to the pair of incident vertices (as illustrated in Fig. 1). Therefore, the nine-point conic can even be deﬁned in the more general setting of projective geometry as this turned out to be the case for many geometric objects associated to triangles (cf. [10]). Clearly, real c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 71–81, 2023. https://doi.org/10.1007/978-3-031-13588-0_6

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projective geometry is not the only framework with a nine-point conic within nine-point conics can be studied. The nine-point conic is well-deﬁned even in Rational Trigonometry, see [8]. P2

H12 P1 n

D1

H13

H23 D3 H14

P3 H34

H24 n

P4

F1 g

Q23

F2

Q13

Q24

Q14 Q34

D2

Fig. 1. The nine-point conic n associated with a complete quadrangle Q = P1 P2 P3 P4 and a straight line g contains the harmonic conjugates H12 , H13 , . . . , H34 of the six points Qij = [Pi , Pj ] ∩ g, the 3 diagonal points D1 , D2 , D3 , and (if they exist) the ﬁxed points F1 , F2 of the Desargues involution induced by Q on g.

In Sect. 2, we shall present an apparently new result on a complete quadrangle whose vertices lie on a conic. Unfortunately, this result can only be shown by means of computation. Indeed, any quadrilateral Q lies on some conic k and any quadrilateral determines a pencil of circumconics. We will see that the pencil of conics on Q deﬁnes a pencil of ten-point conics. In comparison to the deﬁnition and construction of the nine-point conic, the arbitrarily chosen line g is replaced by the point P and the conic k. Nevertheless, a certain line instead of g will show up. Section 3 brieﬂy describes the ten-point conic of cyclic quadrilaterals. It will turn out that the nine-point conic is in fact a special case of a ten-point conic, at least for cyclic quadrilaterals with P being the center of the circumcircle. Section 4 is devoted to the pencil of conics deﬁned by the quadrilateral and the associated pencil of ten-point conics. In Sect. 5, we leave the purely projective setting and even the Euclidean plane and show that the existence of the ten-point conic has implications on totally asymptotic quadrangles in the hyperbolic plane.

Beyond the Nine-Point Conic

2

73

Ten Points on One Conic

Let k be a conic in the projective plane P2 and assume that P1 , . . . , P4 are four (pairwise diﬀerent) points on k. (It is thereby guaranteed that these points form a quadrilateral, i.e., no three points are collinear.) Further, let P be an arbitrarily chosen point in P2 that does not lie on the tangents Ti of k at Pi . The four given points determine six chords pij = [Pi , Pj ] (i = j and i, j ∈ {1, 2, 3, 4}) of k with their respective poles Pij with regard to k. The projections of the poles Pij onto the chords pij shall be denoted by Qij (cf. Fig. 2). Now, we have the rather surprising result: Theorem 1. The six points Qij , the three diagonal points of Q, and the point P lie on a single conic l. This result seems to be unknown and does not appear in the classical literature, neither does it in the newer. Proof. Unfortunately, we can only give an analytic proof. For that purpose, we impose a projective frame (cf. [6]) such that the given points have the homogeneous coordinates P1 = 1 : 0 : 0, P2 = 0 : 0 : 1, P3 = 1 : 1 : 1, P4 = 1 : t : t2 ,

(1)

where t ∈ R \ {0, 1} (hence, P4 = P1 , P2 , P3 ) and P = p0 : p 1 : p 2 . In this case, the conic k circumscribed to Pi is the standard conic k : x0 x2 − x21 = 0.

(2)

(Any other four points can be mapped to these and any arbitrary point will be mapped to P , see [6]). There are some natural restrictions on the position of P . They can be written in algebraic form as follows. (i) P may not lie on any of the six lines pij : p1 (p1 − p2 )(p2 − tp1 )(p1 − p0 )(p1 − tp0 )(p0 t − (1 + t)p1 + p2 ) = 0. (ii) P may not lie on any of the four tangents of k at Pi : p0 p2 (p0 − 2p1 + p2 )(t2 p0 − 2tp1 + p2 ) = 0. (iii) P may not lie on the sides of the diagonal triangle of Q: (tp0 − 2p1 + p2 )(tp0 − 2tp1 + p2 )(tp0 − p2 ) = 0.

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Therefore, none of the latter 13 factors can be zero, and we are allowed to cancel any of them whenever they occur. Any ﬁve of the six points Qij = pij ∩ [P, Pij ] determine a unique conic l whose equation can be derived from a 6 × 6 determinant, see [6, p. 241, eq. (6.13)]. Independent of the choice of the ﬁve points Pij , the equation of l reads l : tp2 x20 + 2(tp1 − tp2 − p2 )x0 x1 + (p2 − tp0 )x0 x2 + 2(p2 − tp0 )x21 + 2(tp0 + p0 − p1 )x1 x2 − p0 x22 = 0.

(3)

The coordinates of any remaining sixth point annihilate (3). This holds also true for the three diagonal points and the point P itself. Figure 2 shows the ten-point conic for a cyclic quadrangle. The cyclicity of the points Pi is not a projective property. However, the contents of Theorem 1 are invariant under arbitrary projective transformations, in particular, that transformation that maps the standard conic (2) to a circle k.

D3 l P4

P34 Q34 Q24

k

Q14

P24

P3

P14

P23 D1

D2 Q23

P2 Q13

P13

Q12

P

P12 P1

l

Fig. 2. The ten-point conic l of a cyclic quadrilateral P1 P2 P3 P4 .

Now, we can close the gap between the nine-point conic and the ten-point conic: Theorem 2. The ten-point conic l described in Theorem 1 is also the nine-point conic of the quadrilateral Q with respect to the polar line p of P with respect to the conic k.

Beyond the Nine-Point Conic

75

Proof. The arbitrarily chosen point P has a polar line p with regard to k. The points Qij on the polars pij are the harmonic conjugates of Qij = p ∩ [Pi , Pj ] with respect to the pairs [Pi , Pj ], cf. [6, Ch. 7.1]. Theorem 2 shows that the results in [4] can be seen from a superordinate standpoint. As an immediate consequence of Theorem 2, we can formulate: Theorem 3. The nine-point conic n of a quadrilateral Q on a conic k equals the ten-point conic l if P is chosen as the center of k. Proof. If P is the center of k, then its polar line with regard to k is the ideal line (line at inﬁnity). The harmonic conjugates of the ideal point of the each of the six lines pij with respect to the pair (Pi , Pj ) (with i = j) are the midpoints of the segments Pi Pj . Moreover, the points Qij are the images of the poles Pij under the projective inversion ι : P2 → P2 (acting on the projective plane P2 sliced along a triangle) in k with center P (cf. [6, p. 343]).

3

Cyclic Quadrilaterals

We ﬁnd a very special situation if the initial conic k is chosen as the Euclidean unit circle and P as its center. The polar line of P (the center of k) is the ideal line ω (line at inﬁnity) of the projectively closed Euclidean plane. Thus, the points Qij are the ideal points of the lines pij and their harmonic conjugates Qij (inverses of the poles Pij ) are the midpoints of the segments Pi Pj . Then, the projective inversion becomes the “ordinary” inversion in a Euclidean circle. In terms of Cartesian coordinates, the unit circle k has the equation x2 + y 2 = 1

(4)

which can be written in terms of complex coordinates as zz = 1, where z = x + iy and z = x − iy is the complex conjugate of z. With the help of complex coordinates, the inversion is described by ι : z → z1 . If the four given points P1 , . . . , P4 are now described by four complex numbers a, b, c, d of norm 1, then the equation of the ten-point conic l can be given in the form l : 2z 2 − 2abcd z 2 − (a + b + c + d)z + (abc + abd + acd + bcd)z = 0.

(5)

The points Qij are then given by 12 (a + b), . . . , 12 (c + d) and it is a matter of simple computations to show that the center of l equals 1 (a + b + c + d), 4

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i.e., that is the centroid of the quadrilateral Q = abcd. In this very special case, the conic l coincides with the ordinary nine-point conic. In any case, l from (5) is an equilateral hyperbola since its ideal points are √ F1,2 = 0 : abcd − 1 : i(1 ± abcd)2 which are conjugate in the elliptic involution on the ideal line (that joins ideal points of pairs of orthogonal directions). We can summarize (cf. Theorem 3): Theorem 4. For a cyclic quadrilateral with P being the center of the circumcircle, the ten-point l conic equals the ordinary nine-point conic n. The conic l is not a notion of inversive geometry since its inverse in the circle k equals the cubic curve ι(l) : 2z 2 − 2abcd z 2 − (a + b + c + d)z 2 z + (abc + abd + acd + bcd)zz 2 = 0 which is a strophoid (circular cubic with orthogonal tangents at the double point, cf. [14]). Its node lies in the center of k, i.e., the center of inversion (see Fig. 3).

D3

s P3

P4

D1

l D2 P P2

k P1

Fig. 3. The ten-point conic l of a cyclic quadrilateral P1 P2 P3 P4 is an equilateral hyperbola. Its inverse in k (center of inversion = center of k) is a strophoid s.

Beyond the Nine-Point Conic

4

77

A Property of Pencils

The four points we have chosen prior to Theorem 1 can be considered as the base points of a pencil of conics of the ﬁrst kind, i.e., the one-parameter family of all conics passing through these four points (cf. [6]). The constructions done above are invariant under projective transformations: Tangents, polars, joins of points, and intersections of lines are not altered under collineations and correlations. Consequently, we can state the following: Theorem 5. Let P1 , . . . , P4 form a quadrilateral in a projective plane, let c be any conic from the pencil of the first kind spanned by the quadrilateral, and let further P be a point not contained in any line [Pi , Pj ] (i = j, i, j ∈ {1, 2, 3, 4}). Then, the projections of the six poles Pij of [Pi , Pj ] with regard to c from P onto [Pi , Pj ] lie on a single conic n which also houses P and the diagonal points of the quadrilateral. If we use the assumptions made on the coordinatization in the proof of Theorem 1, we can span the pencil of conics through P1 , . . . , P4 by k and a singular conic in the pencil, e.g., by the union of the lines [P1 , P3 ] and [P2 , P4 ]. This yields the equations of the conics as B(λ, μ) : λ(x2 − x1 )(x1 − tx0 ) + μ(x0 x2 − x21 ) = 0, [P1 ,P3 ]∪[P2 ,P4 ]

k

where λ : μ = 0 : 0 is a homogeneous parameter. Proceeding in the same way as done in the proof of Theorem 1, we arrive at the equations of the ten-point conics associated with the conics of the pencil l(λ, μ) : λ tx0 − 2tx1 + x2 )(t(p1 − p2 )x0 + (p2 − tp0 )x1 + (tp0 − p1 )x2 μ tp2 x20 + 2(tp1 − tp2 − p2 )x0 x1 + (p2 − tp0 )x0 x2 + 2(p2 − tp0 )x21 + 2(tp0 + p0 − p1 )x1 x2 − p0 x22 = 0 Obviously, the conics l(λ, μ) form a pencil. Since the determinant of the coeﬃcient matrix equals (up to non-vanishing factors) 4μ(tλ − μ)(λ(t − 1) − μ), the singular conics in this pencil correspond to μ1 = 0, μ2 = λt, μ3 = λ(t − 1). These are the pairs of lines s1,2 : (tx0 − 2tx1 + x2 )(t(p1 − p2 )x0 + (p2 − tp0 )x1 + (tp0 − p1 ) = 0, s3,4 : (tx0 − x2 )(tx0 p1 − tx1 p0 − x1 p2 + x2 p1 ) = 0, s5,6 : (tx0 − 2x1 + x2 )(tx0 p1 − tx1 p0 − x0 p2 + x1 p2 + x2 p0 − x2 p1 ) = 0 forming a complete quadrilateral with the diagonal points D1 = s1 ∩ s2 = 2t2 p0 −t(p0 +2p1 )+p2 : t2 p0 +t(p2 −2p1 ) : t2 (p0 −2p1 +2p2 ) − tp2 , D2 = s3 ∩ s4 = tp0 +p2 : 2tp1 : t(tp0 +p2 ), D3 = s5 ∩ s6 = tp0 −2p0 +2p1 −p2 : t(2p1 −p0 ) − p2 : −t2 p0 + t(2p1 +p2 ) − 2p2

78

B. Odehnal P2 =0:0:1 k

1:t:t2 =P4

V2 =1:0:-t P3 =1:1:1 P1 = =1:0:0

V3 =1:t:t

0:1:0

V1 =1:1:t P = V4

Fig. 4. The standard frame attached to k and the base points of the pencil of ten-point conics as described in Theorem 6

and the vertices V1 V2 V3 V4

= s1 ∩ s3 ∩ s6 = s1 ∩ s4 ∩ s5 = s2 ∩ s3 ∩ s5 = s2 ∩ s4 ∩ s6

= 1 : 1 : t, = 1 : 0 : −t, = 1 : t : t, = p0 : p1 : p2 .

We can summarize this in: Theorem 6. Let P1 , . . . , P4 form a quadrilateral Q in a projective plane with the diagonal points D1 , D2 , D3 . The ten-point conics associated with the pencil of conics (of the first kind) defined by Q form themselves a pencil of conics (of the first kind) with the pivot P and the diagonal points D1 , D2 , D3 of Q for its base points (provided that P does not lie on the sides of the diagonal triangle). Figure 4 shows the construction of the base points of the associated pencil of ten-point conics from the base points of the initial pencil of conics on Q. In Fig. 5, besides the quadrilateral Q and a conic k on Q, the thus determined ten-point conic l and its projective inverse s ∩ p in k (with center of inversion P ) is shown. The curve s is a cubic with its node at P and the line p is the polar of P with respect to k.

5

Implications on Non-Euclidean Quadrilaterals

The projective model of the hyperbolic plane H2 is the interior of the absolute conic Ω. The points in the interior of Ω are the points of the hyperbolic plane, the hyperbolic lines are the chords of Ω. (A point is an interior point of a conic if it does not send (real) tangents to the conic.) The points on Ω are called absolute or improper points of H2 . For the sake of simplicity, we choose the Euclidean unit circle (4) as the absolute conic Ω which delivers the well-known Cayley-Klein model of the hyperbolic

Beyond the Nine-Point Conic p

P1

79

l P2

P

l P3

s k P4

Fig. 5. The cubic curve s is a part of the inverse of l in k (with center P ) and can be viewed as a projective version of the strophoid mentioned after Theorem 4.

plane (cf. [5]). A quadrilateral in H2 is called asymptotic or ideal if all its vertices lie on Ω. Now, we can use Theorem 1 to show: Theorem 7. Let Q = P1 P2 P3 P4 be an asymptotic quadrilateral in the hyperbolic plane and let P be an arbitrary point in H2 . Then, the six pedal points of the hyperbolic normals from P to the six sides of the complete quadrilateral on Q are located on a single conic l independent of the choice of P . P is also located on l. Proof. The hyperbolic normal n12 through a point P of a hyperbolic line [P1 , P2 ] passes through the absolute pole PAB of [P1 , P2 ], i.e., the pole of [P1 , P2 ] with regard to Ω. Since all vertices of Q are located on a conic (here it is Ω), the hyperbolic pedals on the lines of the complete quadrangle are the projections of the respective absolute poles onto these lines. Therefore, the hyperbolic pedal points meet the requirements of Theorem 1 and line up on a single conic. The behaviour of the locus C of all points P ∈ H2 with conconic pedal conics shows a completely diﬀerent behaviour than that in the Euclidean plane as shown in [11]. However, in the Euclidean plane this locus is an algebraic curve of degree 7 (in general) or 6 (in special cases) and it can be shown that the degree does not drop below 6 (cf. [11]). In the hyperbolic plane, C is of degree 12 and with each vertex of the initial quadrilateral that happens to lie on Ω, the degree drops about 3. Hence, the degree of C assigned to a completely asymptotic quadrilateral is of degree 0, i.e., C equals the entire hyperbolic plane. Figure 6 illustrates the contents of Theorem 7: In fact, the point P can be chosen freely and the six hyperbolic pedal points on the sides of a completely asymptotic quadrilateral are conconic anyhow. The pedal conic l also contains P and the only hyperbolic diagonal point D. Therefore, l is an eight-point conic.

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B. Odehnal P3 P2 Ω D2 l

l

P1

P

P4

Fig. 6. The six hyperbolic pedal points of P on the sides of a completely asymptotic quadrilateral lie on a single conic l independent of the choice of P . Further, the conic l passes through P and the only (hyperbolic) diagonal point D2 .

6

Final Remarks

In Sect. 4, we have discovered the pencil of ten-point conics associated to a pencil of conics. All the pencils we have met so far are pencils of the ﬁrst kind, i.e., the one-parameter family of conics through four points (forming a quadrilateral). It would be interesting to see whether we can assign pencils of ten-point conics to pencils of the other (four projectively distinguished) types of pencils. The base quadrilateral will then have 3 or 2 vertices, or even 1 vertex. Maybe it is possible to study these cases by means of limit procedures.

References 1. 2. 3. 4.

Allardice, R.E.: On the nine-point conic. Edinb. M.S. Proc. 19, 23–32 (1901) Anonymus: The nine point’s conic of any tetrastigm. Messenger V, 1–2 (1869) Bˆ ocher, M.: On a nine-point conic. Ann. Math. VI 132, 178 (1892) Gates, F.: Some considerations on the nine-point conic and its reciprocal. Ann. Math. VII I, 185–188 (1894)

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5. Giering, O.: Vorlesungen u ¨ ber h¨ ohere Geometrie. Vieweg, Braunschweig/Wiesbaden (1982) 6. Glaeser, G., Stachel, H., Odehnal, B.: The Universe of Conics. Springer Spektrum, Heidelberg (2016). https://doi.org/10.1007/978-3-662-45450-3 7. Holgate, T.F.: On the cone of second order which is analogous to the nine-point conic. Ann. Math. VI I, 73–76 (1893) 8. Le, N., Wildberger, N.J.: Incenter Symmetry, Euler lines, and Schiﬄer points. KoG 20(20), 22–30 (2016) 9. Minthorn, M.A.: The Nine Point Conic. Master’s thesis, Univ. of California, Berkeley (1912) 10. Odehnal, B.: Generalized gergonne and nagel points. Beitr. Algebra Geom. 51(2), 477–491 (2010) 11. Odehnal, B.: A rarity in geometry: a septic curve. KoG 25(25), 1–25 (2021) 12. Pinkerton, P.: On a nine-point conic. Edinb. M.S. Proc. 24, 31–34 (1906) 13. Regan, F., Wilke, R.L.: On the nine point conic associated with a complete quadrangle. Math. Mag. 44, 261–266 (1971) 14. Stachel, H.: Strophoids, a family of cubic curves with remarkable properties. JIDEG (J. Ind. Des. Eng. Graph. ISSN 1843-3766 10, 65–72 (2015) 15. Whitworth, W.A.: Trilinear Coordinates. Cambridge Univ, Press (1866)

Complex Solution of Engineering Problems by Graphic Methods Aleksandr Yurievich Brailov(B) Odessa Professional College of Computer Technologies, Odessa State Ecological University, Odessa, Ukraine [email protected]

Abstract. A geometrical problem is to determine the location of the image equidistant from four points that do not coincide in three-dimensional space. A practical example of this problem is to determine the location of the lamp equidistant from four lamps, arbitrarily located in the hall of an art gallery, a sports hall, on a site in a park and other objects. A method has been developed for the graphical determination of the geometrical place of an image equidistant from four points that do not coinciding in three-dimensional space. A methodology for the graphical solution of engineering geometric problems is proposed. Keywords: Geometric image · Engineering problem · Graphic solution · Point · Sphere · Solution methodology

1 Stereotyped Approach to Problem Solving When creating modern architectural structures and engineering structures, one of the challenging problems is to determine the geometrical place of the image, equidistant from four points that do not coincide in the three-dimensional space. A stereotype formulation of the problem is: “Construct the locus of points equidistant from points A, B, C, D” [1–3]. As an example, a complex drawing of four non-coincident points is given in Fig. 1. The initial assumption that the desired locus of points is the axis of a cylinder, on the surface of which points A, B, C, D are located, may leads to the result shown in Fig. 2. According to Fig. 2, the solution of the problem starts from the straight line segment defined by the points A and D, which is a generatrix line of a circular cylinder. The constructed i-axis is parallel to the generatrix line AD. The i5 is a projection of the axis i equidistant from the projections A5 , D5 , B5 , C 5 of the points A, B, C, D, respectively. For the method described above to solve the problem, the obtained result is not unique. The number of possible results obtained using the described method is the number of unordered subsets {A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D} of two elements (r = 2) of the set {A, B, C, D} containing four elements (k = 4). The number of all possible results C k r is equal to the number of combinations of two (r = 2) different points of the set containing four (k = 4) points: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 82–94, 2023. https://doi.org/10.1007/978-3-031-13588-0_7

Complex Solution of Engineering Problems by Graphic Methods

83

C k r = k!/[r!:(k–r)!] = 4!/[2!(4–2)!] = 4!/(2!:2!) = 24/4 = 6. Thus, the number of all possible results for the described method for solving the problem is six. Using the graphical method of changing the projection planes [1–5], it is easy to demonstrate that all six results of the presented solution are different.

C2

B2

A2

D2

П2 x21 П1

B1 A1

D1

C1

Fig. 1. Complex drawings of mismatched points.

i2

C2

B2 D2

A2

П2 x21 П1

B1

i1 A1

П1 x41 П4

D1

C1

A5≡D5 A4 i4

D4 C4

i5

B4 C5 П4 П5 x45

B5

Fig. 2. Solution of the problem for the assumption that the desired locus of points is the i-axis of the cylinder.

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Let us show that the stereotype approach to constructing the locus of points equidistant from points A, B, C, D, which are arbitrarily located in the three-dimensional orthogonal space and do not coincide with each other, does not give the true result (Fig. 3).

i2

C2

B2 D2

A2

П2 x21 П1

B1

i1 A1

П1 x41 П4

D1

C1

A4 ΔA

i4 C4

D4

A5≡D5

O4

O5 i5

ΔB

ΔC Φ4

B4

C5 П4 П5 Δ C x45

ΔA

A

B Δ B5

B

C

Fig. 3. The natural values of the distances between an arbitrary point O of the i-axis and points A, B, C, D are different.

The plane F intersecting the cylinder is constructed perpendicular to the plane of projections P4 . The plane F passes through point D perpendicular to the i-axis of the cylindrical surface, and point O is the intersection between the plane F and the i-axis of the cylinder. All four points A, B, C, D belong to a cylindrical surface. Since the plane F is perpendicular to the i-axis of the cylinder and is parallel to the plane of projections P5 , the distance between points O and point D in the plane of projections P5 projects in its natural size. To test the validity of the assumption that “the desired locus of points is the axis of the cylinder, on the surface of which points A, B, C, D are located”, it remains to determine the natural values of the distances between an arbitrary point O of the i-axis and points A, B, C. The distances between an arbitrary point O and points A, B, C are determined by the method of a right-angled triangle [1–5].

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Graphical constructions shown in Fig. 3 present the visual proof that, on the “constructed geometrical place of points”, the distances from an arbitrarily chosen point O to points A, B, C, D are different. Consequently, the constructed locus of points i is not the true solution of the problem. A straight line would be indeed the locus of points equidistant from points A, B, C, D if and only if all the original points A, B, C, D are located on the same circle of the cylinder. In this case, the resulting straight line must pass through the center of the original circle and be perpendicular to the plane containing this circle. Any circle on the surface of a circular cylinder is a guide line and all its points are equidistant from an arbitrary point on its axis. The validation test of any of the six possible results (Fig. 3) confirms the need to develop a method for obtaining a (true) real result through a correctly formulation of the problem.

2 Correct Approach to Solving the Problem The development of the required solution begins with the correct formulation of the problem and an assumption about the nature of the resulting geometric image. If the assumption is incorrect, it is impossible to obtain the correct result. It makes sense to generalize the formulation of the problem being solved as follows: Determine the locus of the image equidistant from four points A, B, C, D, which are not coincident, in the three-dimensional space. Provided that any combination of three points out of the given four points does not belong to a straight line and a plane of level [3–5]. Then the initial assumption that the desired locus of points is the axis of the cylinder, on the surface of which points A, B, C, D are located, will take on a different form. Assumption 1. The locus of the desired image equidistant from four non-coincident points A, B, C, D in the three-dimensional space is a point, denominated herein as K, but not a straight line (cylinder axis). Point K is equidistant from the given points A, B, C, D if it is the center of the spherical surface on which points A, B, C, D are located. To construct a complex drawing of the desired image, it is necessary to graphically determine the position of the center K and the actual value of the radius RHB of the spherical surface containing points A, B, C, and D. The solution of such a subproblem is related to the assumption of a possible way of constructing the center K of the spherical surface containing points A, B, C, D. Assumption 2. The center K of the spherical surface with points A, B, C, D is located at the intersection of the mid-perpendiculars restored to all four flat faces of the tetrahedron, the vertices of which are points A, B, C, D that belong to the spherical surface. Each such perpendicular to the flat face of the tetrahedron ABCD is the locus of points equidistant from the three vertices of the corresponding face. The intersection of two mid-perpendiculars of any two flat faces of the tetrahedron inscribed in the sphere is the point K equidistant in three-dimensional space from the given four non-coincident points A, B, C, D.

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If the formulated assumptions 1 and 2 are valid, then three subproblems are solved using the graphic methods of engineering geometry [1–5] as: 1. The construction of complex drawings of the four centers of the circles circumscribed around each triangular face of the tetrahedron. 2. The construction of complex drawings of the midpoint perpendiculars reconstructed through the centers of the circumscribed circles to the flat faces of the tetrahedron ABCD. 3. The construction of a complex drawing of the point of intersection of the midperpendiculars. The construction of a complex drawing of the center of a circle described through the vertices of the triangular face of the tetrahedron is performed by changing the projection planes using the main lines of the flat face [1–5]. The result is OA (O1 A , O2 A ) of the graphical solution of such a subproblem, for example, for the face BCD is shown in Fig. 4.

h2

C2

12

B2

A 2

O

A2 D2

П2 x21 П1 П4 A4

x41

A1 C 1

ΔA

B4 C4

x45

П4 П5

h4

B1

O1A

D4

O4A

h1

11

D1

П1

A5

C5 D5

O5A B5

Fig. 4. Complex drawing of the center OA (O1 A , O2 A ) of the circle described through the vertices of the triangular face BCD of the tetrahedron.

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The algorithm for constructing a complex drawing of the center of a circle described through the vertices of the spatial triangular face of the tetrahedron consists of seven steps. Focusing on the face BCD, we have: 1. A complex drawing h (h1 , h2 ) of the horizontal line belonging to the face BCD is constructed. 2. Perpendicular to the horizontal projection plane P1 , a new projection plane P4 is introduced. Then, the x 14 axis is drawn perpendicular to the horizontal projection h1 of the horizontal h. 3. New lines of projection links are constructed from the horizontal projections A1 , B1 , C 1 , D1 of the tetrahedron vertices perpendicular to the new x 14 axis. 4. On the lines of projection links in the plane of projections P4 , the distances equal in magnitude to the distances from the x 21 axis between the planes of the replaced system P2 /P1 to the corresponding frontal projections A2 , B2 , C 2 , D2 of the tetrahedron vertices are indicated by serifs marks. Graphically, these different distances are marked with dashes or any other graphical signs. 5. At the intersection of the lines of projection links and serifs for the corresponding distances, new projections A4 , B4 , C 4 , D4 of the tetrahedron vertices are graphically indicated and highlighted. Since the horizontal h of the plane of the BCD face is perpendicular to the plane of projections P4 , the projection B4 C 4 D4 of this face is a collective segment of a straight line. Now if the projection plane P5 orthogonal to the plane P4 and parallel to the collective projection B4 C 4 D4 of the flat face BCD is introduced, then the triangular face of BCD is projected onto this new plane P5 in true size. 6. Similarly, repeating steps 2–5 of the developed algorithm, projection B5 C 5 D5 is constructed on the new projection plane P5 . This projection is equal to the true value of the triangular face BCD of the tetrahedron ABCD. The center OA of the circumscribed circle is located at the intersection of the perpendiculars to the sides B5 C 5 , C 5 D5 , B5 D5 of the triangle B5 C 5 D5 . 7. Following the rules of the method of changing projection planes [1–3], projections O4 A , O1 A , O2 A of the center OA of the circumscribed circle for the triangular face BCD of the tetrahedron ABCD are determined using reverse projections. Complex drawings of the centers of OB , OC , OD of the circumscribed circles other faces ACD, ABD, ABC are constructed in the same way (Fig. 5). The construction of a complex drawing of the middle perpendicular, reconstructed through the center of the circle circumscribed around the vertices of the flat face of the tetrahedron, is carried out based on the theorem on the projection of the right angle.

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The result pA (p1 A , p2 A ) of the graphical solution of this subproblem is obtained, for example, for the face BCD (Fig. 6). The algorithm for graphical construction of a complex drawing of the middle perpendicular includes the following three steps. 1. A complex drawing f A (f 1 A , f 2 A ) of the frontal f A belonging to the face BCD is constructed. A complex drawing hA (h1 A , h2 A ) of the horizontal hA belonging to the face BCD is constructed, if it was not constructed earlier. 2. A horizontal projection p1 A of the middle perpendicular pA to the flat face BCD of the tetrahedron ABCD is constructed through the horizontal projection O1 A of the center OA to be perpendicular to the horizontal projection h1 A of the horizontal hA . 3. A frontal projection p2 A of the middle perpendicular pA to the flat face BCD of the tetrahedron ABCD is constructed through the frontal projection O2 A of the center OA to be perpendicular to the frontal projection f 2 A of the frontal f A (Fig. 6).

C2 h2A≡h2D 22

12

A2 B7

D7

O7D x21 x16

A7

B6 C6

Π7 Π6 x67

П1

O1D

Π1

D6 C4 h4A

П4 x45 П5

O2D

O

D2

A1

A6 Π6 hD O6D 6

C7

П2

B2 A 2

21 h1D

A4

A 1

h O4A B4

D4

П4

C1

11

O1A D1

B1

П1 x41

A5

C5 D5

O5A B5

Fig. 5. Complex drawings of centers OA , OD of circumscribed circles for faces BCD, ABC.

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Applying the described three-step algorithm, one can construct the middle perpendicular pD to the face ABC adjacent to the face BCD (Fig. 6). Complex drawings of the mid-perpendiculars pB , pC for the remaining faces ACD, ABD are constructed in a similar way. The construction of a complex drawing K (K 1 , K 2 ) of the point K of the intersection of the middle perpendiculars is performed as the intersection of the corresponding horizontal and frontal projections of any two perpendiculars. On the complex drawings (Fig. 6, 7), the point K (K 1 , K 2 ) of the intersection of the middle perpendiculars pA , pD to the faces BCD, ABC is developed (Fig. 7).

32 42 f D 2 12 O2A

C2 h2A≡h2D 22 A2 П2 x21

O2D p2A

D 2

p

B2

D2 f2A

П1

O1D

A1

O1A D1 f1A 1 1 D C1 p1 p1A

h1D

21 h1A

31

41

B1 f1D

Fig. 6. Complex drawings of the center perpendiculars pA , pD for the faces BCD, ABC.

C2 3 D 2 4 2 f2 h ≡h 22 12 B2 O2A K2 A2 O2D D2 A f2A П2 p2 p2D K1 П1 B1 O1D 41 f1D A A1 O 1 31 21 fA D h1 C1 11 D D1 A 1 p1 p1 h1A A 2

x21

D 2

Fig. 7. Proof of correctness of the correct graphic constructions of the complex drawing K (K 1 , K 2 ) of the point K.

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The same point K (K 1 , K 2 ) of intersection of the middle perpendiculars pB , pC to the faces ACD, ABD can be constructed by determining the position of the centers OB , OC of the circumscribed circles for the faces ACD, ABD. Such graphic work was done and published [1, 2]. The criterion of correctness of the graphical constructions presented in (Figs. 6, 7) is the fulfillment of the first law of projection connections according to which the constructed straight line connecting the horizontal K 1 and frontal K 2 projections of the K point is actually perpendicular to the abscissa axis [1–5]. The truth criterion of the obtained graphic solution is the equality of the values of the coordinates of the point K along the applicate axis for frontal projections and along the ordinate axis for horizontal projections for different pairs of perpendiculars pA , pD (Fig. 7) and pB , pC in various complex drawings [1, 2]. Thus, the full validity of the second assumption about the presence of a point K of intersection of two mid-perpendiculars has been proved graphically. Point K is equally distant from all four specified non-coincident points A, B, C, D, because it simultaneously belongs to the midpoint perpendicular pA , all points of which are equally distant from points B, C, D, and it also belongs to the midpoint perpendicular pD , all points of which equally distant from points A, B, C. To verify the validity of the first assumption about the equidistance of the constructed point K from all four given non-coincident points A, B, C, D graphically, a complex drawing Σ (Σ 1 , Σ 2 ) of the sphere Σ with center K is constructed, on the surface of which points A, B, C, D are located. It is, however, can be constructed if the radius of this sphere RHB is known. Using the method of a right-angled triangle [1–5] on a complex drawing of five points A, B, C, D and K, one can determine the true value of the radius RHB of the sphere with points A, B, C, D equidistant from its center K as shown in Fig. 8.

C2 ΔC

Φ2

K2

A A2 Δ

П2 A HB

ΔB

B2

ΔD D 2

K K1

x21 П1≡Ф1 R B1 B A B Δ A RHB B Δ C A1 RHB D D1 RHB D C ΔC Δ C1 D Fig. 8. Determination of the actual value of the radius RHB of the sphere.

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The validity of the graphic constructions is confirmed by the equality of the true values of the radii RA HB , RC HB , RD HB , RB HB of all four points A, C, D, B—KA = KC = KD = KB = d tv . The actual value of the radius RHB of the sphere, to which all four points A, B, C, D do not coincide in the three-dimensional space, was determined graphically. RHB = RA HB = RC HB = RD HB = RB HB = d tv . For a given spatial arrangement of points, A, B, C, D and a center K relative to these points, a sphere is constructed with a natural value of the radius RHB equal to d tv in a separate complex drawing shown in Fig. 9. Visual construction of a sphere with center K and points A, B, C, D requires moving it away from the frontal plane and raising it above the horizontal plane as shown in Fig. 8. The clarity of the complex drawing Σ (Σ 1 , Σ 2 ) of the sphere Σ is achieved by moving the horizontal projections K 1 , A1 , B1 , C 1 , D1 of the points K, A, B, C, D downward from the x 21 axis by d t1 and front projections K 2 , A2 , B2 , C 2 , D2 points K, A, B, C, D upward from the x 21 axis by d t2 (Fig. 9).

Σ2

C2 B2

K2 A2

D2

П2 x21 П1 Σ1

K1 B1 A1

D1 C1

Fig. 9. Complex drawing Σ (Σ 1 , Σ 2 ) of the sphere Σ with points A, B, C, D.

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To prove that points A, B, C, D belong to the surface of the sphere Σ, complex drawings of the intersection lines a, b, c, d of horizontally projecting planes ΔA , ΔB , ΔC , ΔD with the surface of the sphere Σ are constructed: a = ΔA ∩ Σ, b = ΔB ∩ Σ, c = ΔC ∩ Σ, d = ΔD ∩ Σ as shown in Fig. 10. As known, a point belongs to a surface if and only if it belongs to any line of this surface. Because the frontal projection C 2 of the point C belongs to the frontal projection c2 of the line c of the surface of the sphere Σ and the horizontal projection C 1 of the point C belongs to the horizontal projection c1 of the same line c of the surface of the sphere Σ, then the point C itself belongs to the spherical surface, i.e., CΣ. Similarly, it is graphically proved that point D belongs to the surface of the sphere Σ, since it is obvious from the drawing shown in Fig. 10 that the corresponding projections D2 , D1 of point D belong to the corresponding projections d 2 , d 1 of the line d of the intersection of the plane ΔD with the surface of the sphere Σ. The corresponding graphic proofs are carried out for points A and B.

Σ2 ΔC2

c2 ΔA2 A2

C2

b2 K2

a2

ΔB2 B2

d2

ΔD2

D2 П2 x21 П1 Σ1

K1 b1 ΔA1 a1 c1 (A1) C Δ1

d1

ΔB1 B1 ΔD1 (D1)

C1

Fig. 10. Proof of the truth of the obtained result.

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Therefore, the point K is the locus equidistant from all four given non-coincident points A, B, C, D. Thus, point A is located on the left front lower quarter of the surface of the sphere Σ, point B is located on the right front upper quarter of the surface, point C belongs to the left front upper quarter of the surface, and point D is located on the right front lower quarter of the surface of the sphere Σ centered at K (Fig. 10). Since the points A and D are located on the front lower half of the surface of the sphere Σ, their frontal projections A2 , D2 are visible, and the horizontal projections A1 , D1 are invisible. It is customary to enclose invisible projections of geometric images in parentheses – (A1 ), (D1 ). Since points C and B are located on the front upper half of the surface of the sphere Σ, their frontal projections C 2 , B2 and horizontal projections C 1 , B1 are visible.

3 Methodology for the Graphical Solution of an Engineering Geometric Problem The proposed methodology for the graphical solution of an engineering geometric problem consists of a number of mandatory stages. 1. 2. 3. 4.

Analysis of the essence of the geometric phenomenon (Figs. 1, 2 and 3). Formulation of assumptions about the nature of the solution result. Determination of the number of possible solutions to the problem. Development of a method, technique and algorithm for a graphical solution (Figs. 4, 5, 6, 7, 8 and 9). 5. Verification of the validity of the obtained result (Fig. 10).

4 Conclusions 1. Based on the analysis of the initial assumption, it is proved that the locus of points equidistant from four points A, B, C, D that do not coincide in three-dimensional space is not the “axis of the cylinder, on the surface of which points A, B, C, D are located”. 2. The performed research proves the validity of the assumption that the geometrical place of the desired image equidistant from four points A, B, C, D that do not coincide in the three-dimensional space is a point, denominated herein as K. Provided that any combination of three points out of the given four points does not belong to a straight line and a plane of level [3–5]. 3. We have also proved the validity of the second put forward assumption that the center K of the spherical surface with points A, B, C, D is located at the intersection of the mid-perpendiculars restored to all four flat faces of the tetrahedron, the vertices of which A, B, C, D belong to the surface.

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4. A method for solving a geometric problem is proposed, which consists in graphically constructing the centers of the flat faces of a tetrahedron, the vertices of which are located on a spherical surface, constructing mid-perpendiculars to each face through their centers and determining the point of intersection of these perpendiculars. 5. On the basis of the proposed method, a methodology for the graphical solution of an engineering geometric problem has been developed.

References 1. Brailov, A.: Methods for solving the olympiad geometric problem. Mod. Prob. Model. 18, 38–51 (2020) 2. Brailov, A.Y.: Methodology for graphical solution of engineering geometric problems. In: Materials of the XXI International Conference on Mathematical Modeling. Kherson: KhNTU, pp. 90–91 (2020) 3. Brailov, A.Y.: Engineering Graphics. Theoretical Foundations of Engineering Geometry for Design. Springer International Publishing, p. 340 (2016) https://doi.org/10.1007/978-3-31929719-4. (ISBN 978-3-319-29717-0) 4. Brailov, A.Y.: Engineering geometry [Tutorial]. K.: Caravel, p. 472 (2016). (ISBN 978–966– 222–987–5) 5. Brailov, A.Y.: Engineering geometry [Tutorial]. K.: Caravel, p. 490 (2021). (ISBN 978–966– 8019–68–5)

Integer Sequences from Circle Divisions by Rational Billiard Trajectories Daniel Jaud(B) Gymnasium Holzkirchen, Holzkirchen, Germany [email protected]

Abstract. We study rational circular billiards. By viewing the trajectory formed after each reflection point to another inside the circle as the number of circle divisions into regions we derive a general formula for the number of division regions after each reflection. This will give rise to an p · 2π integer division sequence. Restricting to the special cases ϑ = 2p+1 we show that the number of regions after each reflection n is beautifully related to Gauss’s arithmetic series.

Keywords: Rational billiards Arithmetic series

1

· Circle division · Integer sequence ·

Introduction and Setup

The simplest realization of a billiard table is given by a circular region C. Here we are interested in the trajectory of the billiard ball which is considered to be a point particle moving inside the boundary of the circle with unit velocity and bouncing elastically along the boundary ∂C. Due to the elastic collision the incident and reﬂected angle α with respect to the normal to the boundary are identical. Further, because of rotational symmetry the system is fully described by the angle ϑ made by two consecutive scattering points with the circle [3,6] (see Fig. 1). From the ﬁgure we determine ϑ, depending on the incident angle α, to be ϑ = π − 2α.

(1)

The natural question of this problem is for which values of ϑ closed, i.e. periodic, orbits exist. This means how do we need to choose the value of ϑ such that after a given number of collisions n the billiard ball returns to its initial position P0 . Obviously, due to rotational symmetry this case applies if q · ϑ = 2π · p p ↔ ϑ = · 2π, q c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 95–106, 2023. https://doi.org/10.1007/978-3-031-13588-0_8

(2) (3)

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Fig. 1. Graphical representation of the billiard trajectory, the incident angle α and the angle ϑ between two consecutive reflection points.

where q, p ∈ IN such that gcd(p, q) = 1 and pq < 12 1 . The result tells us that if ϑ is a rational multiple of 2π we always end up with periodic orbits [6]. q determines the number of periods that the circle is surrounded before returning to P0 . The trajectory forms in general regular star shaped ﬁgures consisting of q corners equally distributed along ∂C 2 . Two consecutive star corners Pi and Pi+1 (i ∈ {0; 1; . . . ; q −1}) lying on ∂C form an angle of ϑ/p with the circle (see Fig. 2). Thus the parameter p tells us that from one scattering point to the next p − 1 corner points lie in between. If n is the number of scattering then the billiard trajectory follows the path formed by the star corners Pi = Pp·n mod q ,

(4)

where in polar coordinates (circle radius R) the positions of Pi are given by ϑ ϑ · i · 2π , R sin · i · 2π . (5) Pi R cos p p In the following sections we are interested in deriving a general formula for the number of regions the circle is divided by the billiard trajectory after each scattering n. Thereby, we continue as follows: In Sect. 2 we are going to show that any two billiard trajectories intersect only in a single point, i.e. it doesn’t occur that three or more paths cross in a single common point. This is a rather technical side remark to the main results that we will develop in Sect. 3. In particular, we will derive a general formula for the number of circle divisions 1

2

The cases p/q > 1/2 simply correspond to a billiard ball moving in the opposite direction, i.e. clockwise. Since this gives no new insights to the systems behavior we restrict to the cases smaller or equal 1/2. If p = 1 the trajectory describes regular polygons. Star shaped figures arise for p ≥ 2.

Integer Sequences from Circle Divisions

Fig. 2. Example for a 5−star formed by the trajectory corresponding to ϑ =

97

2 5

· 2π.

after each scattering. The ﬁnal Sect. 4 will be dealing with the special cases when p · 2π p ∈ IN. (6) ϑ= 2p + 1 In these cases the general formula of Sect. 3 will be signiﬁcantly simpliﬁed containing the well known Gauß formula for the integer series n i=1

2

i=

n · (n + 1) . 2

(7)

Number of Intersecting Lines

Theorem 1. For periodic billiard trajectories in any circle holds that at each point of self-intersection no more than two sides are meeting. Proof. For the proof we consider the rotational angle ϑ = pq · 2π, where gcd(p, q) = 1 and pq < 12 . The trajectory forms a regular star shaped geometry with reﬂection points Pi on the circle are determined by the angle Θi = i · ϑp , i ∈ {0; . . . ; p − 1}, i.e. Θi = ∠P0 M Pi . We thus can consider the following setup of possible intersecting lines (see Fig. 3) where without loss of generality due to the rotational symmetry we can restrict to the case when the billiard ball starts at the point P0 characterized by Θ0 . The angle α, i.e. the scattering angle with respect to the circle normal, is π−ϑ . Making use of the law of sines one obtains for the distance given by α = 2 ri of the intersection point Si with respect to the circle center M (radius of the circle R): cos( pq π) sin(α) . (8) = R · ri = R · cos( pq π − qi π) sin(π − α − Θ2i )

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Since holds 0

O together with environ is situated under the based flatness, as the peak of type α2 , if the point of positive curvature W + = 2π – θ > O together with environ is situated over the based flatness, as the peak γ, if the point possesses negative curvature. W − = 2π – θ < 0, as the peak of type β1 , if the point possesses null curvature W = 2π – θ ≈ O and incident to the based flatness, β2 - is situated under the based flatness together with environ, β3 - is situated over the based flatness together with environ. Situation with based flatness is very important, so far as, if in classical differential geometry it always means the possibility to turn, to “shake” the surface and to lead it into comfortable situation for an observer, but while studying the topographical surfaces, such possibility is excluded (Fig. 2). The notion introduction of curvature degree and types of TS points is expedient/advisable and explained by availability of analogical signs on the peaks of real relief. The peaks of type α1 form the vivid forms of relief (mountain, heights, hills, and so on), the peaks of type α2 form concave forms of relief - hollows, gorges, and so on, the peaks of type γ make forms of “saddle back”, of type β1 - calm forms of relief (terraces, plateaus and so on), β1 , β2 make the lines of water-sheds, fractures/breaks, and so on. We may retrace analogy with types of points on the regular surface, well- known from classical differential geometry, as TS is a particular case out of class of irregular surfaces.

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Fig. 2. .

Peaks of type α1 are analogue of elliptical points or umbilic points, peaks of types γ- analogue of hyperbolic points, of type β1 are analogue of parabolic points or points of flattening. Points of type α2 don’t have any analogue. Let’s select out of quantity X the sub-quantity A ⊃ X, which is subspace of space X, on which some metric ρ generates induction topology. Obviously, τ/x = τ/A. So far as properties a) – f) are hereditary, the subspace A is also metric subspace, which can be successively considered as T1 , T2 , T3 , T4 , - space, satisfying all separation axioms. i.e. – consecutively, just as separable, housedorffic, regular and normal space. Let’s define the task operation on the new topology A, to be exact, on plurality of equivalent topologies. Let it be the final or computing number of points {{x1 }N i=1 ⊃ A ⊃ X , according to the property d) each one out of them possesses the local base. Let’s introduce the notion of maximal environ point, apart from provisions of common topology, where the point environ can be of arbitrary size. The notion of maximal environ has another content, apart from the notion of connected/coherent component, and includes it into its volume in some particular cases. Definition 5. The maximal environ of the point is the environ, having the biggest radius among all the environs, giving the local base of the given point. The sizes of the maximal environ O(X)max are defined by properties of TS, about which the narration is mentioned below.

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Let’s introduce the notion of point order. Definition 6. Let’s say, that the point has the first order, if in its maximal environ there are no other points – essential peaks, the second order, if its maximal environ is kept in the maximal environ of point of the first order, n- order, if its maximal environ is kept in the maximal environ of the point of the order (n–1) (Fig. 3)

Fig. 3. .

Fig. 4. .

In further text the symbol W i j± ,i = 1, 4 j = 1, n, means curvature of the peak, where i is the curvature degree, j is the order. Let’s consider the selected earlier sub-quantity A as the space T2 , satisfying to the second separation axiom: 1) {x1 }N i=1 ⊃ A have the fourth degree of positive curvature; 2) each peak xi ∈ A has the environ O(x)max , non-disjunctive with the environ of neighbouring peak. Obviously, selected, by this way, sub-quantity A has the computing base and therefore is the separable sub-space, in addition, deliberately measured. The selected sub-quantity A ⊃ X defines the open field on the topographical surface. Definition 7. Let’s name the arbitrary closed curve, circumscribed around the open field, as the trivial border ∂A of the field A on TS. The trivial border is incident to only non- essential peaks of TS. As an example, the field is given on Fig. 4 Definition 8. As the non-trivial border ∂A of the field A on the TS, let’s name all the essential peaks x ∈ A, having environs ν ∈ τ(x), such ones, that

As an example, the field is given on Fig. 5

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Fig. 5. .

Fig. 6. .

The minimal trivial border of the field is the circumscribing round maximal environs boundary points of the field, i. e. such points xi ∈ A, that O(x)max (x\A) = 0. For the open field A ⊃ X, selected out of quantity X, the following affirmation is true. Affirmation 1. Connection of the nearest peaks of the metric of the first order and the fourth curvature degree { W41+ } by segments, cut-off portions, of straight lines generates Delaunay triangulation [4, 5]. Demonstration of the affirmation follows directly out of definition and properties of Delaunay triangulation Fig. 6. Consecutive junction of boundary points by segments of straight lines generates some new field B ⊃ A ⊃ X with non-trivial border ∂B. The availability of angle ϕ = 120° on the border ∂B of the field B let’ s name as infringement of vividness (Fig. 7).

Fig. 7. .

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For the field B ⊃ A ⊃ X Lemma 1 is reasonable. In the field B ⊃ A, containing essential peaks {xi }N i=1 , of the first order and the fourth curvature degree, non- trivial border ∂B of which contains a single break of vividness, the addition of one more essential peak xi ⊃ A, with curvature W 1+ 4 will not change the length of the border ∂B, if the new border ∂B* is vivid. Really, let’s consider the field B on Fig.7, containing {hi }6i=1 peaks of the fourth degree and the first curvature order, the border ∂V of which connects the quantity points consecutively. The border ∂V has the single break of vividness. Let’s add peak H7 by the way, that field V would remain T1 –T2 - space, but the new ∂V* border would become the vivid one (Fig. 8).

Fig. 8. .

Let’s compare the length of borders ∂B and ∂B*. For this we will consider quadrangle with peaks x3 , x4 , x6 , x7 . Obviously, it is the rhombus with sides, that are equal to the radius of maximal environs’ peaks. Hence: [x6 , x3 ] + [x3 , x4 ] = [x6 , x7 ] + [x4 , x7 ], that proves the lemma to be true. Let’s consider the field with different orders of the peaks. (1)± of the point of Addition to the field G, containing the only peak x1 x1 /W(4) (2)± the second order x2 x2 /W(4) , converts the field into T 1 - space, if to consider both peaks with their maximal environs. At the same time the field G can be considered as T2 −T3 −T4 - space, if to take into consideration the rest “added, invested“ consecutively decreasing environs of peaks Oi (x1 ) ∩ Oi (x2 ) = 0 . For the fields with different orders of the peaks, property 9 takes place. (n)± Inclusion of new points xj xj /W(4)j into the field G, containing A quantity of (1)± peaks {xi }N increases curvature of the field G, but does not increase the i=1 , xi /W(4)i length of the maximal trivial border. Really, on Fig. 23 the field G, containing two peaks of the first order let’s consider (1)± 2 {xi }i=1 xi /W(4)i . The including of new peaks {xi }3j=1 of the second order into the field G increases the summary curvature of the field, because according to the definition, the curvature is W ± = i Wi± ..

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Hence it follows, that curvature of the field can be increased infinitely W ± → ±∞: W± =

(2)± W(4)i +

(2)± W(4)i + ... → ±∞

so, the process of addition of new peaks can be infinite, because the availability of small environs can be “ as much as one wants”. With this, the length of minimal trivial border remains unchanged, as the environs of new peaks are completely contained in the given environs. Let’s take notice, that availability of peaks of various orders is the peculiarity of topographical surfaces, apart from regular surfaces, where each peak has only the first order. Conclusion. The article considers approaches to the creation of the theory of topographic surfaces. From the standpoint of general topology, the definitions of a topographic surface and the curvature of its vertices are given. The properties of sections of an irregular surface are considered and the relationship between their geometric parameters is shown. The proposed approach allows solving both applied problems and theoretical problems of irregular surfaces.

References 1. Kuchkarova, D.F., Achilova, D.A.: Identification method of topographic surfaces models. In: The 18th International Conference on Geometry and Graphics (ICGG2018), Politecnico di Milano, #035 (A) (2018) 2. Aleksandrov, A.D., Zalgaller, V.A.: Two-dimensional manifolds of bounded curvature. In: Proceedings of the Mathematical Institute of the ANSSSR 63, 262 (1962) 3. Aleksandrov P.S.: An introduction to set theory and general topology. Science 368 (1977) 4. Kravchenko, U.A.: Fundamentals of design of geomodeling systems. In: Book 2., Information Geomodeling Models and Methods, part. 1, p. 196. Novosibirsk (2008) 5. Skvorcov, A.V.: Delaunay Triangulation and its Application, p. 128. Tomsk (2002)

Generalizing Continuous Flexible Kokotsakis Belts of the Isogonal Type Georg Nawratil(B) Institute of Discrete Mathematics and Geometry & Center for Geometry and Computational Design, TU Wien, Vienna, Austria [email protected] https://www.geometrie.tuwien.ac.at/nawratil/

Abstract. A. Kokotsakis studied the following problem in 1932: Given is a rigid closed polygonal line (planar or non-planar), which is surrounded by a polyhedral strip, where at each polygon vertex three faces meet. Determine the geometries of these closed strips with a continuous mobility. On the one side, we generalize this problem by allowing the faces, which are adjacent to polygon line-segments, to be skew; i.e. to be non-planar. But on the other side, we restrict to the case where the four angles associated with each polygon vertex fulfill the so-called isogonality condition that both pairs of opposite angles are equal or supplementary. In more detail, we study the case where the polygonal line is a skew quad, as this corresponds to a (3 × 3) building block of a so-called V-hedra composed of skew quads. The latter also gives a positive answer to a question posed by R. Sauer in his book of 1970 whether continuous flexible skew quad surfaces exist. Keywords: Kokotsakis belt · Continuous flexibility · Skew quad surfaces

1 Introduction Let us consider a so-called Kokotsakis belt as described in [1], which is illustrated in Fig. 1a. In general these loop structures are rigid, thus continuous flexible ones possess a so-called overconstrained mobility. Kokotsakis himself formulated the problem for general rigid closed polygonal lines p, but in fact he only studied flexible belts with planar polygons p in [1]. Planarity was only not assumed in the study of necessary and sufficient conditions for infinitesimal flexibility (see also Karpenkov [2]). Clearly, the restriction to planar polygons p makes sense in the context of continuous flexible polyhedra, as this condition has to be fulfilled around faces where all vertices have valence four1 . Our interest in Kokotsakis belts results from our research on continuous flexible polyhedral surfaces; especially those composed of rigid planar quads in the combinatorics of a square grid. A very well known class of these flexible planar-quad (PQ) surfaces are V-hedra, which are the discrete analogs of Voss surfaces2 according to [4]. 1

Assumed that this part of the continuous flexible polyhedra is not rigid. Surfaces on which geodesic lines form a conjugate curve network [3]. Dedicated to my newborn son and his mother on the occasion of his birth. 2

c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 115–126, 2023. https://doi.org/10.1007/978-3-031-13588-0_10

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Fig. 1. Original Kokotsakis belt: (a) A rigid closed polygonal line p (not necessarily planar) with n vertices V0 , . . . ,Vn−1 is surrounded by a belt of planar polygons in a way that each vertex Vi of p has valence four. Moreover, the planar angles δi∗ , γi∗ , λi∗ , μi∗ ∈ (0; π ) are illustrated as well as the orientation of the enclosing line-segments used for the construction of the spherical image, which is illustrated in parts in (b). Note that the edge Vi−1Vi is mapped to the point Ci .

They can easily be characterized by the fact that in each vertex the angles of opposite planar quads are equal. The question whether V-hedra can be generalized by dropping the planarity condition of the quads (cf. Fig. 2) motivated us for the study at hand, which is structured as follows: We proceed in Sect. 1.1 with a literature review on flexible Kokotsakis belts, where we place emphasis on the so-called isogonal type3 , which means that in every polygon vertex both pairs of opposite angles are (1) equal or (2) supplementary. In Sect. 2 we discuss the spherical image of Kokotsakis belts from a kinematical point of view. Based on these considerations we study generalized flexible Kokotsakis belts of the isogonal type in Sect. 3. In Sect. 4 we discuss continuous flexible skew-quad (SQ) surfaces, where we focus on V-hedra composed of skew quads in more detail. The paper is concluded in Sect. 5. 1.1

Review on Continuous Flexible Kokotsakis Belts

Until now only examples of continuous flexible Kokotsakis belts are known where the rigid polygon line p is planar as well as all faces adjacent to its line-segments. Therefore these assumptions hold for the complete review section, which is structured along the number n of vertices V0 , . . . ,Vn−1 of p (cf. Fig. 1a). General results were only obtained by Kokotsakis [1] for the isogonal type, to which for example rigidly foldable origami twists [5] belong as a special case. For n = 3 and n = 4 more results are known, which can be summarized as follows: • Case n = 3: This case implies continuous flexible octahedra, which are very well studied objects dating back to Bricard [7]. Especially, the Bricard octahedra of the 3rd type (cf. [8]) correspond to the isogonal case, which was already pointed out by Kokotsakis [1]. Moreover, the study of these Kokotsakis belts allows also the determination of continuous flexible octahedra with vertices at infinity [9]. 3

This notation is in accordance with [6].

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Fig. 2. Generalized Kokotsakis belt: (a) It is obtained from the original Kokotsakis belt illustrated in Fig. 1 by dropping the planarity condition of the faces adjacent to the polygon line-segments of p. The resulting spatial structures are illustrated as tetrahedra. A part of the corresponding spherical image of the generalized Kokotsakis belt is visualized in (b).

• Case n = 4: These Kokotsakis belts, which are also known as (3 × 3) complexes, are the building blocks of continuous flexible PQ surfaces according to [4, Theorem 3.2]. Based on spherical kinematic geometry [6], a partial classification of continuous flexible (3 × 3) building blocks was obtained by Stachel and the author [10–12]. Inspired by this approach, Izmestiev [13] obtained a full classification containing more than 20 cases. Note that the first classes of continuous flexible PQ surfaces were given by Sauer and Graf [14]; namely the so-called T-hedra (see also [15, 16]) and the already mentioned V-hedra (see also [15, 17]). A rigid-foldable PQ surface which can be developed is a special case of origami. Under the additional condition of flat-foldability (as in case of the popular Miuraori) Tachi [18, 19] developed computational tools to design surfaces, where each vertex is of the isogonal type. Recently, Feng et al. [20] gave a complete analysis of the flat-foldable case, which can also be used for design tasks [21]. Within the field of computational design Jiang et al. [22] presented recently an optimization technique to penalize an isometrically deformed surface with planar quads. Its design space is restricted to rigid-foldable quad-surfaces which can be seen as a discretization of flexible smooth surfaces (e.g. Voss surfaces, profile-affine surfaces [14, 15]).

2 Spherical Image of Kokotsakis Belts In order to get a consistent notation for the construction of the spherical image of the Kokotsakis belt we orient the line-segments meeting at a vertex Vi according to Figs. 1a and 2a, respectively. Taking this orientation of the line-segments into account, the spherical 4-bar mechanism, which corresponds with the arrangement of faces around Vi , has the following spherical bar lengths:

δi = π − δi∗ ,

γi = π − γi∗ ,

λi = π − λi∗ ,

μi = π − μi∗

(1)

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for the index4 i = 0, . . . , n−1. The spherical image of faces around two adjacent vertices Vi and Vi+1 is illustrated in Figs. 1b and 2b, which show the motion transmission from the vertex Ci over Ci+1 to Ci+2 by two coupled spherical 4-bar mechanisms. Note that in the isogonal case these 4-bar mechanisms are so-called spherical isograms fulfilling one of the following two conditions: (1) λi = μi ,

δ i = γi ,

(2) λi + μi = π ,

δ i + γi = π .

(2)

Note that these two types are related by the replacement of one of the vertices of the spherical isogram by its antipodal point, which does not change its motion. In Sect. 3 we show that we can restrict to type (1) without loss of generality by assuming an appropriate choice of orientations. In the following we use the half-angle substitutions sin αi =

2ai , 1+a2i

cos αi =

1−a2i , 1+a2i

sin βi =

2bi , 1+b2i

cos βi =

1−b2i , 1+b2i

(3)

in order to end up with algebraic expressions. It is well known (e.g. [6]) that the input angle αi and the output angle βi of the i-th spherical isogram of type (1) of Eq. (2) are related by δi ±sin λi bi = fi ai with fi = 0 and fi = sin (4) sin (δi −λi ) . The two options in the expression for fi implied by the ± sign refer to the case whether the motion transmission corresponds to that of a spherical parallelogram (⇔ fi > 0) or spherical antiparallelgram (⇔ fi < 0), respectively. Note that the degenerated cases (δi = λi and δi + λi = π ) of the spherical isogram are excluded by the condition fi = 0 given in Eq. (4). The angles βi and αi+1 are related over the offset angle εi+1 ; i.e. βi + εi+1 = αi+1 . This means that εi+1 gives only the shift between the output angle βi of the i-th isogram to the input angle αi+1 of the (i + 1)-th isogram. This yields the relation: tan αi+1 =

tan βi +tan εi+1 1−tan βi tan εi+1 .

Using the half-angles and the Weierstrass substitution ei+1 := tan ai+1 =

(5) εi+1 2

bi +ei+1 1−bi ei+1 .

yield (6)

Note that the spherical arcs BiCi,i+1 and Ai+1Ci,i+1 enclose the angle ζi+1 := εi+1 + τi+1 (cf. Fig. 2b), where the latter angle is the torsion angle of the spatial polygon p, which is defined as the angle enclosed by the spherical arcs CiCi+1 and Ci+1Ci+2 . From the polygon p the angles τi+1 can be computed as the angle of rotation about the oriented axis ViVi+1 , which brings the plane [Vi−1 ,Vi ,Vi+1 ] to the plane [Vi ,Vi+1 ,Vi+2 ]. Therefore τi+1 , which is within the interval (−π ; π ], can be computed as: (ci×ci+1 ) (ci+1×ci+2 ) τi+1 = sign (o) arccos c (7) with o := (ci × ci+1 ) ci+2 i×ci+1 ci+1×ci+2 where ci denotes the vector from Vi−1 to Vi . Remark 1. For the original Kokotsakis belt (cf. Fig. 1) the angle ζi+1 is zero (⇒ εi+1 = −τi+1 ) or π (⇒ εi+1 = π − τi+1 ) for all i = 0, . . . , n − 1. Note that p is a planar curve if all τi+1 are either zero or π . 4

Note that in the remainder of the paper the indices are taken modulo n.

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3 Continuous Flexible Kokotsakis Belts of the Isogonal Type According to [6, Theorem 1] the Kokotsakis belt is continuous flexible if and only if the spherical image has this property. Now we will show that any Kokotsakis belt of the isoganal type can be identified with a spherical mechanism, which is only composed of spherical isograms of type (1) in Eq. (2): We start with the spherical image of p, i.e. the points C0 , . . . ,Cn−1 and construct the spherical points A0 and B0 according to Sect. 2. If the spherical isogram C0C1 B0 A0 is of type (2) then we replace B0 by its antipode. Then we proceed as follows around the spherical image of the polyline p; i.e. for i = 0, . . . , n − 1: a. In the case where the two antipodal points, which are candidates for Ai+1 , correspond with the values zero and π for εi+1 , we have to choose the one which implies εi+1 = 0 as εi+1 = π is not covered by Eq. (6). In any other case Ai+1 can be chosen arbitrary from the corresponding set of two antipodal points. b. Bi+1 has to be chosen from the corresponding set of two antipodal points such that the spherical isogram Ci+1Ci+2 Bi+1 Ai+1 is of type (1). We can end up in two situations; either An = A0 and we are done or An is the antipodal point of A0 . In the latter case we denote by j the highest possible index within the set {0, . . . , n − 1} for which the choice of Ai+1 was done arbitrarily in step (a). Then we replace all Ai+1 and Bi+1 with i ≥ j by their antipodal points which yields An = A0 . Note that such a j has to exist as otherwise we can construct the following contradiction: No j exists if and only if there are no shifts; i.e. e0 = e1 = . . . = en−1 = 0. As a consequence α0 = β0 = 0 implies αi+1 = βi+1 = 0 for all i ∈ {0, . . . , n − 1}, which already shows that in this case An = A0 has to hold. As a consequence of the above considerations one can write down the condition for continuous flexibility of any Kokotsakis belt of the isogonal type, where the rigid polygon p has n > 2 vertices, as (8) a0 − an = 0. In this so-called closure condition we substitute an by ai =

ai−1 fi−1 +ei 1−ai−1 fi−1 ei

(9)

which results from Eq. (6) under consideration of Eq. (4). By iterating this substitution (in total n times) we end up with an expression of the form q2 a20 + q1 a0 + q0 = 0 where q2 , q1 , q0 are functions in f0 , . . . , fn−1 , e0 , . . . , en−1 . This means that the spherical coupler arms A0C0 and AnC0 coincide for all input angles α0 if and only if the following necessary and sufficient conditions for continuous mobility are fulfilled: q2 = 0,

q1 = 0,

q0 = 0.

(10)

This results in the following theorem: Theorem 1. For a given polyhedral curve p with n vertices, there exists at least a (2n − 3)-dimensional set of continuous flexible Kokotsakis belts of the isogonal type over C.

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Fig. 3. Generalized Kokotsakis belt of the isogonal type: (a) Edges with the same absolute values of their rotation angles during the continuous flexibility are illustrated with the same color. (b) For n = 3 we obtain an overconstrained angle-symmetric 6R linkage.

By taking a closer look at q2 = 0 it can easily be seen that the terms linear in ei are given by f0 e1 and f0 . . . fk−1 ek for k = 2, . . . , n. In the equation q0 = 0 the linear terms in ei are e0 , en−1 fn−1 and ek fk . . . fn−1 for k = 1, . . . , n − 2. Therefore each of the two conditions q2 = 0 and q0 = 0 can only be fulfilled independently from the choice of the fi ’s if there are no shifts; i.e. e0 = e1 = . . . = en−1 = 0. Note that these are not only necessary conditions but already sufficient ones as they imply q2 = q0 = 0. In this case the remaining condition q1 = 0 simplifies to f0 f1 · · · fn−1 = 1 and we end up with a (n − 1)-dimensional set of continuous flexible Kokotsakis belts of the isogonal type over C. Note that e0 = e1 = . . . = en−1 = 0 only implies planarity of p if we assume the faces to be planar. For a spatial polyline p with planar faces the spherical coupler arms BiCi+1 and Ai+1Ci+1 are aligned. Therefore all ei+1 are determined (cf. Remark 1) and we get: Theorem 2. For a given polyhedral curve p with n > 3 vertices, there exists at least a (n − 3)-dimensional set of continuous flexible Kokotsakis belts with planar faces of the isogonal type over C. For planar curves p (which is always true for n = 3) this dimension raises to (n − 1).

3.1

Property Regarding the Rotation Angles

According to [1, §8] opposite angles in a spherical isogram are either equal or complete each other to 2π . As a consequence opposite dihedral angles along edges meeting in a vertex Vi have at each time instant t the same absolute value of their angular velocities. Therefore the absolute values of the rotation angles around these two edges are the same (measured from an initial starting configuration). As one of the dihedral angels is the angle about an edge Vi Vi+1 of the polygon p, this property holds for the two spherical 4bars, which have the common point Ci+1 (cf. Figs. 1b and 2b, respectively). Therefore the same absolute values of the rotation angle can always be assigned to three edges within a continuous flexible Kokotsakis belt of the isogonal type (cf. Fig. 3a).

Generalizing Continuous Flexible Kokotsakis Belts of the Isogonal Type

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Example 1. Now we consider the case n = 3. For any choice of δi and γi for i = 1, 2, 3 and γ1 + γ2 + γ3 = 2π (closure condition of central triangle) there exist e0 , e1 , e2 ∈ C such that we get a continuous flexible Kokotsakis belt of the isogonal type. The resulting structure can be seen as an overconstrained 6R loop (cf. Fig. 3b), which belongs to the third class of so-called angle-symmetric 6R linkages [23] due to the above discussed angle property. Note that for e0 = e1 = e2 = 0 we get the already mentioned Bricard octahedron of type III (cf. case n = 3 in Sect. 1.1).

4 Continuous Flexible SQ Surfaces On page 168 of Sauer’s book [15] the following open problem is mentioned: Do there exist continuous flexible SQ surfaces? We answer this question positively by constructing (3 × 3) building blocks of continuous flexible V-hedra with skew quads within this section. The restriction to these substructures is sufficient as Theorem 3.2 of [4] can be generalized in the following way: Theorem 3. A non-degenerate SQ surface is continuous flexible, if and only if this holds true for every (3 × 3) building block. Proof. The arguments used for the proof of Theorem 3.2 of [4] do not rely on the planarity of the involved quads. 4.1 Associated Overconstrained Mechanism We start this section with the definition of reciprocal-parallel quad meshes: Definition 1. Two quad meshes Q and V are called reciprocal-parallel if the following conditions are fulfilled: Q and V are combinatorial dual; i.e. vertices of one mesh correspond to the faces of the other and vice versa. The edges of both meshes are related by the implied bijection that edges of adjacent face are mapped to edges between corresponding adjacent vertices and vice versa. Edges, which are related by this bijection, are parallel. Sauer [15] showed that every infinitesimal flexible quad surface Q possesses in general a unique (up to scaling) reciprocal-parallel quad mesh V with rigid vertices. The reciprocal-parallel surface of the latter mesh V is only uniquely determined (up to scaling) if Q is composed of skew quads, otherwise there exist infinitely many, which are in a parallelism relation5 to each other (cf. [15, Theorem 16.22]). The corresponding deformation of the V mesh during the continuous flexion of Q has to be a conformal transformation, as the vertex stars are rigid. The corresponding kinematic structure of V is composed of rigid vertex stars linked by cylindrical joints (cf. Fig. 5). Note that in general such a structure only has the trivial mobility resulting from the homothetic transformation. This motion can be omitted by fixing the length of one edge in the structure. These modified linkages are in general rigid but those stemming from continuous flexible quad surfaces Q have an overconstrained motion. 5

Corresponding faces and edges of these meshes are parallel.

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4.2

V-hedra with Skew Quads

We study this class in more detail, as it is a generalization of V-hedra with planar quads having many applications to structural engineering practice [17, 24]. For n = 4 the equations q2 = q1 = q0 = 0 of Eq. (10) read as follows: q2 := f0 [e0 f3 (e1 e2 f2 + e2 e3 f1 + e1 e3 − f1 f2 ) + e1 e2 e3 f2 − e3 f1 f2 − e2 f1 − e1 ], q1 := e1 e2 f0 f2 f3 + e2 e3 f0 f1 f3 + e1 e3 f0 f3 − e1 e3 f1 f2 − f0 f1 f2 f3 − e1 e2 f1 − e2 e3 f2 + 1 + e0 (e1 e2 e3 f1 f3 − e1 e2 e3 f0 f2 − e1 f1 f2 f3 + e3 f0 f1 f2

(11)

+ e2 f0 f1 − e2 f2 f3 + e1 f0 − e3 f3 ), q0 := e0 (e1 e3 f1 f2 + e1 e2 f1 + e2 e3 f2 − 1) + f3 (e1 e2 e3 f1 − e1 f1 f2 − e2 f2 − e3 ). We can solve this set of equations explicitly for e1 , e2 , e3 in dependence of e0 , f0 , . . . , f3 , which yields the following two solutions: e1 =

e0 f0 f2 ( f12 −1)( f32 −1)±R1 R2 , e20 ( f0 f2 − f1 f3 )( f0 − f1 f2 f3 )+( f0 f3 − f1 f2 )( f0 f2 f3 − f1 )

e2 =

∓R1 R2 , e20 ( f0 f1 − f2 f3 )( f0 f2 − f1 f3 )+( f0 f1 f3 − f2 )( f0 f2 f3 − f1 )

e3 =

(12)

e0 f1 f3 ( f02 −1)( f22 −1)±R1 R2

e20 ( f0 f2 − f1 f3 )( f0 f1 f2 − f3 )+( f0 f3 − f1 f2 )( f0 f1 f3 − f2 )

with R1 : = R2 : =

e20 ( f0 f1 − f2 f3 )( f0 f2 − f1 f3 ) + ( f0 f1 f3 − f2 )( f0 f2 f3 − f1 ),

(13)

e20 ( f0 f1 f2 − f3 )( f1 f2 f3 − f0 ) + ( f0 f1 f2 f3 − 1)( f1 f2 − f0 f3 ).

Remark 2. Alternatively, the above given equations q2 = q1 = q0 = 0 from Eq. (11) can also be solved explicitly for f1 , f2 , f3 in dependence of f0 , e0 , . . . , e3 , which yield the following two solutions: f1 =

−( f02 +1)e0 e1 (e23 +1)(e22 −1)+ f0 (e20 e21 e22 −e20 e21 e23 −e20 e22 e23 −e21 e22 e23 +e20 +e21 +e22 −e23 )±R3 R4 , 2e2 (e23 +1)(e0 f0 +e1 )(e0 e1 − f0 )

f2 =

( f02 +1)e0 e1 (e23 +1)(e22 +1)− f0 (e20 e21 e22 +e20 e21 e23 −e20 e22 e23 −e21 e22 e23 −e20 −e21 +e22 +e23 )∓R3 R4 , 2e2 e3 f0 (e21 +1)(e20 +1)

f3 =

−( f02 +1)e0 e1 (e23 −1)(e22 +1)− f0 (e20 e21 e22 −e20 e21 e23 +e20 e22 e23 +e21 e22 e23 −e20 +e21 +e22 −e23 )±R3 R4 2e3 (e22 +1)(e1 f0 +e0 )(e0 e1 − f0 )

with

R3,4 := [ f0 (e20 e21 e22 + e20 e21 e23 − e20 e22 e23 − e21 e22 e23 − e20 − e21 + e22 + e23 ) 1

− ( f02 + 1)e0 e1 (e23 + 1)(e22 + 1) ± 2 f0 e2 e3 (e21 + 1)(e20 + 1)] 2 where R3 corresponds to the plus sign and R4 to the minus sign.

(14)

(15)

Note that for a given skew central quad p and a set of real values e0 , . . . , e3 , f0 , . . . , f3 fulfilling the three equations q2 = q1 = q0 = 0, the missing geometric parameters δi can be computed from the equation sin δi ± sin λi − fi sin (δi − λi ) = 0 (cf. Eq. (4)). For the

Generalizing Continuous Flexible Kokotsakis Belts of the Isogonal Type

123

minus sign we get one further real solution beside the excluded degenerate case δi = λi . By shifting these two values obtained for δi by π we obtain the solutions of the equation with respect to the plus sign. Therefore we get a unique value for δi ∈ (0; π ) with δi = λi for each i = 0, . . . , 3. Example 2. The coordinates of the vertices of the skew central quad p are given by: V0 = (5, 0, 0)T ,

V1 = (4, 3, 0)T ,

V2 = (1, 2, 2)T ,

V3 = (0, 0, 0)T ,

(16)

from which the angles λi and τi for i = 0, . . . , 3 can be calculated. Moreover, the input data is completed by the values: e0 = 100,

d0 = 0.3,

d1 = 0.15,

d2 = 0.2,

d3 = 0.25,

(17)

where di = tan δ2i . From that we can compute the fi values according to Eq. (4) with respect to the minus sign for all i = 0, . . . , 3. Then the formulas for the solution set related to the upper sign in Eq. (12) yield e1 = −0.86081001,

e2 = −5.06077939,

e3 = 0.57043281.

(18)

One configuration of the resulting continuous flexible (3 × 3) building block of a Vhedra with skew quads is illustrated in Fig. 4, where also the corresponding spherical image is displayed. The associated overconstrained mechanism implied by the reciprocal-parallelism (cf. Sect. 4.1) is shown in Fig. 5. In the captions of Figs. 4 and 5 we also provide links to gif animations showing the overconstrained motion of these three mechanisms. We close this section by making the following two final comments: • The edges of the V-hedra can be subdivided into two families of discrete parameter lines, which are called u-polylines and v-polylines for short. Due to the property pointed out in Sect. 3.1, the rotation angles along any u-polyline or v-polyline are the same. Note that this property is well known for V-hedra with planar quads (cf. [14, page 529]) but also holds for the skew case. • In view of Sect. 4.1 it should be noted that there is a further remarkable relation to an overconstrained mechanism beside the one illustrated in Fig. 5. As already pointed out by Sauer [15] the vertex star fulfilling the isogonality condition is reciprocalparallel to a skew isogram, which has the following additional property: If the four bars of the isogram are hinged in the vertices by rotational joints, which are orthogonal to the plane spanned by the linked bars (cf. Fig. 6), then one obtains a so-called Bennett mechanism [25]. This is the only non-trivial mobile 4R loop. If the quads of the V-hedra Q are skew then the four axes of the Bennett mechanisms, which can be associated with a vertex of the mesh V , differ from each other (cf. Fig. 6). Only in the case where Q is a V-hedra with planar quads, each vertex of V can uniquely be associated with one rotational axis orthogonal to the planar vertex star. Then the resulting network of Bennett mechanisms is highly mobile6 . Finally it should be noted that in this case V is a discrete pseudospherical surface [4, 15, 26]. 6

The degree of the mobility corresponds to the number of rows plus columns of V minus one (cf. Sauer [15, Theorem 11.18]).

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Fig. 4. A (3 × 3) building block of a V-hedra with skew quads (a) and its spherical image (b). The vertex Vi and the line-segment Vi−1 Vi have the same color, where i = 0 corresponds to black, i = 1 to red, i = 2 to magenta and i = 3 to purple. Note that the illustrations of Figs. 4, 5 and 6 are rendered with respect to the same view. The animations of the mobility of the V-hedra and its spherical image are online available at https://www.dmg.tuwien.ac.at/nawratil/skew quad spatial. gif and https://www.dmg.tuwien.ac.at/nawratil/skew quad spherical.gif, respectively.

Fig. 5. The overconstrained mechanism which results from the reciprocal-parallelism to the (3 × 3) building block of a V-hedra with skew quads illustrated in Fig. 4a. The animation of the mobility of this mechanism is online available at https://www.dmg.tuwien.ac.at/nawratil/ skew quad reciprocal.gif, where we fixed the length of the black edge, which is parallel to V3 V0 .

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Fig. 6. The four Bennett mechanisms associated with the structure illustrated in Fig. 5, where the rotation axes are displayed in yellow.

5 Conclusions, Open Problems and Future Research We generalized continuous flexible Kokotsakis belts of the isogonal type by allowing that the faces, which are adjacent to the line-segments of the rigid closed polygon p, to be skew. In more detail we studied the case where p is a skew quad as it corresponds to a (3 × 3) building block of a V-hedra composed of skew quads, which proves (under consideration of Theorem 3) the existence of continuous flexible SQ surfaces. Open questions in this context regard the smooth analog of continuous flexible Kokotsakis belts of the isogonal type and of V-hedra with skew quads. This study at hand is also the starting point towards a full classification of continuous flexible (3 × 3) SQ building blocks, which is subject to future research. Acknowledgements. The research is supported by grant F77 (SFB “Advanced Computational Design”, subproject SP7) of the Austrian Science Fund FWF.

References ¨ 1. Kokotsakis, A.: Uber bewegliche Polyeder. Math. Ann. 107, 627–647 (1932) 2. Karpenkov, O.N.: On the flexibility of Kokotsakis meshes. Geom. Dedicata. 147, 15–28 (2010) ¨ 3. Voss, A.: Uber diejenigen Fl¨achen, auf denen geod¨atische Linien ein konjugiertes System bilden. M¨unchner Berichte, pp. 95–102 (1888) 4. Schief, W.K., Bobenko, A.I., Hoffmann, T.: On the integrability of infinitesimal and finite deformations of polyhedral surfaces. In: Bobenko, A.I. et al. (ed.) Discrete Differential Geometry Oberwolfach Seminars 38, 67–93 (2008) 5. Evans, T.A., Lang, R.J., Magleby, S.P., Howell, L.L.: Rigidly foldable origami twists. In: Miura, K. et al. (ed.) Origami6 , I: Mathematics, pp. 119–130. American Mathematical Society (2015)

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6. Stachel, H.: A kinematic approach to Kokotsakis meshes. Comput. Aided Geometric Des. 27, 428–437 (2010) 7. Bricard, R.: M´emoire sur la th´eorie de l’octa`edre articul´e. J. de Math´ematiques pures et appliqu´ees, Liouville 3, 113–148 (1897) 8. Stachel, H.: Remarks on Bricard’s flexible octahedra of type 3. In: Proceedings of the 10th International Conference on Geometry and Graphics, 28 July–3 August 2002, Kiev/Ukraine), vol. 1, pp. 8–12 (2002) 9. Nawratil, G.: Flexible octahedra in the projective extension of the Euclidean 3-space. J. Geom. Graph. 14(2), 147–169 (2010) 10. Nawratil, G., Stachel, H.: Composition of spherical four-bar-mechanisms. In: Pisla, D. et al. (ed.)New Trends in Mechanisms Science–Analysis and Design, pp. 99–106. Springer (2010). https://doi.org/10.1007/978-90-481-9689-0 12 11. Nawratil, G.: Reducible compositions of spherical four-bar linkages with a spherical coupler component. Mech. Mach. Theory 46(5), 725–742 (2011) 12. Nawratil, G.: Reducible compositions of spherical four-bar linkages without a spherical coupler component. Mech. Mach. Theory 49, 87–103 (2012) 13. Izmestiev, I.: Classification of flexible Kokotsakis polyhedra with quadrangular base. Int. Math. Res. Not. 2017(3), 715–808 (2017) ¨ 14. Sauer, R., Graf, H.: Uber Fl¨achenverbiegung in Analogie zur Verknickung offener Facettenflache. Math. Ann. 105, 499–535 (1931) 15. Sauer, R.: Differenzengeometrie. Springer (1970). https://doi.org/10.1007/978-3-64286411-7 16. Sharifmoghaddam, K., Nawratil, G., Rasoulzadeh, A., Tervooren, J.: Using flexible trapezoidal quad-surfaces for transformable design. In: Proceedings of the IASS Annual Symposium 2020/21 and the 7th International Conference on Spatial Structures, IASS. (in press) 17. Montagne, N., Douthe, C., Tellier, X., Fivet, C., Baverel, O.: Voss surfaces: a design space for geodesic gridshells. J. Int. Assoc. Shell Spat. Struct. 61(206), 255–263 (2020) 18. Tachi, T.: Generalization of rigid foldable quadrilateral mesh origami. J. Int. Assoc. Shell Spati. Struct. 50(3), 173–179 (2009) 19. Tachi, T.: Freeform rigid-foldable structure using bidirectionally flat-foldable planar quadrilateral mesh. In: Ceccato, C. et al. (ed.) Advances in Architectural Geometry, pp. 87–102. Springer (2010). https://doi.org/10.1007/978-3-7091-0309-8 6 20. Feng, F., Dang, X., James, R.D., Plucinsky, P.: The designs and deformations of rigidly and flat-foldable quadrilateral mesh origami. J. Mech. Phys. Solids 142, 104018 (2020) 21. Dang, X., Feng, F., Plucinsky, P., James, R.D., Duan, H., Wang, J.: Inverse design of deployable origami structures that approximate a general surface. Int. J. Solids Struct. 234–235, 111224 (2022) 22. Jiang, C., et al.: Using isometries for computational design and fabrication. ACM Trans. Graph. 40(4), 42 (2021) 23. Li, Z., Schicho, J.: Classification of angle-symmetric 6R linkages. Mech. Mach. Theory 70, 372–379 (2013) 24. Mitchell, T., Mazurek, A., Hartz, C., Miki, M., Baker, W.: Structural applications of the graphic statics and static-kinematic dualities: rigid origami, self-centering cable nets, and linkage meshes. In: Proceedings of the IASS Annual Symposium (16–20 July 2018, Boston/USA), Symposium: Graphic statics, pp. 1–8(8) (2018) 25. Bennett, G.T.: The skew isogram mechanism. In: Proceedings of the London Mathematical Society, vol. s2-13(1), pp. 151–173 (1914) 26. Wunderlich, W.: Zur Differenzengeometrie der Fl¨achen konstanter negativer Kr¨ummung. ¨ Sitzungsbericht der Osterreichischen Akademie der Wissenschaften, Mathem.-Naturw. Klasse IIa 160(1–5), 39–77 (1951)

The Intersection Curve of an Ellipsoid with a Torus Sharing the Same Center Ana Maria Reis D’Azevedo Breda1(B) , Alexandre Emanuel Batista da Silva Trocado2 , and José Manuel Dos Santos Dos Santos3 1

3

Universidade de Aveiro, Aveiro, Portugal [email protected] 2 Center for Research and Development in Mathematics and Applications, Universidade de Aveiro, Aveiro, Portugal [email protected] Centre for Research and Innovation in Education (inED), Escola Superior de Educação Politécnico do Porto, Porto, Portugal [email protected]

Abstract. The main objective of this work is focused on classifying the families of curves defined by the intersection of an arbitrary ellipsoid with an arbitrary torus, sharing the same center, based on the number of their connected components and on the number of their auto-intersection points. The graphic geometric representation of these curves, in GeoGebra, and the respective algebraic descriptions, supported from a theoretical and computational point of view, were of fundamental importance for the development of this work. In this paper, we describe a procedure and the necessary implementations to achieve the objective outlined. Keywords: GeoGebra · Toric sections · Quadrics · Intersection curves

1 Introduction The need for an accurate description of geometric and algebraic representations of the intersection curves of two surfaces, reported by different scientific areas, highlights the relevance of this research topic. For example, the intersection curves of a quadratic with a Torus is of great interest in robotics applications, [4]. However, the characterization of these curves, heavily based on computational algorithms, is still very scarce. The work we presented here focuses on classifying the intersection curves of an arbitrary ellipsoid with an arbitrary torus, sharing the same center, based on the number of connected components and on the number of their auto intersection points. There are different methods of approaching the intersection curve problems. Recently, the tendency is to combine two or more methods, preserving the advantages of each one of them and eliminating, whenever possible, their disadvantages. We will follow this trend, making use of the dynamic and geometric capabilities of GeoGebra, the Maple’s capability for symbolic computation and the projection/lifting method, [8]. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 127–137, 2023. https://doi.org/10.1007/978-3-031-13588-0_11

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2 Intersection Curve of an Ellipsoid with a Torus Let T be a torus centred at the origin, O, of the Cartesian coordinate system. Denoting the radius from the center of the hole to the center of the torus tube by rM and the radius of the tube by rm, T can be described by the parametric equation: x = (rM + rm cos u) cos v, y = (rM + rm cos u) sin v , and z = rm sin u

(1)

and also by 2 z4 + 2rM 2 − 2rm2 + 2x2 + 2y2 z2 + rM 2 − rm2 + x2 + y2 − 4rM 2 x2 + y2 = 0 (2) where rm and rM are positive real numbers satisfying rm < rM, and u, v ∈ [0, 2π ]. Let us denote by E the ellipsoid described by x2 y2 z2 + + − 1 = 0, (3) a2 b2 c2 where a, b, and c are positive real numbers. The intersection curve CTE = T ∩ E, is ruled by the algebraic equation, rm2 c2 a2 − b2 (cos v)2 + a2 b2 − a2 c2 (cos u)2 − 2 2 2 a b c 2 c2 rM a2 − b2 (cos v)2 − a2 rm cos u − (4) a2 b2 c2 2 2 2 2 2 c rM a − b (cos v) − a2 b2 c2 2 2 2 b − rM c − b2 rm2 a2 − = 0. a2 b2 c2 Under certain assumptions we may solve this equation in order, firstly, to cos u, getting, Δ1 (v)abc − Δ2 (v)abc Δ1 (v)abc + Δ2 (v)abc cos u = or cos u = , (5) Δ3 (v)abc Δ3 (v)abc where, Δ1 (v)abc = rMc2 ((−a2 + b2 )cos v2 + a2 ) Δ2 (v)abc = a2 b2 (−c2 (c2 + rM 2 − rm2 )(a2 − b2 )cos v2 + +a2 (c4 − (b2 − rM 2 + rm2 )c2 + b2 rm2 ))

(6)

Δ3 (v)abc = rm(c2 (a2 − b2 )cos v2 + a2 (b2 − c2 )). Solving 5 in order a u, one has, u1 (v) = arccos

Δ1 (v)abc − Δ2 (v)abc Δ1 (v)abc + Δ2 (v)abc (v) = arccos or u , 2 Δ3 (v)abc Δ3 (v)abc

(7)

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Fig. 1. GeoGebra outputs of cQT .

where v below to [−π , π ]. Finally using (7) in (1) we could get an analytical description of CTE in terms of the parameter v. Using this method we find a stable way to represent CTE in GeoGebra, see Fig. 1(a). We may also represent a relation between the parameters u and v which allow us to analyse the behavior of the functions enrolled in the parametrizations, Fig. 1(b) and, from there classify CTE . Having in mind the characterisation of CTE , we will follow the above procedure, dividing it in several (sub)cases. Firstly, we consider the special case of the sphere, a = b = c, followed by two special cases of a “real” ellipsoid position, that is a = b with c = a or b = c with a = b, ending up with the case a < b < c. 2.1 CTE When a = b = c If a = b = c, the Eq. (3) defines a sphere and CTE corresponds to one of the cases already studied in [3, 5] and in [2], where a distinct methodology from the one we propose here, was applied. Using the procedure described previously, the Eq. 4 takes the form, −a2 + rM 2 + 2 rM rm cos u + rm2 = 0, a2

(8)

−rm , which is a well defined constant, since rM − rm ≤ a ≤ and so, u = arccos a −rM 2rMrm rM + rm. If a = rM − rm, the intersection curve is a circle of radius a, centered at (0, 0, 0), in the xOy plane. When a = rM + rm, the intersection curve is a union of two circles of radius, respectively, rM − rm and a = rM + rm, centered at (0, 0, 0), leaving, also, in the xOy plane. In the remaining cases, the intersection curve is the union of two congruent circles of radius, a2 − rm2 sin u2 , centered at (0, 0, −rm sin u) and (0, 0, rm sin u), leaving on planes, parallel to the xy-plane. 2

2

2

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CTE When a = b and c = a

2.2

Considering a = b with c = a, the corresponding Eq. 4 to CTE is given by:

−a2 + c2 rm2 (cos (u))2 + 2 cos (u) c2 rM rm + −a2 + rM 2 c2 + rm2 a2 = 0, a2 c2

(9)

which is, for each value of the parameters a, c, rm, and, rM, a quadratic equation in √ aac Δaac 1 ± Δ2 2 cos u, whose solutions are , with Δaac 1 = c rM , Δaac 3 2 2 2 2 2 4 2 2 2 2 aac Δaac 2 = −a a c − a rm − c − c rM + c rm , and Δ3 = (a − c)(a + c)rm. Meaning that, CTE is: √ = rM − rm; – one circle, if a = rM + rm, and, c ≥ rMrm + rm2 , or a√ – the union of three circles, if a = rM + rm, and, 0 < c < rMrm + rm2 ; – the union of two circles, if rM − rm < a < rM + rm. Besides, whenever the intersection curve is a union of circles, they leave in parallel planes, perpendicular to the common rotation axis of both surfaces. The study of CTE for b = c ∧ a = b or 0 < a < b < c will be done studying the projection of CTE in the xy-plane and will be the subject of the following sections. 2.3

The Auto-Intersection Points of CTE

In the previous sections we presented a more robust form to represent CTE , graphically. The graphic representations reveal two and only two types of auto intersection points. One of them, notated, from now on, by auto intersection points of type I, occurs when CTE is not a planar curve and contains one of the following eight points: (±rM + rm, 0, 0); (±rM − rm, 0, 0); (0, ±rM + rm, 0, 0), (0, ±rM − rm, 0). A second type, the auto intersection points of type II, occurs in points leaving outside of the coordinate axes but located in the xz-plane (or in the yz-plane). The explicit equation of the projection, PCTE , of CTE in the xy-plane may be obtained by the resultant of the implicit surface Eqs. (1) and (2). For more details see [2]. Since, all the critical points of CTE have null y coordinates, we must look at the intersection points of PCTE in the admissible region, Ar = {(x, y) ∈ R2 : x2 + y2 ≥ 2 2 (rM − rm)2 ∧ x2 + y2 ≤ (rM + rm)2 ∧ ax + by ≤ 1} whose lifting are in the xzplane. Auto intersection points of type I of CTE occur when PCTE is “tangent” either to the circle x2 + y2 = rm2 or to the circle x2 + y2 = (rM + rm)2 contained in Ar. Auto intersection points of type II of CTE will be given by the analysis of the xcoordinates of the critical points in Ar. The critical points of PCTE are, in general, six, and their x-coordinate are given by: √ −arM ± δca a (10) ; a2 − c2

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√ arM ± δca a ; a2 − c2

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− (b2 − c2 ) (a2 − b2 ) δcb ba ; ± (b2 a2 − c2 a2 − b4 + c2 b2 ) c

(11) (12)

where δca = c4 − a − rM 2 − rm2 a + rM 2 − rm2 c2 + a2 rm2

(13)

δcb = −c4 + b − rM 2 − rm2 b + rM 2 − rm2 c2 − b2 rm2

(14)

and

Computer calculations, as expected, confirm that the x-coordinates given by 10 and 11 correspond to he x-coordinates of the critical points. However, 12 only appear when the parameters a, b, and c are pairwise distinct. In the case of a = b ∧ a = c, pcTE is defined by the equation: 4 a − 2a2 c2 + c4 x4 + + 2a4 − 4a2 c2 + 2c4 y2 + 2a4 c2 − 2a4 rM 2 − 2a4 rm2 − 2a2 c4 − 2a2 c2 rM 2 + 2a2 c2 rm2 x2 + 4 + a − 2a2 c2 + c4 y4 + + 2a4 c2 − 2a4 rM 2 − 2a4 rm2 − 2a2 c4 − 2a2 c2 rM 2 + 2a2 c2 rm2 y2

(15)

+a4 c4 + 2a4 c2 rM 2 − 2a4 c2 rm2 + a4 rM 4 − 2a4 rM 2 rm2 + a4 rm4 = 0.

The corresponding path is either one or two circles as, is the case, illustrated in Fig. 2,

Fig. 2. PCTE for a = b = 9, c = 2, rM = 4, and rm = 2.

The x-coordinates of the critical points of PCTE are given by (10) and (11). Considering these, we get, respectively, the radius of one of two circles centered at O, which contain the points of PCTE . Summarizing,

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rM i) if δca = 0 the points of PCTE belong to the circle centred at O and radius aa2 −c 2; ii) if δca = 0 the points of PCTE belong to the union of two circles center at O and radius √ arM ± δca a . In this case, PCTE has no auto-intersections points and the same a2 − c2 happens with CTE .

Considering now, b = c ∧ a = c. PCTE is defined by the equation: 4 a − 2a2 c2 + c4 x4 + 2a4 c2 − 2a4 rM 2 − 2a4 rm2 − 2a2 c4 − 2a2 c2 rM 2 + 2a2 c2 rm2 x2 +a4 c4 + 2a4 c2 rM 2 − 2a4 c2 rm2 + a4 rM 4 − 2a4 rM 2 rm2 − 4a4 rM 2 y2 + a4 rm4 = 0.

(16)

The corresponding paths are similar to the one illustrated in Fig. 3(b),

Fig. 3. GeoGebra 3D output of cQT and graph of PCTE .

and the x- coordinates of the singular points are given by (10) and (11). Observe that, now, there are auto intersection points in PC TE whenever δca = 0, being the coordinates a2 rM of the singular points given by ± a2 −c2 , 0 . As illustrated in Fig. 3(a), CTE has two auto intersection points in (0, ±2, 0), corresponding to the “tangential points” of PCTE with As. Besides, CTE is, roughly speaking,“approaching” of having 4 auto intersection points, all leaving outside the xOy plane. When this situation occurs, necessarily, δca = 0. Solving δca = 0, in order to c, we get, in function of the parameter a, two families of CTE with two connected components either without auto intersections points, or with two auto intersections points (±(rM + rm), 0, 0) corresponding to the case a = rM + rm. However, solving δca = 0, in order to the parameter a, we get, (c2 − rm2 ) c2 + rM 2 − rm2 c , a(c) = c2 − rm2

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and CTE could be describe as: (i) a connected curve with four auto intersection points;

a4 − 2a2 c2 − a2 rM 2 + c4 c a2 rM , 0, ± , if rm < c < rMrm + rm2 ; ± 2 2 2 2 a −c a −c (ii) a connected curve with two auto intersections points, (±(rM + rm), 0, 0), if c = rMrm + rm2 ; (iii) a√curve with two connected components and without auto intersection points, if rMrm + rm2 ) < c < rM + rm. (iv) the set {(0, −rM − rm, 0), (0, rM + rm, 0)} , when c = rM + rm. Geometric illustrations, giving a panorama of the several configurations of CTE , based in our findings are given in Figs. 4 and 5, considering the intersection of the torus T(0,2,4) with the class of ellipsoids E(a,a,c) . Assuming now that a, b, and c are pairwise distinct, PCTE is defined by the equation: a4 b4 c4 + 2a4 b4 c2 rM 2 − 2a4 b4 c2 rm2 + 2a4 b4 c2 x2 + 2a4 b4 c2 y2 + a4 b4 rM 4 −2a4 b4 rM 2 rm2 − 2a4 b4 rM 2 x2 − 2a4 b4 rM 2 y2 + a4 b4 rm4 − 2a4 b4 rm2 x2 + −2a4 b4 rm2 y2 + a4 b4 x4 + 2a4 b4 x2 y2 + a4 b4 y4 − 2a4 b2 c4 y2 − 2a4 b2 c2 rM 2 y2 +2a4 b2 c2 rm2 y2 − 2a4 b2 c2 x2 y2 − 2a4 b2 c2 y4 + a4 c4 y4 − 2a2 b4 c4 x2 − 2a2 b4 c2 rM 2 x2 +2a2 b4 c2 rm2 x2 − 2a2 b4 c2 x4 − 2a2 b4 c2 x2 y2 + 2a2 b2 c4 x2 y2 + b4 c4 x4 = 0. (17) Let us analyze the cases δca = 0, δcb = 0, and δca = 0 ∧ δcb = 0. Considering δca = 0 ∧ δcb = 0 then a = b, ending up in the cases already described. The auto intersection points type II have null x-coordinates, if δcb = 0 and null y-coordinates, if δca = 0. In fact, we do only need to study what happens when δca = 0. When δca = 0 similar shapes as the ones described in Sect. 2.3 appear. There are intersection curves with one connected component and six auto intersection points,

a2 rM (0, ±(rM − rm), 0), ± 2 , 0, ± a − c2

a4 − 2a2 c2 − a2 rM 2 + c4 c a2 − c2

as, in the example, illustrated in Fig. 6(a). Other cases with six auto intersection points also appear, as we may see in figure 6(b), where two of them have coordinates (0, ±(rM + rm), 0).

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Fig. 4. I-Panorama of the configurations of CTE

We propose as a classification criterion for the CTE families, the number of their connected components and components and the number of their auto intersection points. The classification obtained, following this criterion, can be found in Table 1. There is one more case where the ellipsoid is tangencial to the torus in eigth points.

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Fig. 5. II-Panorama of the configurations of CTE

The study carried out confirms the results expressed in [1], although these researchers have used the projective immersion of the problem under analysis. The characterization of the intersection curves in the torus by planes, made by Poncelet and known since 1862 [7], is part of the book [6]. The work presented here is done completely in the Euclidean space, extending the results expressed in [1], where a complete characterization of the intersection curves in terms of the self-intersection points and the connected components is given.

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Fig. 6. GeoGebra 3D output of CTE with six auto intersections points. Table 1. Classification of CTE(a,b,c,rM,rm) CTE

N. of Auto Intersection Points

N. Connex Components

0

2

4

6

1

a = c = rM + rm a = c = rM − rm

a = rM − rm ∧ rM − rm < c < rM + rm rM − rm < a < rM + rm ∧ c = rM − rm rM − rm < a < rM + rm ∧ c = rM + rm a = rM + rm ∧ rM − rm < c < rM + rm

a = rM − rm ∧ c = rM + rm a > rM + rm ∧ rM − rm < c < rM + rm

a > rM + rm ∧ c = rM − rm

2

a < rM − rm ∧ rM − rm < c < rM + rm a < rM − rm ∧ c = rM − rm a < rM − rm ∧ rM − rm < c < rM + rm a = rM − rm ∧ c < rM − rm rM − rm < a < rM + rm ∧ c < rM − rm rM − rm < a < rM + rm ∧ rM − rm < c < rM + rm rM − rm < a < rM + rm ∧ c > rM + rm a = rM + rm ∧ rM − rm < c < rM + rm a > rM + rm ∧ rM − rm < c < rM + rm a > rM + rm ∧ c = rM + rm

a < rM − rm ∧ c = rM + rm a = rM − rm ∧ c > rM + rm a > rM + rm ∧ c < rM − rm a > rM + rm ∧ c = rM − rm

a = rM + rm ∧ c = rM − rm a > rM + rm ∧ c = rM − rm

3 4

a < rM − rm ∧ c > rM + rm a = rM − rm ∧ c < rM − rm a > rM + rm ∧ c < rM − rm a > rM + rm ∧ rM − rm < c < rM + rm

3 Conclusions We present, based on the number of connected components and the number of selfintersecting points of the intersection curve and using a combined approach of methodologies, the classification of all families of curves obtained by the intersection of an ellipsoid with a Torus, sharing the same center. We believe that the presented approach, with the necessary adaptations, may be used to obtain the intersection curve of the Torus with other quadratics. This is one of our future lines of research. Acknowledgements. This research was supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020; The Centre for Research and Innovation in Education (inED), through

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the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the project UIDB/05198/2020; and Organization of Ibero-American States for Education, Science and Culture (OEI).

References 1. Bottema, O., Primrose, E.: Algebraic curves on a torus. In: Indagationes Mathematicae (Proceedings), vol. 77, pp. 333–338. North-Holland (1974) 2. Breda, A.M.R.D.A., Trocado, A.E.B.S., Dos Santos, J.M.D.S.: Torus and quadrics intersection using GeoGebra. In: Cheng, L.-Y. (ed.) ICGG 2021. AISC, vol. 1296, pp. 484–493. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-63403-2_43 3. Kim, K.J.: Torus and simple surface intersection based on a configuration space approach. Ph.D. thesis, Department of Computer Science and Engineering, POSTECH (1998). http://bh. knu.ac.kr/~kujinkim/papers/kjkim_thesis.pdf 4. Michael McCarthy, J., Su, H.J.: The computation of reachable surfaces for a specified set of spatial displacements, pp. 709–735. Springer, Heidelberg (2005). https://doi.org/10.1007/3540-28247-5_22 5. Moroni, L.: The toric sections: a simple introduction. arXiv preprint arXiv:1708.00803 (2017) 6. Nicaise, P.: Courbes algébriques planes, cubiques et cycliques. Editions Publibook (2017) 7. Poncelet, J.V.: Applications d’analyse et de géométrie: qui ont servi de principal fondement au traité des propriétés projectives des figures: comprenant sept cahiers manuscrits rédigés à Sabatoff dans les prisons de Russie (1813–1814), et accompagnés de divers autres écrits, anciens ou nouveaux, vol. 1. Mallet-Bachelier (1862) 8. Vega, L.G.: A subresultant theory for multivariate polynomials. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC (1991). https://doi.org/ 10.1145/120694.120705

Four-Dimensional Visual Exploration of the Complex Number Plane 1 ˇ Jakub Rada 1

and Michal Zamboj2(B)

Faculty of Mathematics and Physics, Mathematical Institute of Charles University, Charles University, Prague, Czech Republic [email protected] 2 Faculty of Education, Department of Mathematics and Mathematical Education, Charles University, Prague, Czech Republic [email protected] https://www2.karlin.mff.cuni.cz/∼zamboj/index en.html

Abstract. A straight line intersects a circle in two, one, or no real points. In the last case, they have two complex conjugate intersecting points. We present their construction by tracing the circle with all lines. To visualize these points, the real plane is extended with the imaginary dimensions to four-dimensional real space. The surface generated by all complex points is orthogonally projected into two three-dimensional subspaces generated by both real and one of the imaginary dimensions. The same method is used to trace and visualize other real and imaginary conics and a cubic curve. Furthermore, we describe a graphical representation of complex lines in the four-dimensional space and discuss the elementary incidence properties of points and lines. This paper provides an accessible method of visualization of the complex number plane. Keywords: Complex number plane · Fourth dimension Multi-dimensional visualization · Complex roots

1

·

Introduction

Let us consider a circle and a line in a real plane. The line intersects the circle in two real points, touches the circle in the point of tangency, or has no real but two complex conjugate points on the circle. This could be an elementary exercise in analytic geometry. However, the geometric construction or visualization of the last case is not as obvious. In this paper, our focus lies on the visualization of complex points. The property of keeping the number of intersecting points of a line and a conic (or algebraic curve in general) and their geometric construction based on the polar properties of conics were described in Poncelet’s early texts on the principle of continuity in the framework of projective geometry. Figure 1, from the second edition of his comprehensive work—Trait´e des propri´et´es projectives des figures [5] shows a construction of a secant and non-secant line intersecting an c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 138–149, 2023. https://doi.org/10.1007/978-3-031-13588-0_12

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ellipse. These ideas were thoroughly revisited and presented by Hatton around one hundred years later (see [4]). In Chap. 6, Hatton described the process of “tracing of conics” along their conjugate diameters and created their planar graphs called Poncelet ﬁgures. We aim to lift this idea into a four-dimensional space to visualize the complex points of all branches at once.

Fig. 1. Poncelet’s Fig. 6 [5]. The points M, N are real points on the ellipse, while M , N are complex points on the same ellipse. The points O and O are poles of the polars M N and M N with respect to the ellipse. Source gallica.bnf.fr/Biblioth`eque nationale de France.

In fact, the presently often used and popular visualizations of complex numbers founded by Argand and Wessel had appeared only two decades before Poncelet’s Trait´e (see [8] pp. 438–439 for historical details). To visualize two complex numbers as coordinates of the complex number plane1 C2 in a similar manner, one has to approach a four-dimensional real space. The advancement of computer graphics brought eﬀective visualization tools in higher dimensions. Several authors displayed complex elements in separate 3-dimensional spaces (see [1,9]). A four-dimensional set of points is plotted in (Re(x), Re(y), Im(x)), (Re(x), Re(y), Im(y)), (Re(x), Im(x), Im(y)), or (Re(y), Im(x), Im(y)). The author of [9] has also created an iOS application [10] that can display each of these graphs. Butler in [3] placed a perpendicular plane with axes (Im(x), Im(y)) at each point of the real plane (Re(x), Re(y)), then the perpendicular plane is rotated to the real plane such that the axes Re(x) with Im(x) and Re(y) with Im(y) are parallel at each point. Bozlee in [2] used 3D-printing to create a 3Dprinted model with complex parts of elliptic curves using Amethyst’s bertini real software. 1

Not to be confused with the term “complex plane”, which usually indicates the (Argand, Wessel, or also Gauss) plane with coordinate axes corresponding to real and imaginary elements of one complex variable.

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In this paper, we contribute to the topic by the visualization of complex points on a circle created in a double orthogonal projection into two mutually perpendicular 3-spaces (4D-DOP, see [12]) and in a four-dimensional perspective (4D-perspective, see [6]). Furthermore, we elaborate the visualization of a line through two points in C2 . All the upcoming ﬁgures are created in Wolfram Mathematica.

2

Tracing a Circle

Fig. 2. (a) A point P with the coordinates P [(Re(px ), Im(px )); (Re(py ), Im(py ))] and graphically represented in 4D-DOP. P1 [Re(px ), Re(py ), Im(px )] P2 [Re(px ), Re(py ), −Im(py )] are its Ξ- and Ω-images in one modeling 3-space. (b) The intersections of a circle c : x2 + y 2 = 1 traced by the lines n : x cos ϕ + y sin ϕ = k for k ∈ IR and ϕ = 0 (red), π6 (yellow), π2 (cyan). Points P and P are complex conjugate intersections on the line for ϕ = π6 .

Let us return to the circle – line problem. Suppose we have a real plane IR2 with the coordinate system (x, y), the circle c : x2 + y 2 = 1, and trace it with the line l : x = k,

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for k ∈ IR parallel to y-axis. For k ∈ (−∞,√−1) ∪ (1, ∞), the roots of the corresponding quadratic√equation in y are ±i k 2 − 1. The intersecting points √ [k; i k 2 − 1] and [k; −i k 2 − 1] are on c and l, but not in the plane IR2 . Let us extend the real plane with the imaginary components such that the real coordinates x and y will be denoted Re(x) and Re(y) and the imaginary parts Im(x) and Im(y). A point P in the complex number plane has coordinates P [px , py ] in C2 and P [(Re(px ), Im(px )); (Re(py ), Im(py ))] in IR4 , for px = (Re(px ), Im(px )) and py = (Re(py ), Im(py )). Consequently, we identify the complex number plane C2 with IR4 with the orthogonal system of axes (Re(x), Im(x), Re(y), Im(y)). For example, the above-mentioned com√ 2 − 1)] and k plex intersecting points of l and c have coordinates [(k, 0); (0, √ 4 2 [(k, 0); (0, √ the purely real points for k ∈ −1, 1 are √ − k − 1)] in IR (while [(k, 0); ( 1 − k 2 , 0)] and [(k, 0); (− 1 − k 2 , 0)]). For visualization, we use the 4D-DOP method, which is a generalization of Monge’s projection. Each point P [(Re(px ), Im(px )); (Re(py ), Im(py ))] is orthogonally projected into the reference 3-spaces Ξ(Re(x), Im(x), Re(y)) and Ω(Re(x), Re(y), Im(y)) with the common plane π(Re(x), Re(y)) (see Fig. 2a). Both 3-spaces Ξ and Ω are represented in one modeling 3-space such that a perpendicular line to the plane π(Re(x), Re(y)) creates axes Im(x) and Im(y) with the opposite orientations (Im(x) upwards, Im(y) downwards). Let us have a closer look at the example above. Observe the locus of intersecting points of c and l, in 4D-DOP (Fig. 2b). For k ∈ −1, 1, Im(x) and Im(y)-coordinates are zero, the real parts are obviously related by the equation Re(x)2 + Re(y)2 = 1, representing the circle in the plane (Re(x), Re(y)). However, for k ∈ (−∞, −1) ∪ (1, ∞), equations Im(x) = 0; Re(y) = 0; Re(x)2 − Im(y)2 = 1 represent a hyperbola in the plane (Re(x), Im(y)) and hence also in the 3-space Ω(Re(x), Re(y), Im(y)). Both branches of this hyperbola are projected into two rays in the 3-space Ξ(Re(x), Re(y), Im(x)). Tracing the circle c with a line m : y = k, for k ∈ IR, parallel with the√x-axis (back in π(Re(x), Re(y))), we obtain the 2 points √ of intersection [(0, ± k − 1); (k, 0)] 2for k ∈ (−∞, −1) ∪ (1, ∞) and 2 [(± 1 − k , 0); (k, 0)] for k ∈ −1, 1 in C . Apart from the same circle in the plane π(Re(x), Re(y)), the complex points lie on a hyperbola in the plane (Im(x), Re(y)) and so in Ξ(Re(x), Im(x), Re(y)). The Ω-image of the hyperbola consists of two rays in (Re(x), Re(y), Im(y)). For a general case, assume a line n given by the following equation n : x cos ϕ + y sin ϕ = k for ϕ ∈ 0, 2π), k ∈ IR.

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Fig. 3. The surface of a real circle c generated by its complex points visualized in 4D-DOP. Views in special positions are on the right side. The circle is shifted in the directions Im(x) and Im(y) so that images in the 3-spaces (Re(x), Im(x), Re(y)) and (Re(x), Re(y), Im(y)) do not overlap in the ﬁgure.

Its intersection points with c are [k cos ϕ − (1 − k 2 ) sin2 ϕ; k sin ϕ + cot ϕ (1 − k 2 ) sin2 ϕ], [k cos ϕ + (1 − k 2 ) sin2 ϕ; k sin ϕ − cot ϕ (1 − k 2 ) sin2 ϕ]. For k ∈ −1, 1 the points must lie on the circle in π(Re(x), Re(y)). For k ∈ (−∞, −1) ∪ (1, ∞) the values of coordinates always contain an imaginary

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element, and they represent a hyperbola rotated along the circle and twisted in C2 . Orthogonal images of the surface generated in Fig. 3 are created by the extraction of the real and imaginary parts of the intersection points. At last, the image in 4D-perspective is in Fig. 4.

Fig. 4. The surface of a real circle c generated by its complex points visualized in 4D-perspective.

2.1

Further Issues

The method used in the previous section is theoretically applicable for any algebraic curve over IR. At ﬁrst, the curve is traced by all real lines in the real plane to obtain complex intersections. Next, we extract the real and imaginary parts of the complex points of intersection and plot the ﬁnal image embedded in IR4 . The surfaces corresponding to some other conics: a hyperbola, parabola, imaginary regular conic; and a cubic are depicted in Figs. 5a–5d. However, raising the order of the curve, the computational complexity (equation solving, plotting) increases rapidly.

3

Lines in CIP2

We have been constructing complex points of real curves and lines until now. On top of that, we can construct any point with coordinates in C2 . In this section, we will move a little further and explore the construction of an arbitrary line in C2 . Since our visualizations are created in the four-dimensional real space, we should be aware that images of lines in C2 will behave diﬀerently from the real

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Fig. 5. Surfaces of curves generated by their complex points visualized in 4D-DOP. All the surfaces are shifted in Im(x) and Im(y) directions so that they do not overlap.

lines. For example, one linear equation represents a hyperspace in IRn . While this holds well for lines in IR2 , one equation in IR4 represents a 3-space. Therefore,

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each line in C2 will generates a 3-space in IR4 . Furthermore, the lines Re(x) = 0 and Im(x) = 0 are equivalent in C2 , due to multiplication by a constant i, but they seem distinct in IR4 . To avoid such confusion, we approach lines through the projective extension CIP2 . A point P in CIP2 has homogeneous coordinates P (p1 ; p2 ; p0 ) = (0; 0; 0) for p1 ; p2 ; p0 ∈ C such that (p1 ; p2 ; p0 ) ∼ (λp1 ; λp2 ; λp0 ) for λ ∈ C \ {0}. Expanding real and imaginary parts of the point P , the coordinates will be in the form P ((Re(p1 ), Im(p1 )); (Re(p2 ), Im(p2 )); (Re(p0 ), Im(p0 ))). For the sake of visual representation, we always factorize the coordinates by the last nonzero coordinate. Therefore, proper points in C2 will be represented by points with coordinates ((Re(px ), Im(px )); (Re(py ), Im(py )); (1, 0)) and directions or improper points as ((Re(px ), Im(px )); (1, 0); (0, 0)) or ((1, 0); (0, 0); (0, 0)). Conveniently using the duality in projective spaces, the same holds for the coordinates of lines. Let ((Re(lx ), Im(lx )); (Re(ly ), Im(ly )); (1, 0)) be (factorized) coordinates of a line l, then its equation in the expanded form in IR4 is Re(lx )Re(x) + Im(lx )Im(x) + Re(ly )Re(y) + Im(ly )Im(y) + 1 = 0. Similarly for lines with coordinates ((Re(lx ), Im(lx )); (1, 0); (0, 0)) Re(lx )Re(x) + Im(lx )Im(x) + Re(y) = 0 or for ((1, 0); (0, 0); (0, 0)) Re(x) = 0. Such equations represent 3-spaces in IR4 . To visualize 3-space in orthogonal projection, we construct its traces, i.e., intersecting planes with the 3-spaces Ξ(Re(x), Im(x), Re(y)) and Ω(Re(x), Re(y), Im(y)). Substituting Im(y) = 0 and Im(x) = 0 into the equation of the line, we obtain the respective Ξ- and Ωtraces (see also [11] for synthetic constructions of traces of 3-spaces). As a consequence, the real part of the line is its intersection with the plane π(Re(x), Re(y)) obtained by vanishing the terms with Im(x) and Im(y). Let us examine the visual representations of lines with several examples in Fig. 6. 1. A line l with coordinates ((1, 0); (−1, 0); (1, 0)) and the equation Re(x) − Re(y) + 1 = 0 is depicted in Fig. 6a. Observe, that the intersection of l with the plane π(Re(x), Re(y)) does not change the equation. Furthermore, it is arbitrary in Im(x) and Im(y). The extension of the line in the directions Im(x) in the 3-space Ξ(Re(x), Im(x), Re(y)) and in Im(y) in Ω(Re(x), Re(y), Im(y)) generates the trace planes of the 3-space of l. Therefore, the trace planes are perpendicular to π in the modeling 3-space. Additionally, we should remind the reader that, due to equivalence, the same representation will have all lines multiplied by a nonzero complex scalar, e.g.: Im(x) − Im(y) + i = 0 ∼ Re(x) − Im(x) − Re(y) + Im(y) + 1 − i = 0 . . . .

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Fig. 6. Lines in CIP2 represented as 3-spaces in IR4 in 4D-DOP. The 3-spaces are given by their intersections with reference 3-spaces Ξ(Re(x), Im(x), Re(y)) (red) and Ω(Re(x), Re(y), Im(y)) (blue).

2. See Fig. 6b for l((1, −1); (0, 0); (1, 0)) with the equation Re(x) − Im(x) + 1 = 0. Apparently, the line Re(x) + 1 = 0 is the intersection with π. The Ξ-image could be reconstructed from the image in the plane (Re(x), Im(x)), and the Ω-image is, again, perpendicular to π in the modeling 3-space.

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3. See Fig. 6c for l((1, 0); (0, −1); (1, 0)) with the equation Re(x) − Im(y) + 1 = 0. The situation is similar to the previous case. Now, the Ξ-image is perpendicular to π. 4. See Fig. 6d for l((1, −1); (1, −1); (1, 0)) with the equation Re(x) − Im(x) + Re(y) − Im(y) + 1 = 0. In this case, none of the trace planes are perpendicular to π. The traces could be generated separately by setting the imaginary components to 0 in the respective 3-spaces. 3.1

Joins and Intersections

In complex homogeneous coordinates in CIP2 , a point P (p1 ; p2 ; p0 ) lies on a line l(l1 ; l2 ; l0 ) if p1 l1 + p2 l2 + p0 l0 = 0. Using the dot product P · l = 0. Another point Q lies on l if

Q · l = 0.

Hence l = P × Q. Dually, a point P is the intersection of distinct lines p and q, only if P =p×q (see [7], Chap. 3 for details). Graphical representation in IR4 of lines and points in C2 will work slightly diﬀerently, too. This is because the multiplication of imaginary components changes sign. For example, the dot product of a point P (p1 ; p2 ; p0 ) and a line l(l1 ; l2 ; l0 ) ∈ CIP2 is p 1 l1 + p 2 l2 + p 0 l0 . However, after the expansion into real and imaginary components, we have P ((Re(p1 ), Im(p1 )); (Re(p2 ), Im(p2 )); (Re(p0 ), Im(p0 )))· l((Re(l1 ), Im(l1 )); (Re(l2 ), Im(l2 )); (Re(l0 ), Im(l0 ))) = Re(p1 )Re(l1 ) − Im(p1 )Im(l1 ) + Re(p2 )Re(l2 ) − Im(p2 )Im(l2 )+ Re(p0 )Re(l0 ) − Im(p0 )Im(l0 ).

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Therefore, in the visualizations in IR4 the point P will not lie in the 3-space representing line l. On the other hand, the complex conjugate P¯ ((Re(p1 ), −Im(p1 )); (Re(p2 ), −Im(p2 )); (Re(p0 ), −Im(p0 )) of the point P lies on the 3-space of the line l through P . And oppositely, the point P lies in the 3-space representing the complex conjugate ¯l of the line l in IR4 . In Fig. 7 the line l has coordinates l = P × Q, ¯ of Q lie on the 3-space of the but the complex conjugate points P¯ of P and Q line l. This is also veriﬁed in the ﬁgure by the construction of the plane in the 3-space of l through P parallel to Ω(Re(x), Re(y), Im(y)).

Fig. 7. A line l in CIP2 passing through points P, Q represented as a 3-space in IR4 . ¯ of P, Q lie on the 3-space, which is given by its planar The complex conjugates P¯ , Q intersections with Ξ(Re(x), Im(x), Re(y)) and Ω(Re(x), Re(y), Im(y)) in IR4 using the 4D-DOP method.

4

Conclusion

We have revisited the method of ﬁnding complex points on a circle by tracing the circle with a line. Intersecting points generate a surface in the 4-dimensional space (Re(x), Im(x), Re(y), Im(y)). The ﬁnal visualization of the images was

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plotted in a double orthogonal projection into 3-spaces (Re(x), Im(x), Re(y)) and (Re(x), Re(y), Im(y)) and in four-dimensional perspective projection. The method was applied to visualize complex points of other conics and a cubic curve. Moreover, it can be used for many other real curves; however, it is very limited by computational complexity. A further possibility of application is, for instance, in ﬁnding graphical solutions of complex intersections of real curves. Furthermore, through a projective extension, we have described how to visualize a complex straight line as a three-dimensional subspace of a fourdimensional real space. We have also discussed how to verify the incidence of a point and a line and how to visualize the join of two points. These concepts are easily extendable and applicable for further research in visualizing a complex number plane identiﬁed with a four-dimensional real space.

References 1. Banchoﬀ, T.F.: Complex Function Graphs. http://www.tombanchoﬀ.com/ complex-function-graphs.html. Accessed 20 Feb 2020 2. Bozlee, S., Amethyst, S.V.: Visualizing complex points of elliptic curves. https:// im.icerm.brown.edu/portfolio/visualizing-complex-points-of-elliptic-curves/. Accessed 20 Feb 2020 3. Butler, D.: Where the complex points are. https://blogs.adelaide.edu.au/mathslearning/2016/08/05/where-the-complex-points-are. Accessed 20 Feb 2020 4. Hatton, J.L.S.: The Theory of the Imaginary in Geometry: Together with the Trigonometry of the Imaginary. Cambridge University Press (1920). https://doi. org/10.1017/CBO9780511708541 5. Poncelet, J.V.: Trait´e des propri´et´es projectives des ﬁgures: ouvrage utile ` a ceux qui s’ occupent des applications de la g´eom´etrie descriptive et d’op´erations g´eom´etriques sur le terrain, vol. 1, 2nd edn. Gauthier-Villars, Paris (1865) ˇ 6. Rada, J., Zamboj, M.: 3-Sphere in a 4-Perspective. In: Jeli, Z. (ed.) Proceedings moNGeometrija 2020, Belgrade, Serbia. Planeta Print, Belgrade. pp. 52–61 (2021) 7. Richter-Gebert, J.: Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry. Springer, Heidelberg (2011). https://doi.org/10. 1007/978-3-642-17286-1 8. Scriba, C.J., Schreiber, P.: 5000 Years of Geometry. Springer, Basel (2015). https:// doi.org/10.1007/978-3-0348-0898-9 9. Avitzur, R.: Visualizing Functions of a Complex Variable. http://www.nucalc.com/ ComplexFunctions.html. Accessed 20 Feb 2020 10. Avitzur, R.: Paciﬁc tech graphing calculator for iOS. https://apps.apple.com/us/ app/paciﬁc-tech-graphing-calculator/id1135478998?ls=1. Accessed 20 Feb 2020 11. Zamboj, M.: Double orthogonal projection of four-dimensional objects onto two perpendicular three-dimensional spaces. Nexus Network J. 20(1), 267–281 (2018). https://doi.org/10.1007/s00004-017-0368-2 12. Zamboj, M.: Visualizing objects of four-dimensional space: from ﬂatland to the hopf ﬁbration (invited talk). In: Szarkov´ a, D., Richt´ arikov´ a, D., Pr´ aˇsilov´ a, M. (eds.). Proceedings of the 19th Conference on Applied Mathematics Aplimat 2020. Bratislava, Slovakia, pp. 1140–1164. Slovak University of Technology, Bratislava (2020)

Regularity Conditions for Voronoi Diagrams in Hyperbolic Space Alˇzbeta Mackovov´ a(B)

and Pavel Chalmoviansk´ y

Comenius University in Bratislava, Bratislava, Slovakia {alzbeta.mackovova,pavel.chalmoviansky}@fmph.uniba.sk

Abstract. Voronoi diagrams belong to frequently used structures in computational geometry with application in many ﬁelds of science. The properties of Voronoi diagram are already studied in various metric spaces – Euclidean, Manhattan, Minkowski, Hausdorﬀ, or Karlsruhe and also in the hyperbolic metric. In this paper, we focus on the Voronoi diagram and its dual in the Poincar´e ball model of the three-dimensional hyperbolic space. We ﬁrst present some basic tools from the Poincar´e ball model needed to construct a Voronoi diagram and for a closer observation of its properties. We have determined certain conditions for the position of the generators controlling the behavior of the hyperbolic Voronoi diagram. In the last section, we demonstrate this eﬀect on the dual graph of a hyperbolic Voronoi diagram, i. e., on hyperbolic Delaunay tessellations. Keywords: Poincar´e ball model diagram · Delaunay tessellation

1

· hyperbolic space · Voronoi

Introduction

The ﬁrst known mention of Voronoi diagram was in the study of quadratic forms by Peter G. L. Dirichlet in 1850. However, Voronoi diagrams are named after Ukrainian mathematician Georgy F. Voronoi, the author of a paper about Voronoi diagram in general m-dimensional case [12]. Naturally, Voronoi diagrams were ﬁrst studied in Euclidean geometry for point sets. But since then, there are results about Voronoi diagrams in other metric spaces, in diﬀerent dimensions and for general types of generators. Applications of Voronoi diagrams are known in several ﬁelds of science. For detailed study of Voronoi diagrams, Delaunay triangulations, algorithms for their construction and applications we recommend to look into the book by Okabe, Boots, Sugihara, and Chiu [10] or the book by Aurenhammer, Klein, and Lee [2]. We are interested in Voronoi diagrams in three-dimensional hyperbolic space, which may be represented in several isometrically equivalent models, each useful in its context. The most commonly used models are: the hyperboloid model, the hemispherical model, the Poincar´e ball model, the Beltrami-Klein model, or the Poincar´e upper half-space model. Each of them has its own metric, geodesics, c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 150–162, 2023. https://doi.org/10.1007/978-3-031-13588-0_13

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and so on. The deﬁnition, metrics of each model, and the isometries between them are summarized in [4]. There are some results already known in the study of Voronoi diagrams and Delaunay triangulation in hyperbolic geometry. In [8], the authors give an incremental algorithm to construct Voronoi diagrams in the hyperbolic plane represented by Poincar´e disk model. An analogous approach is elaborated in [9] for Voronoi diagrams in three-dimensional hyperbolic space represented by Poincar´e upper half-space model. In [6], Nielsen and Nock have shown that the hyperbolic Voronoi diagram in the Klein projective disk model is merely an aﬃne diagram that can be conveniently computed as an equivalent power diagram. They also illustrated how these techniques come in handy for designing user interfaces of an interactive image browser application. In [7] are illustrated some cases in diﬀerent models of the hyperbolic plane, when the dual Delaunay complex is not a triangulation. In this paper, we work with the Poincar´e ball model. We specify our results on the Poincar´e disk model and we give a simple reason why the dual graph to a particular hyperbolic Voronoi diagram is not a triangulation in general. For the sake of simplicity, we replace the adjective hyperbolic with h-. If we use both Euclidean and hyperbolic terms in the text, we use the preﬁx h-. If no mistake can be made, we can omit this preﬁx. To understand the use of the spherical inversion as an auxiliary tool for the construction of elements of the Poincar´e ball, resp. disk, model such as the midpoint or the bisector of the given h-line segment etc., we refer to [1] or [11].

2

Poincar´ e Ball Model

This section deals with basic concepts of the Poincar´e ball model which we use for the representation of the three-dimensional hyperbolic space. The Poincar´e ball model is deﬁned as B3 = {(x, y, z) ∈ R3 : x2 + y 2 + z 2 < 1} with the particular hyperbolic metric ds2 =

4(dx2 + dy 2 + dz 2 ) . (1 − x2 − y 2 − z 2 )2

The model is represented with an open ball of unit radius. The boundary S2 = {(x, y, z) ∈ R3 : x2 + y 2 + z 2 = 1} of the model is called an absolute sphere and its points are called ideal points. Clearly, the ideal points do not belong to the hyperbolic space and they somehow play the role of the points at inﬁnity. Let us denote the center of the absolute sphere by S. Let us start with deﬁnition of planar elements in the Poincar´e ball model. By considering any Euclidean plane ρ : ax + by + cz = 0 incident with the center S of the absolute sphere we get as the result of the intersection of the plane ρ with the open unit ball an open unit disk. So locally, we get a Poincar´e disk

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model, the boundary of which is called an absolute circle. The hyperbolic lines are represented by arcs of Euclidean circles in the plane ρ that are perpendicular to the corresponding absolute circle (1) or as an open unit line segment incident with the center S of the unit ball (2). A hyperbolic line may be expressed for some non-zero and linearly independent (a, b, c) and (k, l, m) as ax + by + cz = 0 x + y + z − 2kx − 2ly − 2mz + 1 = 0, where k 2 + l2 + m2 > 1 or 2

2

2

ax + by + cz = 0 kx + ly + mz = 0

(1)

(2)

such that (k, l, m) gives a sphere perpendicular to the absolute sphere in (1). Obviously, there is a unique hyperbolic line passing through two given points A, B ∈ B3 . Two lines in hyperbolic space either lie in one hyperbolic plane (then they might be intersecting, parallel, or ultraparallel) or do not lie in one hyperbolic plane (then these lines are skew). Let us stay in the plane ρ. We classify the basic planar curves of hyperbolic geometry which are represented by Euclidean circles with a certain mutual position to the absolute sphere. Let k be a Euclidean circle in ρ. Then the circle k represents in hyperbolic geometry – an h-circle, if the circle k is inside the corresponding absolute circle; but its Euclidean and hyperbolic centers are, in general, diﬀerent; – a horocycle, if the circle k is tangent from inside to the corresponding absolute circle at one ideal point; – an h-line, if the circle k is intersecting the corresponding absolute circle and it is perpendicular to it; – a hypercycle, if the circle k is intersecting the corresponding absolute circle and it is not perpendicular to it (Fig. 1). Other cases of mutual position of the circle k and the absolute circle are not required in consideration, since they do not have points inside B3 . The hyperbolic plane is represented by a part of the sphere that is perpendicular to the absolute sphere (3) or as the open unit disk incident with the center S of the absolute sphere (4). A hyperbolic plane may be expressed for some non-zero (a, b, c) and (k, l, m) as x2 + y 2 + z 2 − 2kx − 2ly − 2mz + 1 = 0, where k 2 + l2 + m2 > 1 or

(3)

ax + by + cz = 0.

(4)

Two hyperbolic planes can be intersecting, the intersection is an hyperbolic line; parallel, their common point is one ideal point; ultraparallel, the intersection is neither inside nor on the absolute sphere.

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Now, we classify the basic surfaces of hyperbolic three-dimensional space which are represented by Euclidean spheres with a certain mutual positions to the absolute sphere. Let S be an Euclidean sphere. Then the sphere S represents – an h-sphere, if the sphere S is inside the absolute sphere; but its Euclidean and its hyperbolic centers are, in general, diﬀerent; – a horosphere, if the sphere S is tangent to the absolute sphere at one ideal point from inside and its h-center equals this ideal point; – an h-plane, if the sphere S intersect the absolute sphere and it is perpendicular to it; – a hypersphere, if the sphere S intersect the absolute sphere and it is not perpendicular to it; – other possible mutual positions do not need to be considered (Fig. 2).

Fig. 1. Hyperbolic curves in plane ρ represented by Euclidean circle: h-circle (blue), horocycle (yellow with the red ideal point), h-line (green), hypercycle (pink).

Fig. 2. Hyperbolic surfaces represented by Euclidean sphere: h-sphere (blue), horosphere (yellow with the red ideal point), h-plane (green), hypersphere (pink).

Further, we will need the notion of the perpendicular bisector plane to construct an h-Voronoi diagram. We could provide an algorithm for the construction

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of perpendicular bisectors between given pairs of points analogous to the algorithm given in [9]. But an easier way for us is to use the properties of the spherical inversion. Algorithm 1: Hyperbolic perpendicular bisector between two given points Input: Two points A = (xA , yA , zA ) and B = (xB , yB , zB ) in the Poincar´e ball model. Output: The hyperbolic perpendicular bisector plane between A and B. 1. Step: The formulae for the coordinates (x∗ , y ∗ , z ∗ ) of the image for the given point with the coordinates (x, y, z) under the inversion λ by the sphere with the center S = (0, 0, 0) and the radius r = 1 are x∗ =

x y z , y∗ = 2 , z∗ = 2 . x2 + y 2 + z 2 x + y2 + z2 x + y2 + z2

Let us denote the images A∗ , B ∗ of points A and B in inversion λ. Now, we construct the Euclidean circle c incident with the points A, B, A∗ , B ∗ . The part of the circle c inside the absolute sphere represents the h-line AB. 2. Step: We are looking for the h-midpoint M of the h-line segment with endpoints A and B. Let the point E be the intersection of the lines AB ∗ and BA∗ . Then, h-midpoint M is the intersection of the line SE with the circle c, that lies inside the absolute sphere. 3. Step: The perpendicular bisector plane μ of the h-line segment AB is represented by the part of the sphere perpendicular to the absolute sphere with the center O and the radius r = |OM |. It is true, that the sphere inversion by the sphere μ maps A to B and vice versa. The same is true for the points A∗ and B ∗ . Therefore, the center O is the intersection of the lines AB and A∗ B ∗ .

3

Hyperbolic Voronoi Diagram

Definition 1. If a finite set P = {p1 , . . . , pn } ∈ B3 of distinct points is given, the region given by V (pi ) = {x ∈ B3 : d(pi , x) ≤ d(pj , x), for i = j; i, j ∈ In = {1, . . . , n}}

(5)

is called the hyperbolic Voronoi polyhedron associated with pi , and the set given by V = {V (p1 ), . . . , V (pn )} is the hyperbolic Voronoi diagram generated by P . We call pi the generator of the i-th hyperbolic Voronoi polyhedron V (pi ) and the set P is the generator set of the hyperbolic Voronoi diagram V.

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For the two generators pi , pj , the set of points satisfying the equality in the condition (5) form a hyperbolic bisector plane which one can construct according to the steps of algorithm 1. Hence, the boundaries of h-Voronoi polyhedrons are formed by some parts of h-planes and they are called h-Voronoi facets. H-Voronoi facets are intersecting in h-Voronoi edges represented by h-lines or parts of them. Each h-Voronoi edge is an intersection of three or more h-Voronoi facets. The endpoints of h-Voronoi edges are called h-Voronoi vertices and each vertex is a common point of four or more h-Voronoi polyhedrons. Two h-Voronoi polyhedrons V (pi ), V (pj ) are called adjacent if they share a face. Most of the properties of Euclidean Voronoi diagrams formulated in [10] might be analogously formulated also in hyperbolic geometry. For our next observations, we have to mention the following properties of h-Voronoi vertices and edges. Property 1. For every h-Voronoi vertex qi ∈ Q, where Q is the set of all vertices of an h-Voronoi diagram, there is a unique h-sphere Gi centered at qi which passes through the generators pi1 , . . . , pik ∈ P , for (k ≥ 4), and contains no other generator in its interior. Such an h-Voronoi vertex is common to the h-Voronoi polyhedrons V (pi1 ), . . . , V (pik ). Property 2. For every h-Voronoi edge ei ∈ E, where E is the set of all edges of hVoronoi diagram, there is a unique h-circle ci lying in an h-plane perpendicular to the h-Voronoi edge ei with the h-center incident with the edge ei passing through the generators pi1 , . . . , pik ∈ P , for (k ≥ 3), and containing no other generator in its interior. Such an h-Voronoi edge is common to the h-Voronoi polyhedrons V (pi1 ), . . . , V (pik ). We call the h-Voronoi diagram degenerate if – there is an h-Voronoi edge common to four and more h-Voronoi polyhedrons or – there is an h-Voronoi vertex common to ﬁve and more h-Voronoi polyhedrons. Otherwise, we call the h-Voronoi diagram non-degenerate. Non-degenerate and degenerate h-Voronoi diagrams are illustrated in the Figs. 3 and 4, respectively. Generators are arbitrary ﬁnite sets in B3 . In order to test these situations when the h-Voronoi diagram is degenerate, we make the following conditions: Condition 1. The 3D non-degeneracy hyperbolic conditions: 1. Every h-Voronoi edge in an h-Voronoi diagram has exactly three h-Voronoi facets. 2. Every h-Voronoi vertex in an h-Voronoi diagram has exactly four h-Voronoi edges. If the condition 1 holds, then the h-sphere Gi with the property 1 passes through exactly four generators and the h-circle ci with the property 2 passes though exactly three generators. Since in the h-Voronoi diagram in Fig. 4 there

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exists an h-sphere passing through six generators and the h-Voronoi diagram is degenerate. The case, when all generators of an h-Voronoi diagram are collinear is trivial. Furthermore, if all generators of an h-Voronoi diagram are coplanar, then all hVoronoi polyhedrons are unbounded. Such an h-Voronoi diagram has no vertex. To avoid these trivial cases, we have to add the following conditions on the positions of the generators in hyperbolic space.

Fig. 3. (a) The position of the generators. (b)–(f) Non-degenerate hyperbolic Voronoi diagram of nine generators from ﬁve diﬀerent points of view.

Condition 2. The 3D Non-collinearity and Non-coplanarity Condition: For the given set P = {p1 , . . . pn } of generators of the h-Voronoi diagram we assume, that the points in P are not on the same h-line or in the same h-plane. Now, we introduce the main ideas needed for further observations. Let us consider any three generators of an h-Voronoi diagram and an h-triangle given by these three points. There is a unique Euclidean circle incident with these generators and an unique Euclidean sphere which has this circle as an equator. If this sphere is empty, we focus on which hyperbolic surface this Euclidean sphere represents. It aﬀects the intersection of h-bisector planes between the vertices of the h-triangle. This intersection is a potential h-Voronoi edge. If it represents a) an h-sphere, then the intersection is an h-line; b) a horosphere, then the intersection is an ideal point; c) an h-plane, then the intersection is outside the absolute sphere; d) a hypersphere, then the intersection is also outside the absolute sphere. Of course, it may occur that some other generators are incident with this circle, but the consequences are the same.

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Now, we consider four arbitrarily chosen non-coplanar points (the case, when these generators are coplanar, is included in the previous paragraph) of an h-Voronoi diagram and the h-tetrahedron given by these four points. If the Euclidean sphere incident with the vertices of the h-tetrahedron is empty, then we are interested in which hyperbolic surface it represents. Depending on that, the point of intersection of h-bisector planes between the vertices of the h-tetrahedron might be on or outside the absolute sphere, i. e., it does not belong to the hyperbolic space for the same reasons as in previous paragraph. This point of intersection is a potential h-Voronoi vertex.

Fig. 4. (a) The position of the generators. (b)–(f) Degenerate hyperbolic Voronoi diagram from ﬁve diﬀerent points of view. There is an Voronoi vertex (purple) common to six hyperbolic Voronoi polyhedrons.

To detect these situations we test whether the following conditions are true or not for the set of generators of hyperbolic Voronoi diagram. We call the h-Voronoi diagram generic if and only if all of the conditions 1-6 hold. Otherwise, we call the h-Voronoi diagram non-generic. Condition 3. The hyperbolic non-coplanarity condition: For a given set of points P = {p1 , . . . , pn } ∈ B3 , (3 ≤ n < ∞), shall be no h-plane μ such that the points pi1 , . . . , pik ∈ P , (k ≥ 3) lie on the h-plane μ and all the other points of P \ {pi1 , . . . , pik } are in the same half-space given by the h-plane μ. Condition 4. The Horosphere Condition: For a given set of points P = {p1 , . . . , pn } ∈ B3 , (3 ≤ n < ∞), shall be no horosphere α such that the points

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pi1 , . . . , pik ∈ P , (k ≥ 3) lie on the horosphere α and all the other points of P \ {pi1 , . . . , pik } are the external points of the horosphere α. Condition 5. The Hypersphere Condition: For a given set of points P = {p1 , . . . , pn } ∈ B3 , (3 ≤ n < ∞), shall be no hypersphere β such that the points pi1 , . . . , pik ∈ P , (k ≥ 3) lie on the hypersphere α and all the other points of P \ {pi1 , . . . , pik } are the external points of the hypersphere β (Fig. 5). Condition 6. The Hyperbolic Sphere Condition: For a given set of points P = {p1 , . . . , pn } ∈ B3 , (3 ≤ n < ∞), shall be no h-sphere δ such that the points pi1 , . . . , pik ∈ P , (k > 4) lie on the h-sphere δ and all the other points of P \ {pi1 , . . . , pik } are the external points of the h-sphere δ.

Fig. 5. Hyperbolic Voronoi diagram for 16 generators from nine diﬀerent points of view. This hyperbolic Voronoi diagram does not meet the hypersphere condition – seven generators lie on the purple hypersphere.

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The validity of the conditions 3–5 ensure that corresponding h-Voronoi facets meet inside the B3 . The condition 6 follows from the non-degeneracy condition 1 and from the properties 1 and 2. Otherwise there are k facets meeting at this vertex, resp. edge so this h-Voronoi diagram is degenerate.

4

2D Hyperbolic Voronoi Diagram in the Poincar´ e Disk Model

We dealt with the three-dimensional case before the two-dimensional one, because we consider it easier to reduce the dimension than to increase it. In this section, we shortly illustrate how the results from the hyperbolic space look like in the hyperbolic plane represented in the Poincar´e disk model D2 . By reducing the dimension by one, we make use of the hyperbolic non-collinearity, horocycle, hypercycle condition for k ≥ 3 and hyperbolic circle condition for k ≥ 4.

m a

D2

c

b

u c

a b

d

d

e

e

D2

f

f g

g

h

(a) generators a, b, c on an h-line m D v

a

c d g

(b) generators a, b, c on a horocycle u

2

b e

h

a f

b g

l

e d c

D2

h f h

(c) generators a, b, c on a hypercycle v

(d) generators a, b, c, d, e on an h-circle l

Fig. 6. Conditions for the degeneracy of a hyperbolic Voronoi diagram.

In Fig. 6, we illustrate the conditions 3–6 for the given h-Voronoi diagrams generated by the set P = {a, b, c, d, e, f, g, h} of distinct points. In a 2-dimensional space, we have a better opportunity to illustrate what causes the validity or invalidity of the previous conditions. We take a closer look at the concept of a hyperbolic Delaunay triangulation as a dual tessellation of the given hyperbolic Voronoi diagram which is constructed in hyperbolic plane

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in an analogous way as in Euclidean geometry – by joining the generators of adjacent h-Voronoi polygons with an h-line segment for every edge in the given h-Voronoi diagram. If the resulting tessellation consists of triangles only, we call this tessellation a hyperbolic Delaunay triangulation (Fig. 7a). However, in the general case, we cannot call the result of this construction a triangulation without considering any of the previous condition, which have been formulated in the previous section. Firstly, for the same reason as in Euclidean geometry – this tessellation can also contain a polygon which should be triangulated. In this case, we call the resulting tessellation a Delaunay pretriangulation (Fig. 7b). Secondly, the resulting tessellation may also be a tessellation as in Fig. 7c which obviously is not a triangulation and cannot be triangulated. Moreover, it does not have to be true, that an h-Voronoi diagram and its dual graph have to have the same number of facets.

(a) triangulation

(b) pretriangulation

(c) tessellation

Fig. 7. Hyperbolic Delaunay tessellation as dual graph to the given hyperbolic Voronoi diagram.

This means that, if we want to get an h-Delaunay triangulation as a result of the usual construction of a dual graph, the h-line condition, the hypercycle condition, and the horocycle condition must be valid. Knowing this, allows us to approach triangulations in a diﬀerent way than in [3], where they described in more detail the algorithms from their previous work [5] in the space of spheres. They compute the h-Delaunay triangulation of the given set P in two steps: ﬁrstly, they compute the Euclidean Delaunay triangulation and then, secondly, they extract the polygons from the Euclidean Delaunay triangulation that also belong to the h-Delaunay tessellation. We plan to show in our next work which approach to this issue is more eﬀective depending on the calculations. If any of the conditions is not met, there is no reason to construct an h-Delaunay triangulation, if we do not know in advance whether it is a triangulation or not. If all these conditions are met, the construction and calculation of the h-Delaunay triangulation is comparable with the calculation of Euclidean triangulation in computational sense.

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Conclusion

In this paper, we introduced several conditions for the position of the generators to help us to describe the behavior of the Voronoi diagram in hyperbolic threedimensional space represented by the Poincar´e ball model. After reducing the dimension by one, we also illustrated in the Poincar´e disk model hyperbolic Voronoi diagrams that do not satisfy these conditions. We further demonstrated the eﬀect of invalidity of these conditions on the dual graph of Voronoi hyperbolic diagram. Now, when we know the behavior of Voronoi diagrams in hyperbolic geometry, we would like to dynamically illustrate these situations when some generator moves along an h-line segment, an h-circle or another simple curve in the Poincar´e disk, resp. ball, model. Dynamic illustrations of Voronoi diagrams in three-dimensional hyperbolic space from this paper are available as animations here: https://www.youtube. com/playlist?list=PLHGj3kTMk3z1Jn-0h3IE5sAYxvH48S TS. Acknowledgements. This work was supported by the Slovak Research and Development Agency under the contract No. APVV-16-0053 and by the grant VEGA 1/0596/21. We also thank anonymous reviewers for their comments leading to the improvement of the ﬁnal version of this article.

References 1. Anderson, J.: Hyperbolic Geometry. Springer Undergraduate Mathematics Series, Springer, London (2005) 2. Aurenhammer, F., Klein, R., Lee, D.S.: Voronoi diagrams and Delaunay triangulations. World Scientiﬁc Publishing Co., Singapore (2013) 3. Bogdanov, M., Devillers, O., Teillaud, M.: Hyperbolic delaunay complexes and voronoi diagrams made practical. J. Comput. Geometry Carleton Univ. Comput. Geometry Laboratory 5(1), 56–85 (2014) 4. Cannon, J.W., Floyd, W.J., Kenyon, R., Parry, W.R.: Hyperbolic geometry. Flavors Geometry 31, 59–115 (1997) 5. Devillers, O., Meiser, S., Teillaud, M.: The space of spheres, a geometric toll to unify duality results on Voronoi diagrams. Research Report RR-1620, INRIA (1992) 6. Nielsen, F., Nock, R.: Hyperbolic Voronoi diagrams made easy. In: CoRR abs/0903.3287 (2009) 7. Nielsen, F., Nock, R.: The hyperbolic Voronoi diagram in arbitrary dimension. In: CoRR abs/1210.8234 (2012) 8. Nilforoushan, Z., Mohades, A.: Hyperbolic voronoi diagram. In: Gavrilova, M.L., Gervasi, O., Kumar, V., Tan, C.J.K., Taniar, D., Lagan´ a, A., Mun, Y., Choo, H. (eds.) ICCSA 2006. LNCS, vol. 3984, pp. 735–742. Springer, Heidelberg (2006). https://doi.org/10.1007/11751649 81 9. Nilforoushan, Z., Mohades, A., Rezaii, M.M., Laleh, A.: 3D hyperbolic Voronoi diagrams. Comput. Aided Des. 42, 759–767 (2010)

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10. Okabe, A., Boots, B. Sugihara, K., Nok Chiu, S.: Spatial tessellations: Concepts and applications of Voronoi diagrams. John Wiley and Sons, Hoboken, New Jersey, USA (2010) 11. Foundations of Hyperbolic Manifolds. GTM, vol. 149. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-31597-9 9 12. Voronoi, G.: Nouvelles applications des param`etres continus ` a la th´eorie des formes quadratiques. J. f¨ ur Reine Angew. Math. 133, 97–178 (1908)

Poncelet and the (Arquimedean) Twins Liliana Gabriela Gheorghe(B) Federal Univ. of Pernambuco, Recife, Brazil [email protected]

Abstract. A new and sharp construction for twins in arbelos, foreseen as solutions of two Apollonius problem is given. To perform it, we use polar reciprocity, the method forged by Poncelet to prove his Porism. In the process, other circles, displaying archimedean aﬃnities, came into scene.

Keywords: Arbelos Poncelet

· Apollonius’s problem · Poles · Polar duals ·

2020 MSC: 51A05, 51A30, 51M15

1

Introduction

Arbelos is the Greek word for ‘shoemaker’s knife’, the shape obtained by halving three pairwise tangent circles with collinear centres, by their common diameter. As a geometric object, it was ﬁrst brought into scene by Archimedes in his Book of Lemmas, hence it dates back more than 2200 years. Archimedes proved that the two circles inscribed into the pieces of the arbelos cut oﬀ by the perpendicular to the baseline through the tangency point of the two inner arbelos’ circles are congruent; these are the “twins” (Fig. 1).

Fig. 1. Twins in arbelos as solutions of Apollonius’ problem; their diameters are con2R1 R2 . gruent, and equal to the harmonic mean of the arbelos’ i-circles: 2r = R 1 +R2 c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 163–174, 2023. https://doi.org/10.1007/978-3-031-13588-0_14

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Twins’ diameters are the harmonic mean of the radii of the two arbelos internal circles; as a tribute, any circle in arbelos, congruent with the twins, is called “archimedean”. An elementary construction of the twins, due to Bankoﬀ [2], uses the fact that the circle (G0 ), whose diameter is the parallel through M at the bases of the (rectangular) trapezium [O1 N1 N2 O2 ] in Fig. 2, I), is archimedean; two circles, (G1 ) and (G2 ) of radius ρ can be drawn touching the arbelos’ internal circles, as well as their common tangent; the latter prove to be tangent to the external arbelos’ circle, as well; thus, (G1 ) and (G2 ) are “the twins.”

Fig. 2. I) (G0 ) is an archimedean circle and (G1 ), (G2 ) are the twins. II) The tangents from O1 , O2 meets at I, the center of the inscribed arbelos’ circle (I).

A more natural path is to foresee the twins as solutions of two (distinct) degenerated Apollonius’ problems, ﬁnd their centres, and compute their radii. We follow this path and obtain the twins’ centres as an intersection of two special conics. Though two conics in general position cannot be (geometrically) intersected, these conics have a common focus and our naive idea works due to polar reciprocity; and this is where Poncelet came into scene. By polar duality, we convert the problem of intersecting two conics with a common focus, into the problem of drawing the pole of the common tangent to their dual curves, which are two circles. Main Results: Our main result is a simple geometric construction resumed in Theorem 1. We then attach to a classic arbelos another pair of mutually tangent circles, and study a tern of circles in this frame: one of these new circles is archimedean; this one is not new; yet we give a new proof in Theorem 4. The other two circles are new and verify an archimedean-type relation, proved in Theorem 5. Related Work: Chasing archimedean circles in arbelos is a recurrent theme; [8] and the references therein. The literature on arbelos is so rich and the references are so abundant, that we cite a small sample just to show that interest for this problem is still vivid. In [2] a famous construction for the i-circle in arbelos, much simpler than the one from Archimedes’ proof, due to the existence of a third archimedean circle is given; [14] also provides a simple construction of the

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i-circle in (classic) arbelos; a beautiful and elementary inversive treat is in [3]; in [4] the authors study inversions mapping that switch two given circles and apply it to arbelos; [10] studies a generalization of arbelos; [9] study twin circles in skewed arbelos. For an album of arbelos’ archimedean circles, [6] is a rich and handy source. Notations: We shall note by (O) a circle centered in O. By reﬂection in (O) we mean the symmetry (or inversion) with respect to circle (O). We freely interchange the terms “dual curve”, “polar dual” or “reciprocal curve”.

2

Twins’ Centers and the Apollonius’ Problem

The two circles that touch both the arbelos’ nested circles and their radical axis are called “twins”. Each twin is a solution of an Apollonius problem; their centres can be obtained by intersecting two conics. Let a circle (O2 ) and a tangent to it, at a point M be given, as in Fig. 3, right. Lemma 1. The locus of the centres of all circles, externally touching both to a circle and line (which are mutually tangent) is a parabola focused at circle’s center and whose vertex is the tangency point of the circle and the line, apart from said vertex. Let (O) and (O1 ) be two circles, that are internally tangent, as in Fig. 3, left.

Fig. 3. I) (left) The locus of the centres of the circles that touch two internally tangent circles (O) and (O1 ) is an ellipse (orange) which has the foci at the centres of the two circles and one vertex at their common tangency point. II) (right) The locus of the centres of the circles that touch tangents externally a circle (O2 ) and a line (that is tangent in M to the circle) is a parabola, which has the focus at O2 and vertex at M

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Lemma 2. The locus of the centres of the circles that touch two nested circles that are internally tangent at two distinct points is an ellipse focused at the centres of the circles and passing through their common tangency point, apart from said tangency point. Both Lemmas 1 and 2 have elementary proofs that we omit. When we specialize to arbelos, we get a sharper result. Refer to Fig. 3. Proposition 1. The center of each twin is the intersection between an ellipse and a parabola. Each ellipse has one vertex at the common tangency point of each internal arbelos circle with the external one, (points A1 and A2 respectively), has one focus in O and the other focus in O1 and O2 , respectively. The vertex of the parabolas is the common tangency point of the internal arbelos’ circles (O1 ), (O2 ) (point M ) and their focus is in O1 and O2 , respectively. At this point, any drawing software is able to perform the intersection of these two conics. Nevertheless, we are “not done”, since the task is to perform a geometric construction: a construction with a straight-line and a compass only. In a geometric construction, one cannot intercept “continuous curves”, other then circles and lines. In fact, while two arbitrary conics cannot be (geometrically) intersected, the conics that aroused here, have a special feature: a common focus. And here is where polar reciprocity comes into scene. The reader not acquainted with this topic, may see Appendix. Lemma 3. The intersection of two regular curves are the poles of the common tangents of their duals. In particular, the intersection of two conics which have a common focus, w.r. to an inversion circle centered at their (common) focus are the poles of the (common) tangents to their dual circles. For more details on poles, polars and polar reciprocity, see [1,5,12]. Now we may specify who these duals are. Refer to Fig. 4 and choose as the inversion circle (O1 ). Lemma 4. The polar duals (w.r. to (O1 )) of the ellipses E1 and the parabola P1 in Fig. 4, are two circles that: 1. are tangent to (O1 ) at A1 and M respectively; 2. the diameter of the parabola’s dual is [O1 M ] = R1 and the diameter of the 2R2 +R R ellipse’s dual is [A1 O2 ] = R11 +R12 2 where O2 is the reflection of O2 in (O1 ); 3. the similitude centres of these two circles is O2 . Proof. The proof uses known facts on the polar of a conic, that we collect in Appendix. Both the parabola and the ellipse in Fig. 4 have one common focus in O1 ; therefore, their duals w.r. to (O1 ) are circles. The dual of a parabola w.r. to an inversion circle centered at its focus is a circle, whose diameter is [O1 M ], where M is the reﬂection of the parabola’s vertex and O1 is the center of

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inversion. Since the vertex M is located on the inversion circle, its is invariant by reﬂection; thus, the dual of the parabola is a circle with diameter [O1 M ] = R1 . The ellipse focused in O1 and O, and passing through A1 has its second vertex at O2 . Therefore, its polar dual w.r. to (O1 ) is a circle whose diameter is

Fig. 4. I) The polar dual w.r. to (O1 ), of the ellipse focused in O1 and O the center of the external arbelos’ circle (not shown), and with vertices in A1 and O2 is a circle passing through A1 (dotted orange). II) The polar dual w.r. to (O1 ) of the parabola focused in O1 and vertex M is the circle of diameter [O1 M ]. III) The similitude center of these dual circles is O2 . IV) The two (real) intersection points of the parabola and the ellipse are the poles of their common tangents at their dual curves.

Fig. 5. Twins in arbelos: a construction via tangents and poles.

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[A1 O2 ], where O2 is the reﬂection of O2 in (O1 ); the vertex A1 is invariant, since it is a point of the inversion circle. Thus, the diameter of the dual is A1 O2 = A1 O1 + O1 O2 = R1 +

R12 2R12 + R1 R2 = . R1 + R2 R1 + R2

Let o1 and S1 be the centres of these dual circles; their radius are, respectively r1 =

2R12 + R1 R2 A1 O2 = 2 2(R1 + R2 )

and rm =

R1 . 2

To prove that the similitude center is O2 , we prove that 2 and In fact, S1 O2 = R21 + R2 = R1 +2R 2 o1 O2 = A1 O2 − A1 o1 = 2R1 + R2 − hence

o1 O2 S1 O2

=

(2R1 +R2 ) (R1 +R2 ) ;

o1 O2 S1 O2

=

r1 rm .

(2R1 + R2 )(R1 + 2R2 ) R1 (2R1 + R2 ) = , 2(R1 + R2 ) 2(R1 + R2 )

therefore

r1 R1 (2R1 + R2 ) 2 (2R1 + R2 ) · , ending the proof. = = rm 2(R1 + R2 ) R1 (R1 + R2 ) As in Fig. 5, we may attach to any arbelos a pair of touching circles, “the siblings”: two circles, that are mutually tangent at the common tangency points of the arbelos circles, and passing through the centres of the arbelos’ internal circles. The results proved above justify the following new and sharp construction of the centres of the twins. Theorem 1. The centres of the twins are the poles of the tangents drawn from the center of a arbelos’ circle, to the opposite sibling.

3

Twins’ Radii

We compute the radii of the twins, via polar reciprocity. Refer to Fig. 5. Lemma 5. Let (O1 ) and (O2 ) the (internal) arbelos circles and let (S1 ) and (S2 ), be two circles whose diameters are [O1 M ] and [O2 M ]. Let O1 T2 the tangent from O1 to circle (S2 ); let D2 be its pole w.r. to (O2 ); construct similarly D1 . Then O2 D2 − R2 = O1 D1 − R1 . Proof. Let rk = Rk /2, k = 1, 2. By hypothesis, O1 O2 P2 ∼ O1 S2 T2 , hence

r2 2r1 + r2 2r2 (r1 + r2 ) = , therefore O2 P2 = . O2 P2 2r1 + 2r2 2r1 + r2

Since D2 , the pole of O1 T2 is the reﬂection of P2 , the projection of O2 to the line O1 T2 , with respect to (O2 ). O2 P2 · O2 D2 = 4r22 , hence O2 D2 =

2r2 (2r1 + r2 ) 4r22 4r2 (2r1 + r2 ) = = 2 , which gives O2 P2 2r2 (r1 + r2 ) r1 + r2

Poncelet and the (Archimedean) Twins

169

(2r1 + r2 ) 2r2 (2r1 + r2 ) 2r1 r2 O2 D2 − R2 = − 2r2 = 2r2 −1 = . r1 + r2 r1 + r2 r1 + r2 By interchanching the indices 1 and 2, we get O2 D2 − R2 = O1 D1 − R1 Lemma 3 and Lemma 5, prove the result on the twins; refer to Fig. 5. Theorem 2. The circles centered in D1 , D2 and of radius r are archimedean twins in arbelos. Proof. The fact that the points D1 and D2 are the centers of the twins is guaranteed by Lemma 3. Thus, the circle centered in D2 and whose radius is D2 O2 −R2 is tangent to the line l, as well, hence is one of the twins. The relation fact that the radius is arquimedean ends the proof.

Fig. 6. The center of the arbelos’ i-circle as intersection of two ellipses: one focused in O and O1 and passing through O2 , and the other focused in O and O2 and passing through O1 . Their intersection is the pole w.r. to (O) of the common tangent to their reciprocal circles.

Arbelos i-circle For the sake of completeness, we show how to draw the arbelos’ i-circle I, the circle that tangents the three arbelos’ circles. Refer to Fig. 6. Theorem 3. 1. The center of I is obtainable as the intersection of two ellipses: one focused in O, and O1 , and axis [O2 A1 ] and the other focused in O and O2 and axis [O1 A2 ].

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2. The intersection points of these ellipses are the poles of the common tangents to their reciprocal circles, w.r. to (O). 1 R2 (R1 +R2 ) . 3. The radius of the arbelos i-circle is R = R R2 +R1 R2 +R2 1

2

Proof. The two ellipses have a common focus in O and are tangent to the inversion circle (O) at their vertices A1 and A2 , respectively. Hence their duals are two circles of diameters [A1 O2 ] and [A2 O1 ] respectively, where O1 is the reﬂection of O1 in (O) and O2 is the reﬂection of O2 in (O). Let H be the similitude center of the reciprocal circles. Then HC1 P1 ∼ HOP ∼ HC2 P2 , hence (R +R )(R2 +R R +R2 ) OP = 1 2 R21+R21 2 2 . Since D is the pole of HP, OD · OP = (R1 + R2 )2 . 1 2 The radius of the i-circle is R = (R1 + R2 ) − OD = (R1 + R2 ) −

4

R1 R2 (R1 + R2 ) (R1 + R2 )2 = 2 . OP R1 + R1 R2 + R22

Archimedean Circles in Doubling Arbelos

We now associate to a classic arbelos, two other circles, passing through the common tangency points M and centered at each of the end-point of arbelos’ diameter; we call it a doubling arbelos; see Figs. 7 and 9. By an i-circle in a doubling arbelos, we mean the circle that is externally tangent to the aforementioned circles and internally the arbelos’ external circle. Such circle was ﬁrst detected by Scotch, in [13], who proved that it is archimedean; above, we give another proof for this fact, as well as a new construction for the center of this circle. Theorem 4. The i-circle in a doubling arbelos is archimedean: 1s = R11 + R12 . Proof. The i-circle associated to a doubling arbelos’ is a solution of an Apollonius’ problem; therefore, its center S, can be obtained as an intersection between an ellipses focused in O and A2 , passing through O1 and a hyperbola focused in A1 and A2 , and passing through M ; see Fig. 7. Since these two conics have a common focus in A2 , a polar dual w.r. to (A2 ) maps them into a pair of circles, whose diameters are the reﬂection in (A2 ) of their vertices. Morover, S, the center of the i-circle in a doubling arbelos is the pole of the common tangent to their reciprocal circles. Straightforward computations, which we skip, as they are similar to those in the proof of Theorem 2, ﬁnishes the proof. As earlier, Theorem 4 itself embeds the geometric construction of S, as a pole of the common tangents to these reciprocal circles. Finally, consider two new circles associated to a doubling arbelos: “the twincousins.” These are the two circles that solves the Apollonius problem for the region delimited by one internal arbelos circle, by the external arbelos circle by the recently-added doubling circles, as in Fig. 9, bottom. The twin-cousins are not congruent; nevertheless they verify an archimedean-type relation.

Poncelet and the (Archimedean) Twins

171

Fig. 7. The cousin circle (S) tangents internally (O) and externally (A1 ) and (A2 ). Its center obtains as intersection of an ellipse (violet) and a hyperbola, focused in at their A2 , O and A1 , A2 , respectively. Its center S is the pole of the common tangent reciprocal circles, w.r. to (A2 ); its radius equals those of the twins: 1s = R11 + R12 .

Theorem 5. Let s1 , s2 be the radius of the twin-cousins; then 1 1 1 1 + =3 + s1 s2 R1 R2 Proof. Refer to Fig. 8. S1 , the center of twin-cousin, obtains as intersection of two ellipses: E1 , the ellipses focused in O and O1 , and passing through A1 , and, E , the ellipses focused in O1 and A1 , and passing through M. Perform a dual transform w.r. to (O1 ); since O1 is the common focus of the ellipses E1 and E , their duals are two circles. The dual of E1 is the circle C2 of diameter [A1 O2 ], where O2 is the reﬂection of O2 in (O1 ); denote by o2 and R2 its center and radius. Similarly, since M ∈ (O1 ), the dual of E w.r. to (O1 ) is the circle C2 of diameter [B1 M ], where B1 is the reﬂection of the (second) ellipse’s vertex B1 in (O1 ); denote by o1 its center and let R1 its radius. Then let p, the common tangent of these dual circles intersect the line of centres in Ω1 , their similitude center. Let P1 , P1 , P2 be the projections of O1 , o1 , o2 on the tangent p. Since E , is focused in O1 and A1 , pass through M, and since by the construction of doubling circles, O1 B1 = 2R1 , we have O1 B1 · O1 B1 = R12 , O1 B1 =

R1 3R1 R1 ; therefore R1 = , O1 o1 = . 2 4 4

In order to calculate O1 o2 note that A1 O2 is the diameter of the dual E1 ; thus A1 O2 = O1 A1 + O1 O2 = R1 +

R12 R1 (2R1 + R2 ) = ; hence R1 + R2 R1 + R2

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Fig. 8. S1 , the center of a twin-cousin is the intersection of two ellipses. S1 is also the pole of the common tangent to their dual circles (dotted), centered at o1 and o2 , respectively.

R2 =

R1 (2R1 + R2 ) A1 O2 = ; ﬁnally, 2 2(R1 + R2 )

O1 o2 = O1 A1 − o2 A1 = R1 − R2 = R1 − Hence o1 o2 = O1 o2 + O1 o1 =

R1 R2 R1 (2R1 + R2 ) = . 2(R1 + R2 ) 2(R1 + R2 )

R1 R2 + 3R1 R2 R1 R2 + = 1 . 2(R1 + R2 ) 4 4(R1 + R2 )

Note that Ω1 o1 P1 ∼ Ω1 O1 P1 ∼ Ω1 O2 P2 . Let x = Ω1 O1 and p = O1 P1 ; by standard manipulation, we get x=

R1 (R1 + 2R2 ) 3R1 (R1 + 3R2 ) 3R1 2R12 + R1 R2 , p= , R1 = , R2 = . 4(R1 − R2 ) (R1 + 3R2 ) 4 2(R1 + R2 )

Since S1 is the pole of p, (w.r. to (O1 )) O1 S1 · O1 P1 = R12 ,

O1 S1 =

R1 (R1 + 3R2 ) R12 = . p R1 + 2R2

With these ingredients in place, the radius of the cousin-twin centered at S1 is R1 R2 R1 R2 , s2 = , (1) R1 + 2R2 R2 + 2R1 where the later obtains by interchanging the indices 1 and 2 in the former formula 1, ﬁnally proving the statement. s1 = O1 S1 − R1 =

Poncelet and the (Archimedean) Twins

173

Fig. 9. Metric coincidences in arbelos: I) The (classic) twins (solid golden) and the cousin i-circle (blue) are congruent and archimedean: s = r1 = r2 and 1s = R11 + R12 . II) The two cousin-circles (solid blue) veriﬁes 3s = s11 + s12 hence s11 + s12 = 3 R11 + R12

5

Final Remarks

The theory of polar loci has a long and spicy history. The terminology pole and polar goes back at least to Servois (1811) and Gergonne (1813). Yet, Poncelet used the dual of a conic w.r. to a circle, in order to prove his Porism, that dates back to 1812–1813, though published in 1822. Poncelet’s more systematic treatment of reciprocal polars was presented in his Trait´e de propri´et´es projectives des figures, ﬁrst published in 1824. Curiously, the controversy between Poncelet and Gergonne concerning the paternity of the principles of “reciprocity” versus that of “duality” had a beneﬁcial collateral eﬀect: it drove the attention toward this new fundamental principle to a wider audience; see [11] and [7]). In fact, for the following decades, at least in the French school, polar duality was a widespread tool. While the method itself have profoundly impacted many ﬁelds in modern mathematics, nowadays it is hardly used in classic, euclidean geometry. Our hope is that our modest contribution could help to change this state of facts. Acknowledgements. The author want to thank both referees for their thorough and careful reading of the manuscript and for the kind and pertinent suggestions, that improved its ﬁnal form; in particular, for identifying several unfortunate mathematical mistyping. We also want to thank Dan Reznik, whose enthusiasm and proﬁciency in seeking and highlighting new phenomenons in classic frames inspired this work.

Appendix. Brief recall on polar reciprocity Fix C(Ω, R), a circle centered in Ω and of radius R, which we shall call inversion circle.

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If p0 is a line that does not pass through Ω, its pole is the inverse of the projection of the center Ω, on the line p0 . If P0 is a point (P0 = Ω), the polar of P0 is the perpendicular line on ΩP0 , that pass through P1 , the inverse of P0 . The polar dual (or a reciprocal curve) of a regular curve (w.r. to an inversion circle) deﬁnes as the curve whose points are the poles of the tangents of the original curve. Proposition 2. (see e.g. [12], art. 306 and 309) The dual of a circle γ = C(O, r), w.r. to an inversion circle C(Ω, R), is a conic, Γ ; if d denotes the distance between the centres of the reciprocated and inversion circles, d = ΩO, then: i) Γ is an ellipse, if d < r; ii) Γ is a parabola, if d = r; iii) Γ is a hyperbola, if d > r. Moreover, (one of ) the the focus of the dual conic Γ is precisely Ω, the center of the inversion circle; its directrix is the polar of O, the center of the reciprocated circle and the eccentricity is e = dr . Proposition 3. The dual of a conic Γ, w.r. to an inversion circle centered into its focus, is a circle, γ. The symmetric of the vertices of the conic Γ, are a pair of diametrically opposite points of the dual circle, γ. The pole of the directrix of Γ, is the center of the circle γ.

References 1. Akopyan, A.V., Zaslavsky, A.A.: Geometry of Conics. Amer. Math. Soc, Providence, RI (2007) 2. Bankoﬀ, L.: Are the twin circles of archimedes really twins? Math. Mag. 47(9), 214–218 (1962) 3. Bergsten, C.: Magic circles in the arbelos. The Montana Mathematics Enthusiast 7(2–3), 209–222 (2008) 4. Danneels, E., van Lamoen, F.: Midcircles and the arbelos. Forum Geometricorum 7(2), 53–65 (2007) 5. Glaeser, G., Stachel, H., Odehnal, B.: The Universe of Conics: From the ancient Greeks to 21st century developments. Springer (2016) 6. van Lamoen, F.: Archimedean adventures. Forum Geom. 6, 77–96 (2006). http:// forumgeom.fau.edu/FG2006volume6/FG200609index.html 7. Lorenatt, J.: Polemics in public: Poncelet, gergonne, pl¨ ucker, and the duality controversy. Sci. Context 28(4), (2015), 545-585 (2015) 8. Nguyen, N.G., Viet, A.L.: Some archimedean circles in arbelos. Intl. J. Geom. 8(2), 84–98 (2019) 9. Okumara, H., Watanabe, M.: The twin circles of archimedes in a skewed arbelos. Forum Geom. 4, 229–251 (2004) 10. Oller-Marcen, A.: The f-belos. Forum Geometricorum 13, 103–111 (2013) 11. Poncelet, V.: Note sur divers articles du bulletin des sciences de 1826 et de 1827, relatifs ` a la th´eorie des polaires r´eciproques, ` a la dualit´e des propri´et´es de situation de l’´etendue, etc. Annales de Math´ematiques pures et appliqu´ees tome 18, 125– 142 (1827-1828) 12. Salmon, G.: A treatise on conic sections. Longman, Green, Reader and Dyer, London (1869) 13. Schoch, T.: A Dozen More Arbelos Twins. http://www.retas.de/thomas/arbelos/ biola/ 14. Woo, P.: Simple constructions of the incircle of an arbelos. Forum Geometricorum 1, 133–136 (2001)

A Note on Local Intersection Multiplicity of Two Plane Curves Adriana Bos´ akov´ a(B)

and Pavel Chalmoviansk´ y

Comenius University in Bratislava, Bratislava, Slovakia {adriana.bosakova,pavel.chalmoviansky}@fmph.uniba.sk

Abstract. Let O = (0, 0) be the intersection point of two plane algebraic curves F and G. According to existing results, we know that their intersection multiplicity IO at O satisfies the inequality IO (F, G) ≥ mn + t, where m and n are the multiplicities of O on F and G respectively, and t is the number of their common tangents at O (counted with multiplicity). The aim of this paper is to investigate under which conditions the equality occurs. These conditions are given in terms of individual common tangents of F and G at O and their relations to the polynomials defining these curves. Keywords: Plane curves geometry

1

· Intersection multiplicity · Algebraic

Introduction

Let F, G ∈ A2k be two aﬃne plane curves, let k be an algebraically closed ﬁeld. By IO (F, G), we denote the intersection multiplicity of F and G at O = (0, 0). It is a natural number, which helps to describe the complexity of this intersection. It reﬂects the multiplicities of O on both curves and the local behavior of both curves. More about intersection multiplicity can be found for example in [3–5]. Let the point O be of multiplicity m on F and n on G. According to the local B´ezout inequality, we have IO (F, G) ≥ mn.

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Our paper was inspired by B. Bydˇzovsk´ y, who has improved this result to IO (F, G) ≥ mn + t, where t is the number of their common tangents at O (counted with multiplicity) in his book [1]. By adding a correction term l = IO (F, G) − mn − t, we can change it to an equality IO (F, G) = mn + t + l.

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Results about this term can be found in [2], where an algebraic description is given, but with no geometric interpretation. In Theorem 1, we give necessary and suﬃcient conditions for l = 0, i.e., for IO (F, G) = mn + t, c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 175–184, 2023. https://doi.org/10.1007/978-3-031-13588-0_15

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in terms of the properties of the individual common tangents of F and G at O. We focus on the case where both F and G are singular at O and inspect their behavior in a small neighborhood of O. In order to keep notations sparse, we use the same capital letter for both the curve and the polynomial it is deﬁned by. The number of tangents of a curve at the point (or number of common tangents of two curves) is always counted with multiplicity. If Fm = Π(αi x − βi y)ei (αi , βi ∈ k, ei ∈ N) is the non-zero homogeneous part of the polynomial F of the smallest degree, we say that the tangent αi x − βi y = 0 is of multiplicity ei on F at O. In a small neighborhood of a point, each curve can be decomposed into a ﬁnite union of branches. Each branch of the curve can be parameterized via a Puiseux expansion.This parametrization is in form of a (possibly inﬁnite) power ∞ series B(t) = (tk , i=1 ai ti ). We always assume that in such parametrization, a generic point corresponds to only one value of the parameter. A proof of the existence of this parametrization and its properties can be found in [4,5]. By order of a polynomial, we mean the degree of its non-zero term of the lowest degree.

2

Intersection of a Curve with a Branch

We are interested in the conditions under which the intersection of two curves satisﬁes the property (3). To determine them, we study the intersection of a curve F and a single branch B in Lemma 1. Then, we use it to build up the main result of this paper, i.e., Theorem 1. Lemma 1. Let F be a curve deﬁned by the polynomial F = Fm + Fm+1 + · · · ,

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where Fi is either homogeneous of degree i, or equal to 0 and Fm = 0. Let B be a branch at O. Let L : αx − βy = 0 be a common tangent of F and B at O of the multiplicity r and k respectively. Let r > k. Then, 1. IO (F, B) ≥ mk + k, 2. IO (F, B) = mk + k if, and only if, Fm+1 is not divisible by L. Proof. The ﬁrst part is proved in [1] by calculating the order of the polynomial F (B(t)). We use the same idea to prove the second part. First, let us assume that L : y = 0. The intersection multiplicity of F and B at O is equal to the order of the polynomial F (B(t)), where B(t) is a parametrization of B. Since y = 0 is a tangent to F of the multiplicity r, we know that Fm is divisible by y r . We can write Fm = 0 + · · · + 0 + ar xm−r y r + ar+1 xm−r−1 y r+1 + · · · + am y m , Fm+1 = b0 xm+1 + b1 xm y + · · · + bm+1 y m+1 , m+2

Fm+2 = c0 x

m+1

+ c1 x

y + · · · + cm+2 y

m+2

(5) .

A Note on Local Intersection Multiplicity of Two Plane Curves

177

The multiplicity of the tangent y = 0 on the branch B equals k. Therefore the parametrization of B can be written as B(t) = (tk , tk (γ1 t + γ2 t2 + · · · ))

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= (tk , tk+1 Γ ). Substituting B(t) into F (x, y) gives F (B(t)) = F (tk , tk+1 Γ ) =

ar tkm+r Γ r + ar+1 tkm+r+1 Γ r+1 + · · · + am tkm+m Γ m + b0 tkm+k + b1 tkm+k+1 Γ + · · · + bm+1 tkm+k+m+1 Γ m+1 + c0 t

km+2k

+ c1 t

km+2k+1

Γ + · · · + cm+2 t

km+2k+m+2

Γ

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m+2

+ ··· . Since r > k, the term of the lowest degree is b0 tkm+k , every other term is of higher degree than km + k. The lowest possible intersection multiplicity in this case is equal to km + k and is achieved if, and only if, b0 = 0. Since b0 is the coeﬃcient of the homogeneous part Fm+1 = b0 xm+1 + · · · bm+1 y m+1 , we know that b0 = 0 if, and only if, Fm+1 is not divisible by y, which completes the proof for L : y = 0. For any other L, we apply a suitable change of coordinates and proceed the same way.

3

Conditions for a Zero Correction Term

Now, we use Lemma 1 to prove the main theorem. Theorem 1. Let F and G be two plane aﬃne algebraic curves deﬁned by the polynomials F = Fm + Fm+1 + · · · and G = Gn + Gn+1 + · · · , where Fi , Gi are either homogeneous polynomials of degree i, or equal to 0. Let Fm = 0, Gn = 0. Let the point O be of multiplicity m and n on F and G respectively. Let t be the number of their common tangents at O (counted with multiplicity). Then, IO (F, G) = mn + t,

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if, and only if, F and G satisfy the following condition for each common tangent L of multiplicity r and s respectively. – If r > s, then Fm+1 is not divisible by L. – If r < s, then Gn+1 is not divisible by L. – If r = s, then v0 as = b0 us , where v0 , us , b0 , as are the following coeﬃcients of F and G, after applying a change of coordinates, which maps L to the line y = 0, F = Fm + Fm+1 + · · · = as xm−s y s + · · · + am y m + b0 xm+1 + · · · bm+1 y m+1 + · · · , G = Gn + Gn+1 + · · · = us xn−s y s + · · · + un y n + v0 xn+1 + · · · vn+1 y n+1 + · · · .

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Remark 1. The case r = s in Theorem 1 is also closely related to the divisibility of Fm+1 and Gn+1 by L. In particular, – If (L|Fm+1 and L|Gn+1 ), the condition v0 as = b0 us never holds. – If (L|Fm+1 and L Gn+1 ) or (L Fm+1 and L|Gn+1 ), the condition v0 as = b0 us always holds. – If (L Fm+1 and L Gn+1 ), we actually need to check the coeﬃcients. Using Lemmas 2 and 3, the proof of Theorem 1 follows afterwards. Lemma 2. Let F be a curve deﬁned by the polynomial F = Fm + Fm+1 + · · · , where Fi is homogeneous of degree i and Fm = 0. Let Fm = a0 xm + · · · + am y m ,

Fm+1 = b0 xm+1 + · · · + bm+1 y m+1 .

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Further, let y = 0 be a tangent of F at O of multiplicity k and y Fm+1 . Then, there is exactly one branch B of F with the tangent y = 0 and it can parametrized by k −b0 k k 2 3 B(t) = t , t t + γ2 t + γ3 t · · · , (11) ak for some γi ∈ k. Proof. This can be seen from the direct construction of the parametrization as a Puiseux series. The multiplicity of the tangent y is k, which means we can write Fm as Fm = y k (· · · ), and a0 = · · · = ak−1 = 0, ak = 0. Since y Fm+1 , the coeﬃcient b0 in Fm+1 has to be nonzero. Now, we construct the Newton polygon of F . Each side of the Newton polygon is deﬁned by a pair of points (p1 , q1 ), (p2 , q2 ). By a slope of a side we mean the 2 number σ = pq11 −q −p2 . A side of a Newton polygon corresponds to a branch with a tangent y, if, and only if, σ ∈ (−1, 0). There is only one such side, the side k , connecting the points which correspond to the monomials with the slope − k+1 m−k k m+1 ak x y and b0 x . After the ﬁrst step, the substitution equals −b 0 k y = xk+1 + y1 . (12) x = xk1 , 1 ak After substituting it into F , we get k −b0 k+1 k + y1 = xmk+k (F ). F x1 , x1 ak

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k+1 k 0 The coeﬃcient of y in F is −b , which is nonzero (because b0 = 0). ak Therefore, its substitution in every other step of the Puiseux expansion algorithm is in the form y = txz , where z ∈ N. We get no new Puiseux pairs and the ﬁnal parametrization is in the form k −b0 k k 2 3 t + γ2 t + γ3 t · · · , (14) B(t) = t , t ak

A Note on Local Intersection Multiplicity of Two Plane Curves

for some γi ∈ k.

179

Lemma 3. Let F be a curve with exactly one branch B with the tangent y = 0 at O, let this tangent be of multiplicity r and let y|Fm+1 . Then, this branch can be parametrized by B(t) = (tr , γ2 tr+2 + γ3 tr+3 + · · · )

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for some (not necessarily nonzero) γi ∈ k. Proof. If y|F , the proof is obvious. Otherwise, it can be seen from the construction of the Puiseux expansion. Since there is only one branch with the tangent y = 0, there can be only one segment of the Newton polygon with a slope σ ∈ (−1, 0). Let i be the smallest number, such that ci xm+i y 0 is nonzero in F . r , hence the parametrization of the said branch is of Then, this slope equals − r+i r r+i + (terms of higher degree)). Since y|Fm+1 , we know that form B(t) = (t , γi t i ≥ 2. Proof (of Theorem 1). Due to the additivity of the intersection multiplicity, we can take care of each common tangent of F and G separately and check its contribution to the whole result. To achieve IO (F, G) = mn + t, we need for each tangent (of multiplicities r and s on F and G respectively) to have minimal possible contribution. First, we take care of the case r = s. Let L be a common tangent of F and G (of multiplicities r and s respectively), let s < r. Then, we split G into branches. The branches Bi of G with the tangent L, are of multiplicities si , where si = s. Since si < r for each i, we can use Lemma 1 and get the condition IO (F, Bi ) =si m + si if, and only if, L Fm+1 . If this is satisﬁed, then IO (F, Bi ) = i si m + si = sm + s, which is exactly what we need. Analogously, if r < s, we get the condition L Gn+1 . If r = s, we need to check the three possible cases. 1. First, let both Fm+1 and Gn+1 be not divisible by L. Let L : y = 0. By Lemma 2, each of these curves has exactly one branch with the tangent y = 0. This branch is of multiplicity r = s for both curves and for the curve G, it can be parametrized by

−v0 t + γ2 t2 + γ3 t3 · · · , (16) B(t) = ts , ts s us for some γi ∈ k (where v0 and us are coeﬃcients of Gn+1 and Gn respectively, as in (9)). Substituting this into the polynomial F , we get

−v0 t + γ2 t2 + γ3 t3 · · · F (B(t)) = F ts , ts s us −v0 s(m+1) (17) = as t + b0 ts(m+1) + (terms of higher degree) us

v0 = ts(m+1) b0 − as + (terms of higher degree). us

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Therefore, IO (F, B) = sm + s(= rm + r) if, and only if, v0 as = b0 us which completes the proof for L : y = 0. If L is a tangent diﬀerent from y, we apply a suitable change of coordinates at the very beginning and then proceed in the same way. 2. Now, let exactly one of the polynomials Fm+1 , Gn+1 be divisible by L. Without loss of generality, let it be the polynomial Fm+1 . Then we split the curve G into branches. By Lemma 2, there is only one branch B of G with the tangent L. We substitute B(t) into F and continue in the same way as in the case 1. We end up with the same condition v0 as = b0 us , but in this case, b0 = 0 and v0 , us , as are all nonzero. Therefore, in this case, the condition v0 as = b0 us always holds and IO (F, B) = sm + s. 3. Finally, let both Fm+1 and Gn+1 be divisible by L. This means that b0 = v0 = 0 and the condition v0 as = b0 us never holds. Let us decompose G into branches. If there is more than one branch with the tangent L, each of these branches Bi is of multiplicity less than si < s. Hence we can use the proof from the r = s case, and get the condition IO (Bi , F ) = si m + si if, and only if, L Fm+1 . This never holds, because in this case, L|Fm+1 . If there is only one branch B of G with the tangent L, we check the conditions for L : y = 0 (in any other case, we use an appropriate change of coordinates and proceed analogously). In this case, the parametrization of B can be written as B(t) = (ts , ts (0 + 0t + γ2 t2 + γ3 t3 + · · · )) by Lemma 3 (for some γi ∈ k). After substituting it in the polynomial F , we can see that all the terms are of degree greater than sm + s, therefore, the condition never holds. All these three subcases of r = s can be characterized by the same condition i.e., v0 as = b0 us , which completes the proof.

4

Examples

We demonstrate Theorem 1 in three diﬀerent situations. Example 1 shows the case where the requirements of Theorem 1 are met, unlike Example 2. In Example 3, we demonstrate a case where the curves have more than one diﬀerent tangents in common. Example 1. Let F and G be curves (Fig. 1) deﬁned by the polynomials F = (x − y)(x + y) + y 3 ,

G = (x + y)3 + x4 .

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Clearly, m = 2 and n = 3. The tangents of F at O are the lines x − y = 0 and x + y = 0, each with multiplicity 1. The only tangent of G at O is x + y = 0 and its multiplicity is 3. Therefore F and G have one tangent in common and IO (F, G) = mn + t + l = 2 · 3 + 1 + l.

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A Note on Local Intersection Multiplicity of Two Plane Curves

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O F G

Fig. 1. The curves F (solid) and G (dashed).

Now, we can check the requirements for l = 0 according to Theorem 1. The tangent x + y = 0 is of higher multiplicity on the curve G, than on F , therefore, we need to check if Gn+1 = G4 is divisible by x + y. Since gcd(G4 , x + y) = gcd(x4 , x + y) = 1,

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the requirements of Theorem 1 are met, which means l = 0 and IO (F, G) = 2 · 3 + 1 + 0 = 7.

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Example 2. Let F and G be curves (Fig. 2) deﬁned by the polynomials F = y 2 + y 3 − x4 ,

G = y 5 − x7 .

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F O

G

Fig. 2. The curves F (solid) and G (dashed).

Then, m = 2 and n = 5. The only common tangent of F and G is the line y = 0 and it is of multiplicity 2 and 5 on F and G, respectively. Since 2 < 5, we need to have a look at Gn+1 = G6 . It is equal to 0, hence it is divisible by any linear polynomial, so the requirements of Theorem 1 for l = 0 are not met. For this case, Theorem 1 doesn’t give us the exact value of the intersection multiplicity, but it tells us that it is strictly greater than mn + t = 12.

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Example 3. Let F and G be curves (Fig. 3) deﬁned by the polynomials F = x2 (x − y)(x + y)y 3 + x7 (x − y) − y 10 ,

G = x5 (x − y)y + y 8 − x9 . (23)

F G O

Fig. 3. The curves F (solid) and G (dashed).

In this example, m = 7 and n = 7. These two curves have three diﬀerent tangents in common, the lines L : x = 0, L : y = 0 and L : x − y = 0. We need to check the requirements for each one separately. – The tangent L : x = 0 is of multiplicity 2 on F and 5 on G. Since 2 < 5, we need to check the divisibility of Gn+1 = G8 by x. gcd(G8 , x) = gcd(y 8 , x) = 1,

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therefore the requirements for l = 0 for the tangent L are met. – The tangent L : y = 0 is of multiplicity 3 on F and 1 on G. Since 3 > 1, we need to check the divisibility of Fm+1 = F8 by y. gcd(F8 , y) = gcd(x7 (x − y), y) = 1,

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therefore the requirements for l = 0 for the tangent L are met. – The tangent L : x − y = 0 is of multiplicity 1 on both F and G. Since Fm+1 = F8 = x7 (x − y) is divisible by (x − y) while Gn+1 = G8 = y 8 is not, we know that the requirements for this tangent are also met (according to Remark 1). The requirements for l = 0 for F and G are met for all their common tangents, therefore l = 0 and IO (F, G) = mn + t = 7 · 7 + 4 = 53.

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A Note on Local Intersection Multiplicity of Two Plane Curves

5

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The Case of a Nonzero Correction Term

If the requirements of Theorem 1 are not met for some tangent L, then obviously l > 0, but we can give some better estimate for this number. Let L be a common tangent of F and G of multiplicity r and s respectively, let r < s. Theorem 1 suggests a connection between the branches of F with the same tangent L. It says that if at least one of the branches Bi (of multiplicity ri ) has an intersection multiplicity with G greater than nri + ri , then all of them do. This is formulated in Theorem 2. For future work, we would like to investigate this connection, which could help with the description of geometric properties of the correction term l. Theorem 2. If the requirements of Theorem 1 are not met for some common tangent L (of multiplicities r and s on the curve F and G respectively), then the correcting term l = IO (F, G) − mn − t is increased at least by e, where – if r < s, then e is the number of branches of F with the tangent L. – if r > s, then e is the number of branches of G with the tangent L. – if r = s, then e is the minimum of the numbers of branches of F and G with the tangent L. Proof. Without loss of generality, let r ≤ s. Then we split F into branches and calculate their intersection multiplicity with G. For each branch Bi (of multiplicity ri ) with the tangent L, we have IO (Bi , G) ≥ nri + ri + 1 (by Lemma 1). Therefore, the intersection multiplicity is increased at least by the number of these branches. Example 4. Let F and G be curves (Fig. 4) deﬁned by the polynomials F = y 4 − 5x4 y 2 + 4x8 − y 9 ,

G = y 5 − x7 .

F G

Fig. 4. The curves F (solid) and G (dashed).

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Then, m = 4, n = 5 and the line y = 0 is their only common tangent (of multiplicities r = 4 and s = 5 on F and G respectively). Since Gn+1 = G6 is a zero polynomial, it is divisible by y, and by Theorem 1, we know that IO (F, G) > mn + t = 24.

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The curve F splits into four branches (with the tangent y = 0) at the origin:

11 B1 (t) = t, t2 − 16 t11 + · · · , B3 (t) = t, 2t2 − 128 + ··· , 3 t (29)

11 B4 (t) = t, −2t2 − 128 + ··· , B2 (t) = t, −t2 − 16 t11 + · · · , 3 t and therefore, by Theorem 2, l ≥ 4 and IO (F, G) ≥ mn + t + 4 = 28.

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Acknowledgements. This work was supported by the Slovak Research and Development Agency, under the contract No. APVV-16-0053 and by the grant VEGA 1/0596/21.

References ´ 1. Bydˇzovsk´ y, B.: Uvod do algebraick´e geometrie. Knihovna spis˚ u matematick´ ych a ˇ fysik´ aln´ıch. Jednota Ceskoslovensk´ ych matematik˚ u a fysik˚ u, Prague (1948) 2. Bod’a, E., Schenzel, P.: Local B´ezout estimates and multiplicities of parameter and primary ideals. J. Algebra 488, 42–65 (2017) 3. Fulton, W.: Algebraic Curves: An Introduction to Algebraic Geometry. https:// dept.math.lsa.umich.edu/∼wfulton/CurveBook.pdf 4. Brieskorn, E., Kn¨ orrer, H.: Plane Algebraic Curves, Birkh¨ auser Basel (1986) 5. Algebraic Geometry. LNM, vol. 732. Springer, Heidelberg (1979). https://doi.org/ 10.1007/BFb0066642

Locally Flat and Rigidly Foldable Discretizations of Conic Crease Patterns with Reflecting Rule Lines Erik D. Demaine1 , Klara Mundilova1(B) , and Tomohiro Tachi2 1

Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] 2 University of Tokyo, Tokyo 153-8902, Japan

Abstract. Conic curved creases with reﬂected rule lines is a style of curved origami design, ﬁrst explored by David Huﬀman, that is attractive in that it gives one-DOF folding motions with rigid rule lines (i.e., the rule lines remain the same throughout the motion). We show how to discretize any such curved crease pattern into a similar straight-line crease pattern that has a one-DOF rigid folding motion. We develop two general methods for such discretization, where each curve is replaced by an inscribing or circumscribing polygonal line, respectively, and show in both cases that the resulting discretized crease patterns are rigidly foldable. In the case of the circumscribed discretization, the crease pattern is also locally ﬂat foldable. On the other hand, only careful sampling in the inscribed method results in locally ﬂat-foldable crease patterns.

Keywords: Origami

1

· Conics · Curved creases

Introduction

Many of David Huﬀman’s curved-crease origami designs use conics sharing focal points as the creases and lines passing through the focal points as rulings of the curved surfaces [4]. Demaine et al. [1] call these natural rule lines. Because of the properties of conics, natural rule lines reﬂect at the creases, and if such creases can fold, they fold with constant fold angles [3]. Demaine et al. [1] characterize valid combinations of conic creases with natural rule lines and show that compatible conic creases produce a continuous folding motion that keeps the rulings rigid, called a rigid-ruling folding motion. Rigid-ruling folding is a curved-crease analogue of rigid folding—i.e., a rigid panel-and-hinge mechanism (with straight creases, see Fig. 1 for example). Therefore, rigid-ruling folding of conics naturally raises a question: can we discretize conic curved creases such that they rigidly fold? Tachi [6] and Lang et al. [5] discretize (using a locally ﬂat-foldable inscribed discretization as explained later) a speciﬁc family of multi-crease curved origami and a single-crease curved origami, respectively. However, whether such a discretization exists for multiple-crease conic curved foldings, and if it exists whether it is rigidly foldable, were unknown. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 185–196, 2023. https://doi.org/10.1007/978-3-031-13588-0_16

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Fig. 1. A locally ﬂat and rigidly foldable discretization of one of David Huﬀman’s designs with scaled and reﬂected parabolas.

In this paper, we propose two general algorithms for discretizing smooth combinations of conics, and analyze the rigid foldability of the resulting discretized piecewise-linear origami. Speciﬁcally, we propose two methods of discretization: (1) the inscribed discretization samples curve points and connects them consecutively to form an inscribed polyline, and (2) the circumscribed discretization samples conic tangents and ends them at consecutive intersections to form a circumscribed polyline. Our main result (Theorem 1 below) is that both discretizations result in rigidly foldable crease patterns. With both methods, the sampling parameters of a single curve can be freely speciﬁed, but need to be propagated to other creases with discrete natural rule lines, that is, the vertices of neighboring compatible creases and their common focal point need to be collinear; see Fig. 4. We also show that circumscribed discretization ensures that each vertex is locally ﬂat foldable, whereas inscribed discretization achieves this property only with careful sampling. When each vertex is locally ﬂat foldable, the fold angle along the original curved crease is constant, preserving the original properties of conic curved crease folding with natural rule lines. A common approach to proving rigid foldability of a quadrivalent pattern (as produced by our discretization methods) is to show the compatibility of fold angles based on fold angle multipliers [2,7], but this approach requires that each vertex is locally ﬂat foldable. As our inscribed discretization method does not always produce locally ﬂat-foldable vertices, we instead prove rigid foldability by using a constructive approach based on a compatible series of planar linkages using the properties of conics.

2 2.1

Notation Conic Sections

In the following, we will work with the polar parametrization of conic sections, that is, a conic section c(t) with focal point f = (0, 0) can be parametrized by cos t a , (1) c(t) = 1 + ε cos t sin t

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Fig. 2. Illustration of Lemma 1.

where ε ∈ R is the eccentricity and a ∈ R is a scale factor. The conic is a circle if ε = 0, an ellipse if 0 < |ε| < 1, a parabola if |ε| = 1, and a hyperbola if |ε| > 1. The other focal points of the conics are 2aε − , 0 , if c is an ellipse or hyperbola, 1−ε2 f = the point at inﬁnity of the x-axis , if c is a parabola. We review the following result about intersections of tangents; see Fig. 2: Lemma 1. Let c(t) be a conic section with focal points f and f . The intersection p of the its tangents at parameter values t and t is the intersection of the angle bisector of {c(t), f , c(t )} and the angle bisector of {c(t), f , c(t )}. Furthermore, ∠(f , p, c(t)) = ∠(c(t ), p, f ) in case of an ellipse or parabola, and ∠(f , p, c(t)) + ∠(f , p, c(t )) = π in case of a hyperbola. 2.2

Compatible Conic Sections

Demaine et al. show that naturally ruled crease pattern of two conic creases c1 and c2 sharing a focal point can fold if and only if the conics have the same or reciprocal eccentricity. Figure 3 shows the nine distinct cases of two conics: (E), (H), (P1), (P2), (P3), (EH1), (EH2), (EH3), and (EH4). If two conics have the same eccentricity, they are scaled versions of each other if the common focal point is real (cases (E), (H), and (P1)), and are translated or reﬂected if the common focal point is a point at inﬁnity (cases (P2) and (P3)), which can only happen in case of a parabola. If two conics have reciprocal eccentricities ε and 1 ε where 0 < |ε| < 1, they are a combination of an ellipse and a hyperbola. We use the parametrization in Eq. (1) with scale factors a1 and a2 and eccentricities ε1 = ε and ε2 = 1ε . Up to re-parametrization there are only four possible cases; see Fig. 6: – – – –

Case Case Case Case

(EH1): (EH2): (EH3): (EH4):

ε>0 ε>0 ε 0, a1 > a2 ε > 0, a2 ε > a1 > 0, a1 > a2 ε > 0.

Note that if a1 > a2 ε, the two conics intersect. This eﬀects the direction of the reﬂected outer rule lines. In the following we only consider parameter intervals (tmin , tmax ), where either |c1 (t)| ≤ |c2 (t)| < ∞ for all t ∈ (tmin , tmax ) or |c2 (t)| ≤ |c1 (t)| < ∞ for all t ∈ (tmin , tmax ).

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(E)

(EH1)

(H)

(EH2)

(P1)

(EH3)

(P2)

(P3)

(EH4)

Fig. 3. The nine distinct cases of compatible conic creases with reﬂecting rule lines sharing a focal point. Red and blue indicate mountain and valley creases, respectively. Note that the entire MV assignment can also be inverted.

2.3

Discretizations

Let c be a conic with focal points f and f and T = {t0 , t1 , . . . , tm } an ordered list of parameter values with ti ∈ (tmin , tmax ). In this paper, we study the following two discretizations; see Fig. 4: – Inscribed discretization: The vertices of the inscribed discretization method are points on the conic corresponding to parameters in T , and vertices corresponding to adjacent parameters are connected with edges. – Circumscribed discretization: The vertices of the circumscribed discretization method are intersections of tangents of the conics at parameters in T , and vertices corresponding to adjacent parameters are connected with edges.

Fig. 4. Left: Conic crease pattern with reﬂecting rulings (case (E)). Center: Inscribed discretization. Right: Circumscribed discretization.

We propagate the two discretizations with discrete natural rule lines: In case of the inscribed discretization, the sampling of the conics is propagated such that the sampled vertices of two compatible conics and the shared focal point are collinear. In case of the circumscribed discretization, the sampling of the points of contact is propagated such the sampled points of contact of two compatible conics and their shared focal point are collinear. It follows from Lemma 1 that the vertices obtained by intersecting neighbouring tangents and the shared focal point are again collinear. Thus the discretized rulings in both discretization methods essentially sample the rulings in the smooth case.

Locally Flat and Rigidly Foldable Discretizations of Conic Crease Patterns

(a)

(b)

(c)

(d)

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(e)

Fig. 5. Illustration of the proof of rigid foldability. (a) Crease pattern. (b) Extracted set of linkages. (c) Conﬁgurations of the set of extracted linkages. (d) Conﬁgurations positioned in pencil of planes with appropriate opening angles. (e) Extracted folded state.

The discretization of a curved crease is a polyline connecting the constructed vertices. The discretization of rulings are the line segments between the corresponding vertices of consecutive discretized curved creases. The discretization of the curved crease pattern is the union of the discretization of curved creases and rulings. Note that the discretized crease pattern is a quadrivalent mesh, that is, it consists of degree-4 interior vertices. The original curved crease and its discretization have the same MV assignment, and so do the original rulings and their discretization.

3

Rigid Foldability

In this section, we prove that both proposed discretizations of compatible conics result in rigidly foldable crease patterns. We provide a geometric construction for the folded states of a crease pattern. In particular, we show that consecutive rule lines remain coplanar and that focal points remain collinear during the folding motion, a property that reﬂects the smooth case behavior as observed by Wunderlich [9] and [8]. Intuitively, for the proof of either discretization method, we ﬁrst consider the linkages obtained by consecutive discretized rule lines, as in Fig. 5. By keeping the common focal point ﬁxed and moving the other incident focal points in a speciﬁc way, we show that there exist planar conﬁgurations of the considered linkages, such that they share the location of focal points. Finally, we show how to position the modiﬁed planar linkages in planes sharing the line spanned by the collinear focal points. This results in a folded state of the crease pattern. More formally, we consider two compatible conic creases c1 and c2 with common focal point. Let p1 and p2 be two neighboring points corresponding to an inscribed or circumscribed discretization. In the following, we study conﬁgurations of the linkage L consisting of ﬁve vertices {f , f1 , f2 , p1 , p2 } connected with three bars, that is, (f1 , p1 ), (f2 , p2 ), and a bar connecting the vertices {f , p1 , p2 }. First, we consider cases with all real focal points, namely cases (E), (H), and (EH1)–(EH4). Points f and fi are deﬁned as the common focal point and the other focal points of ci , respectively. We constrain f to the origin and fi to a one-parameter family of locations on the x-axis, inducing a one-DOF linkage.

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p2 p1 f2

f1

f

f

P2 ιi F2

F1

f1

p1 f1

f2

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P1 F P2 P1

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(a) Compatible conic combinations obtained by scaling: (E), (H), (P1)–(P3).

p2

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P1 ι1

p1

F1 F2

F

ιi

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(b) Compatible ellipse-hyperbola combinations (EH1)–(EH4).

Fig. 6. First row: crease pattern. Second row: extracted linkage {f , f1 , f2 , p1 , p2 }. Third row: conﬁguration of linkage {F, F1 , F2 , P1 , P2 }. Forth row: folded state.

In the cases where the common focal point is at inﬁnity, namely cases (P2) and (P3), we assume without loss of generality that the conics are parametrized

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such that the y-axis bisects p1 and p2 . In these cases, we deﬁne f to be the midpoint of p1 and p2 . We constrain the motion of the linkage to keep the location of the point corresponding to f on the y-axis and the bar p1 p2 horizontal. Also, we let fi be the other focal points, which are constrained on x-axis. In the case where the other focal points are at inﬁnity, namely case (P1), we let f denote the common focal point pinned to the origin and choose points fi such that fi pi are horizontal. We constrain the motion of the linkage to keep the bars fi pi horizontal. The following two lemmas specify the folded conﬁgurations {F, F1 , F2 , P1 , P2 } of the linkage L for the proposed discretizations of compatible conic curves, showing that the positions of Fi are independent of the initial location of pi or thus parameter t. This implies the compatibility between the folded linkages sampled at diﬀerent points on the conic curves as used in Theorem 1. Lemma 2. Given two conics that are hyperbolas, namely the cases (E), (H), for s ∈ [1, smax ], define f + s(fi − f ), Fi = f + 1s (fi − f ),

compatible combinations of ellipses or (EH1)–(EH4), let F be the origin, and if ci is an ellipse, if ci is a hyperbola,

dist(f

where

i ,pi )+dist(f ,pi )

smax =

dist(f ,fi ) dist(fi ,pi )−dist(f ,pi ) dist(f ,fi )

if ci is an ellipse, if ci is a hyperbola.

1 In case of the inscribed discretization method, smax = |ε| . Then there exist point locations P1 and P2 such that {F, F1 , F2 , P1 , P2 } is a configuration of the linkage L.

Proof. We assume that the two compatible conics c1 (t) and c2 (t) are parametrized as in Eq. (1) with eccentricities εi ∈ {0, 1} and scale factors ai . If c1 and c2 are both ellipses or both hyperbolas, we assume without loss of generality that ai > 0 and ε1 = ε2 = ε > 0. If the conic sections are of diﬀerent type, we let c1 be the ellipse and c2 the hyperbola. In this case, we deﬁne ε = ε1 = ε12 and assume without loss of generality that a1 > 0 and a2 ε > 0. Deﬁne ui = dist(fi , pi ), vi = dist(f , pi ), and fi = dist(f , fi ). Note that −sgn(ε) 2ai ε2 s if ci is an ellipse, fi = 2ai ε 1 1−ε if ci is a hyperbola. 1−ε2 s The lengths {fi , ui , vi } specify triangles that we position such that two corners are F and Fi and the third corner Pi lies above the x-axis. Furthermore, we deﬁne ι1 = ∠(P1 , F, F1 ) and ι2 = ∠(F2 , F, P2 ). We use the law of cosines in the triangles (F, Pi , Fi ), that is, cos ιi =

fi2 + vi2 − u2i , 2fi vi

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to prove that (1) cos ι1 = cos ι2 , for (E), (H), (EH3), and (EH4), (2) cos ι1 = − cos ι2 , for (EH1) and (EH2). Inscribed Discretization Method: In the case of the inscribed discretization, the lengths corresponding to pi = ci (t) are ⎧ 2 +2ε cos t) ai ⎨ ai (1+ε , 2 (1−ε )(1+ε cos t) 1+ε cos t , if ci is an ellipse, (ui , vi ) = a ε(1+ε2 +2ε cos t) a ε i ⎩ i if ci is a hyperbola. (1−ε2 )(ε+cos t) , ε+cos t , Equation 2 then simpliﬁes to cos ιi = −sgn(ε)

ε(s2 − 1) + (ε2 s2 − 1) cos t . (1 − ε2 )s

Circumscribed Discretization Method: Recall that intersecting tangents of a conic at parameters t1 and t2 results in ai cos τ+ (cos τ+ , sin τ+ ) , if ci is an ellipse, pi = cos τ− a+ε iε ε cos τ− +cos τ+ (cos τ+ , sin τ+ ) , if ci is a hyperbola. where τ− =

t2 −t1 2

and τ+ =

t1 +t2 2 .

Then the corresponding lengths simplify to

⎧ √ a (1+ε2 +2ε cos t0 )(1+ε2 +2ε cos t1 ) ⎪ ai ⎪ , , ⎨ i 2 cos τ− +ε cos τ+ (1−ε )(cos τ− +ε cos τ+ ) (ui , vi ) = √ 2 2 a ε (1+ε +2ε cos t0 )(1+ε +2ε cos t1 ) ⎪ ε ⎪ , , ε cos τ ai+cos ⎩ i τ (1−ε2 )(ε cos τ +cos τ ) −

−

+

+

if ci is an ellipse, if ci is a hyperbola.

Equation 2 then simpliﬁes to cos ιi = −sgn(ε)

ε(s2 − 1) cos τ− + (ε2 s2 − 1) cos τ+ . (1 − ε2 )s

Note that in case of scaled conics, this implies cos ι1 = cos ι2 . Furthermore, in case of reciprocal conics, cos ι1 = − cos ι2 if ε > 0 and cos ι1 = cos ι2 if ε < 0. Furthermore, cos ι2 = −1 for s = smax . This yields the claimed result. Lemma 3. Given two compatible parabolas with scale factors ai , namely the cases (P1)–(P3), define for s ∈ [0, smax ] ⎧ ⎪ ⎨(−ai s, 0), Fi = fi + (ai s, 0), ⎪ ⎩ (±ai s, 0),

in case (P1), in case (P2), in case (P3),

where

smax =

p1,x − f1,x + dist(p1 − f1 ) , a1

In case of the inscribed discretization method, smax = 1. Then there exist point locations P1 and P2 such that {F, F1 , F2 , P1 , P2 } is a configuration of the linkage L.

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Proof. Case (P1): Consider the two adjacent points p1 and p2 corresponding to a parameter t for the inscribed discretization method or the intersection of a1 1| tangents at t − δ and t + δ. Note that in both cases |p |p2 | = a2 . During rotation in counterclockwise direction by ι−t, the new points are Pi = |pi | (cos ι, sin ι). Thus the ratio of the x-coordinates remains constant and equal to aa12 . This implies that we can ﬁnd a one-parameter family of conﬁgurations in which translations in x-directions of points Fi has constant ratio aa12 , implying the claimed result. Case (P2) and (P3): These cases can be reduced to constructing an appropriate parallelogram or trapezoid, respectively. We now combine the above claims to show rigid foldability. Theorem 1. The inscribed and circumscribed discretization methods result in rigidly foldable crease patterns. In particular, the locations of the cone apices remain collinear during folding. Proof. For a ﬁxed value of s, we consider pairs of compatible conics and use the previous two lemmas to ﬁnd the conﬁgurations {F1 , P1 , P2 , F2 } and {F1 , P1 , P2 , F2 } of two neighboring ruling polylines {f1 , p1 , p2 , f2 } and {f1 , p1 , p2 , f2 }. To obtain the location of the 3D ruling polylines, we position the incident planes such that dist(p1 , p1 ) = dist(P1 , P1 ). From the two possibilities, we chose the one which preserves the correct MV assignment of the ruling polylines. If the shared focal point is real, it follows that triangles (F, Pi , Pi ) are congruent to (f , pi , pi ). If on the other hand the shared focal point is a point at inﬁnity, the horizontal distance between F and F is the same as the horizontal distance between f and f (in case (P3) it is zero). Therefore, both cases imply that {P1 , P2 , P2 , P1 } is a planar quad congruent to {p1 , p2 , p2 , p1 }. Doing this construction for all pairs of neighboring ruling polylines results in a folded state of the crease pattern. Note that this construction allows combinations of multiple pairs of compatible conics for same s with an analogous argument for the compatibility of opening angles. To obtain a rigid folding motion, we let s vary in [smin , smax ], where smin is 0 in the parabolic cases (P1)–(P3), and 1 otherwise. Moreover, smax is the minimum of all smax of the individual rule line polylines.

4

Local Flat Foldability

For a degree-4 vertex to be ﬂat foldable, the sum of opposite angles must be π. It is known that when folding a ﬂat-foldable degree-4 vertex, the fold angles along the opposite creases with same MV assignment are the same. Thus, if a pattern consist of ﬂat-foldable degree-4 vertices, the fold angles along polylines with same MV assignment are constant. As the rule lines of smooth conic curved creases are reﬂected on the crease, the resulting folded creases will have constant fold angle. In this section, we show

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Fig. 7. A vertex in the inscribed discretization is locally ﬂat foldable, if the discretized creases are reﬂected on the normal of the conic. Note that this allows a decomposition in three pairs of equal angles around a vertex as in Lemma 1.

when this property is preserved in our studied discretizations, that is, when all vertices are ﬂat foldable. We show that carefully chosen parameter samplings in the inscribed discretization method result in locally ﬂat-foldable crease patterns. Furthermore, we show that the circumscribed discretization gives ﬂat-foldable vertices for any parameter sampling. 4.1

Locally Flat-Foldable Inscribed Discretizations

Construction of Flat-Foldable Vertices of a Single Crease. It is known, that the bisectors of opposite creases in a ﬂat-foldable degree-4 vertex are perpendicular, and thus are referred to as axes of the ﬂat-foldable vertex. In case of the inscribed discretization method, it follows from the reﬂection property of the conic that the axes of a ﬂat-foldable inscribed vertex are the tangent and normal of the conic; see Fig. 7. Lemma 4. A vertex p = c(t) of an inscribed discretization of a conic section c is flat-foldable if and only if its axes are the tangent t(t) and normal direction n(t) of the conic. For a single crease, we can therefore construct a set of parameters that result in ﬂat-foldable vertices in the following way. Initially, we choose two sampling parameters. Reﬂecting the connecting lines on the incident conic normal and choosing the intersection with the conic as the next vertex, makes the central vertex ﬂat-foldable. Iterating this construction results in a discretization with ﬂat-foldable vertices. Propagation of Flat-Foldable Vertices to Compatible Creases. It remains to show that ﬂat-foldability of vertices propagates to compatible conics. In case of scaled or translated conics, the discretized creases of neighboring conics are parallel. Thus, opposite enclosing angles still sum up to π. In case of reﬂected parabolas with ﬂat-foldable vertices, the reﬂected vertices are also still ﬂat foldable. The only non-trivial case is the combination of conics with reciprocal eccentricities. We therefore prove:

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Fig. 8. Notation used in the proof of Lemma 5 (case (EH1)).

Lemma 5. A vertex in an inscribed discretization of two conics with reciprocal eccentricities is flat-foldable if and only if its neighboring vertex on the other conic is flat-foldable. Proof. Refer to (Fig. 8). Let 0 < |ε| < 1 and let the two compatible conics be an ellipse c1 and a hyperbola c2 with eccentricity ε and 1ε , respectively. Assume without loss of generality that both conics are parametrized as in Eq. (1) and denote their normal vectors by ni . Let pi = ci (t) for t ∈ (tmin , tmax ) be two adjacent vertices of the crease pattern. For δ+ > 0 and δ− > 0 such that t ± δ± ∈ (tmin , tmax ), we denote the neighboring vertices on the crease by pi,± = ci (t±δ± ). In the following, let cos βi,± := cos(∠(ni (t), pi,± − pi )). We show p1 is ﬂatfoldable if and only if vertex p2 by showing cos β1,− = cos β1,+

⇐⇒

cos β2,− = cos β2,+ .

Using the parametrization in (1), we obtain cos β1,± =

cos β2,±

√ 2(1 + ε cos t) sin δ2±

(1 + ε2 + 2ε cos t)(2 + ε2 (1 + cos δ± ) + 2ε(cos t + cos(t ± δ± ))) √ 2ε(ε + cos t) sin δ2± . =

(1 + ε2 + 2ε cos t)(2ε2 + 1 + cos δ± + 2ε(cos t + cos(t ± δ± )))

Note that cos β1,− = cos β1,+ is equivalent to B1,− = B1,+ , where sin δ2± , B1,± =

2 + ε2 (1 + cos δ± ) + 2ε(cos t + cos(t ± δ± )) and cos β2,− = cos β2,+ is equivalent to B2,− = B2,+ , where B2,± =

sin δ2± 2ε2 + 1 + cos δ± + 2ε(cos t + cos(t ± δ± ))

.

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Note that Bi,± > 0, and thus Bi,− = Bi,+ if and only if 1 2 B1,+

−

1 2 B2,+

=

1 2 B1,−

−

1 2 B2,−

1 2 Bi,−

=

1 . 2 Bi,+

Because

= 2(1 − ε2 ),

we conclude that cos β1,− = cos β1,+ ⇐⇒

1 1 1 1 = 2 ⇐⇒ 2 = 2 ⇐⇒ cos β2,− = cos β2,+ . 2 B1,− B1,+ B2,− B2,+

4.2

Circumscribed Discretization Method

With Lemma 1, the angle around a vertex in the circumscribed discretization method decomposes into three pairs of equal angles, that is, 2π = 2α + 2β + 2γ. Note that for discrete natural rule lines the sum of opposite angles is α+β+γ = π, resulting in ﬂat-foldable vertices for any sampling of points of contact. This trivially holds for all compatible conics.

References 1. Demaine, E., Demaine, M., Huﬀman, D., Koschitz, D., Tachi, T.: Conic crease patterns with reﬂecting rule lines. In: Origami7 : Proceedings of the 7th International Meeting on Origami in Science, Mathematics and Education (OSME), vol. 1, pp. 573–590 (2018) 2. Evans, T.A., Lang, R.J., Magleby, S.P., Howell, L.L.: Rigidly foldable origami twists. Origami 6(1), 119–130 (2015) 3. Fuchs, D., Tabachnikov, S.: More on paperfolding. Am. Math. Monthly 106, 27–35 (1999) 4. Koschitz, D.: Computational Design with Curved Creases: David Huﬀman’s Approach to Paperfolding. Ph.D. thesis, Massachusetts Institute of Technology (2014) 5. Lang, R.J., Nelson, T., Magleby, S., Howell, L.: Kinematics and discretization of curved-fold mechanisms. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 5B. American Society of Mechanical Engineers (2017) 6. Tachi, T.: Composite rigid-foldable curved origami structure. In: Proceedings of Transformables (2013) 7. Tachi, T., Hull, T.C.: Self-foldability of rigid origami. J. Mechanisms Robotics 9(2), 021008 (2017) 8. Wunderlich, W.: Raumkurven, die pseudogeod¨ atische Linien eines Zylinders und eines Kegels sind. Compos. Math. 8 (1950) 9. Wunderlich, W.: Raumkurven, die pseudogeod¨ atische Linien zweier Kegel sind. Monatsh. Math. 54, 55–70 (1950)

Transformations Between Developments and Perspectives of Three and Four Dimensional Cubes Takafumi Otsuka and Akihiro Matsuura(B) Tokyo Denki University, Ishizaka, Hatoyama-Machi, Hiki 350-0394, Japan [email protected], [email protected]

Abstract. In this paper, we present methods of dissection and transformation between developments and perspective models of three and four dimensional cubes. In the three dimensional case, several types of developments of a cube are divided into polygonal pieces and transformed to the perspective model with equal area, where some transformations are possible with hinged dissection. In the four dimensional case, two methods are presented for dissection and transformation between the development of a hypercube and its perspective model, where one method is based on hinged dissection using subcubes and the other is based on dissection using tetrahedra. We materialized some of the dissections using planar sheets of EVA resin for the cube, and paper and ABS resin for the hypercube. We finally confirmed that transformations based on these dissections are done smoothly by physical manipulation. Keywords: Cube · Hypercube · Development · Perspective · Transformation · Dissection · Graphic education · Puzzle

1 Introduction Given two polygons of equal area, it was first posed by Wallace in the early 19th century [1] if one of the polygons can be divided into small non-overlapping sub-polygons and recombined into the other polygon. This problem was positively solved by Wallace [2], Bolyai [3], and Gerwien [4], so the result is called the Wallace-Bolyai-Gerwien Theorem, or Bolyai-Gerwien Theorem using two of the contributors. For the problem of dividing and recombining sub-polygons with the constraint of using hinges on the vertices, Dudeney [5] showed a method of dissecting an equilateral triangle to four pieces and transforming the triangle to the square with equal size using hinges. Recently, Abbott et al. [6] showed that any two polygons with equal size can be always dissected and transformed to each other using hinges. We note, however, that sub-polygons in such hinged dissections are not always easy to handle because the resulting pieces could be tiny or sharp and the number of pieces tends to be large. This two dimensional dissection problem was extended to the three dimensional case, and notably, Hilbert posed a question in 1900 as his third problem if any two given tetrahedra with the same base area and height and equal volume can be transformed to each other by dissecting © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 197–208, 2023. https://doi.org/10.1007/978-3-031-13588-0_17

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and recombining the divided solids. This problem was solved negatively by Dehn [7]. We note again that for some specific pairs of solids, there can exist (hinged) dissections that enable transformation between the given solids. In our research, we focus on the three and four dimensional cubes and consider dissections and transformations between their developments and perspective models, both of which are representative ways for expressing objects in lower dimensions. For the cube, we consider several types of developments and the perspective model with equal area and first present two symmetric types of dissections for the development containing a cross that use 13 and 17 pieces. Then, we present hinged dissections for three types of developments that use six or seven pieces. For the four dimensional cube, i.e., the hypercube, we show two methods for dissections and transformations of the development and the perspective model with equal volume. In the first method, cubes in the development are divided into subcubes and are transformed to the perspective model only using rotations along hinges on some edges. In the second method, the perspective model is divided into identical tetrahedra and transformed into the development. We materialized some of the dissections using planar sheets of EVA resin in the three dimensional case, and using material such as paper (or origami) and ABS resin in the four dimensional case. Then, we confirmed that in all of these cases, transformations can be done smoothly with physical manipulation. We expect that these actual objects are used for learning and presenting relevant geometric concepts and can contribute to geometry and graphic science education, as well as being played as geometric puzzles.

2 Preliminaries on Geometry 2.1 Two Dimensional Dissection Problem and Basic Transformation For any two polygons of equal area, it is known that we can divide one polygon into a finite number of disjoint sub-polygons and recombine them into the other polygon. This result is known as the (Wallace-)Bolyai-Gerwien theorem as stated in Sect. 1. The sketch of one of the constructive proofs is the following: We first divide a given polygon to a disjoint union of triangles by dissecting at diagonals connecting pairs of vertices. Then, each triangle is divided into four pieces by dissecting at the parallel and perpendicular lines of the lower base segment and are recombined into a rectangle by turning the upper two triangles upside down and moved to the lower part of the parallel line. Further, it is known that a rectangle of arbitrary aspect ratio can be dissected and recombined into a square with equal area as shown in Fig. 1. By repeating this operation and also merging two squares into a square without changing the total area, we finally obtain the square with area equal to the original polygon. Similar operations are applied to transform the target polygon into the same square, which implies that any two polygons with equal area are transformed to each other. Here, we note that the number of sub-polygons needed in this proof is not always the minimum and the resulting dissected pieces are not always easy to manipulate physically. Therefore, there is room for finding explicit dissections with reasonable numbers of pieces that are suitable for materialization and physical manipulation.

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Fig. 1. Dissection of a rectangle and transformation into a square with equal area.

2.2 Developments and Perspectives of Three and Four Dimensional Cubes From now on, we assume that the length of one side of the three and four dimensional cubes is always one; namely, we focus on unit (hyper) cubes. A three dimensional cube is known to have 11 types of developments, among which we use the four types shown in Fig. 2, which we call Types A, B, C, and D. The Type A development is symmetric with a cross. Other three types have the common property that they can be transformed into a rectangle of size 2 × 3 by making only two rotations around hinges attached on vertices. As a perspective of a cube, we use the front view of a square surface. In order to match the area of the perspective with the development, which is six, the side length √ of the outer square of the perspective is set to 6. Furthermore, we set the view point so that the side length of the inner square is one as shown in Fig. 3. As for the development and perspective model of the unit hypercube, the development we use is shown in the left-hand side of Fig. 4. Since it has volume eight, the side length of the outer cube of the perspective model is set to two. We further set the inner cube to have side length one.

Fig. 2. Four types of developments of a cube (types A, B, C, and D).

Fig. 3. Perspective model of a cube.

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Fig. 4. The development and perspective model of a hypercube.

3 Transformations of Developments and Perspective of a Cube 3.1 Symmetric Dissections for Type a Development In this section, we present two methods for dissecting the Type A development and the perspective square in Fig. 3, having rotational symmetry. Consider the cross contained in the Type A development shown in the upper left-hand side of Fig. 5. By √ rearranging the four small triangles in the figure, we obtain√ a square with side length 5. We divide 5/2 this square into four squares with side length and dissect and transform each of √ 6 ± 1 /2 using the basic dissection method them into the rectangle with side lengths in Fig. 1 in √ reverse order. Lastly, we arrange these rectangles into the square ring with side length 6 as shown in the bottom of Fig. 5 and insert the unit square in the center of the ring, which results in the perspective model of the cube. This dissection requires 4 × 4 + 1 = 17 pieces in total. The dissected cross, the intermediate square, and the square ring with the 17 pieces are summarized in the upper part of Fig. 6. We can further reduce the number of pieces to 3 × 4 + 1 = 13 by merging the four pairs of adjacent pieces in the ring as shown in the lower part of Fig. 6.

Fig. 5. Cross in type A development of a cube and transformations into the square ring.

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Fig. 6. Two types of dissections of crosses, intermediate squares, and square rings.

3.2 Hinged Dissections for Type B, C, and D Developments We first transform these developments into rectangles of size 2 × 3 and then transform them into perspective squares using hinges as follows. First, we draw a circle of diameter

Fig. 7. Dissection of 2 × 3 rectangle and dissected developments of Types B, C, and D.

Fig. 8. Transformation of hinged type B development into the perspective square.

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√ 6 with the common center of the aforementioned rectangle and dissect the rectangle by the diameter as shown in the upper left-hand side √ of Fig. 7. We further divide the rectangle by the two perpendicular lines of length 6/2. It is known that the resulting four polygonal pieces can be rearranged into a square [8]. When the Type B development is used, the upper-left and lower-right squares of the rectangle are rotated to transform into the Type B development. If we simply do it, the number of required pieces for the Type B development is nine. In order to reduce this number, we move the original three lines of dissections horizontally to the left to match the endpoint of the lowerright oblique line to one of the vertices of the lower-right square as shown in the lower left-most side of Fig. 7, which reduces the number of pieces to six. The same type of dissections are also applied to Types C and D developments and the resulting dissections are shown in the right-hand side of Fig. 7. Here we note that the gray area of Type C is divided into two parts and the number of pieces becomes seven, while the number of pieces for Type D is six. Finally, Fig. 8 demonstrates the transformation from the Type B development to the perspective square using hinges. 3.3 Materialization and Demonstration We implemented the presented dissections using planar sheets made of EVA resin with thickness 2 mm. Figure 9 shows the result for the Type A development using 17 pieces. Figure 10 shows the result for the Type B development using 6 pieces, where the perspective square is drawn on the back. We note that the physical hinges have not been attached at this point although rotated under restraint.

Fig. 9. Demonstration of transformation of type A development to the perspective square.

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Fig. 10. Demonstration of transformation of type B development to the perspective square.

4 Transformations of Development and Perspective of A Hypercube In this section, we present two methods of dissection and transformation between the development and perspective model of a hypercube in Fig. 4. The first one is based on small subcubes and the second one is based on tetrahedra as primitive solids. 4.1 Dissection and Transformation Based on Small Subcubes We first show basic dissections for the development and the perspective model of the hypercube. As for the development, we first consider the partial solid shown in the leftmost side of Fig. 11(a), which consists of seven unit cubes. We divide this solid into eight equal parts by the three horizontal and vertical planes as shown in the center of Fig. 11(a). Here, each part consists of seven cubes with side length 1/2. Similarly, we divide the perspective model into eight equal parts by the three perpendicular planes as shown in Fig. 11(b), where each part also consists of seven cubes. The key property of these two types of parts with seven cubes is that they can be transformed to each other by rotating three cubes using hinges on the edges as shown in Fig. 12.

(a) Dissection for the development

(b) Dissection for the perspective model

Fig. 11. Basic dissections for the development and perspective model of a hypercube.

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Fig. 12. Hinged transformation between solids with seven cubes.

We achieve the transformation between the development and perspective model based on these basic dissections along with some additional hinges and reattachment of the rightmost unit cube of the original development. The whole transformation is shown in Fig. 13, where 3D models and animation were created using Fusion 360. The cubes were printed using ABS resin, where small cubes have side length 2 cm and some of them have neodymium magnet on their faces to make them stick to the neighboring cubes. As shown in Fig. 14, it is confirmed that the development of the hypercube is transformed into the perspective model by actual manipulation.

Fig. 13. Transformation from the development of a hypercube to the perspective model.

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Fig. 14. Demonstration of transformation using 3D-printed cubes.

4.2 Dissection and Transformation Based on Tetrahedra Outline of Transformation. Here we present a method of dissection and transformation of the perspective model of a hypercube to its development shown in Fig. 4. The outline of the method is illustrated in Fig. 15. The perspective model is divided into six regular quadrangular frustums (or truncated square pyramids) with the inner unit cube. Each quadrangular frustums is then transformed to a house-like solid that consists of a unit cube and a regular quadrangular pyramid. The six house-like solids are combined at their triangular faces, and finally the development of a hypercube is completed by attaching the unit cube at the right end. We found out that the whole transformation is possible using only one type of tetrahedron, which we call α. This tetrahedron is obtained by vertically dissecting the square pyramid of the house-like solid into four identical parts as shown in Fig. 16, where the tetrahedron A-BCH represents α whose volume is 1/24. Since the volume of the house-like solid is 7/6, the house-like solid is constructed using 28 tetrahedra α s. On the other hand, the regular quadrangular frustum of the perspective model has upper and lower squares of side lengths one and two, respectively, with height 1/2; therefore, its volume is 7/6. The total volume of the development and perspective

Fig. 15. Outline of transformation from the perspective to the development of a hypercube.

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model of the hypercube is eight and the total number of required tetrahedra α s is 28 × 6 + 24 = 192. In the sequel, we show a method for transforming the perspective model to the development using hinged dissection using tetrahedra α s as primitive solids.

Fig. 16. The square pyramid with tetrahedron α (A-BCH) and regular quadrangular frustum.

Transformation from Quadrangular Frustum to House-Like Solid. We first show how to construct a quadrangular frustum of the perspective model using tetrahedron α s. In the construction, we frequently use the following three connected units of α s. The first one, which we call β, is made by connecting two α s at the triangle ABH in Fig. 16. The second one, which we call γ , is made by connecting two α s at the triangle BCH. The third one, which we call δ, is made by connecting two β s using hinge. These solids that are made of paper are shown in Fig. 17. A sequence of construction of the regular quadrangular frustum is shown in Fig. 18. First, three δ s are connected and fixed at the triangular faces. Then, two β s are connected by the hinge shown with red lines. In the lower left figure, unit γ is connected and fixed to the triangular face at the shaded area. In the lower left figure, one unit γ is fixed with the triangular surface shown in the shaded area. Then, the solid with four sequentially connected α s with three hinges is connected at the left end of the seventh image in Fig. 18. Finally, the regular quadrangular frustum is completed by turning the pyramid made by four α s with one γ upside down, placing it on top, and connecting with the horizontal hinge shown in the upper part of the last image. Transformation from the regular quadrangular frustum to the house-like solid is illustrated in Fig. 19. The solid consisting of four α s and γ is first raised and recombined into a different shape in the first row of the figure. In the second row, another set of four α s is raised and combined with the first four α s to make the roof part of the house-like solid. In the last row, units β and δ on the right are raised, rearranged, and connected to the left part to complete the house-like solid. Finally, Fig. 20 summarized the whole transformation.

Fig. 17. Tetrahedron α and three units β, γ , and δ.

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Fig. 18. Construction of the regular quadrangular frustum.

Fig. 19. Transformation from the regular quadrangular frustum to the house-like solid.

Fig. 20. Transformation from the perspective to the development of a hypercube.

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5 Concluding Remarks In this paper, we presented several methods for dissecting and transforming the developments and perspective models of three and four dimensional cubes, and confirmed their feasibility by implementing with several materials such as paper and EVA and ABS resins. We expect these objects to help learners and puzzle lovers to engage in making and changing shapes by themselves and understand related geometric concepts. We also expect educators to use the objects in their teaching concepts of developments and perspective views in a tactile manner.

References 1. Fredkerson, G.N.: Dissections: Plane and Fancy. Canbridge University Press (1997) 2. Wallace, W. (ed.).: Elements of Geometry. Bell & Bradfute, Edinburgh, 8th edition (1831) 3. Bolyai, F.: Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi. Typis Collegii Refomatorum per Josephum et Simeonem Kali, Maros Vásárhely (1832–1833) 4. Gerwien, P.: Zerschneidung jeder beliebigen Anzahl von gleichen geradlinigen Figuren in dieselben Stücke. J. Reine Angew. Math. (Crelle’s J.) 10, 228–234 and Taf. III (1833) 5. Dudeney, H.E.: Puzzles and Prizes. Weekly Dispatch. The puzzle appeared in the April 6 issue and the solution appeared on May 4 (1902) 6. Abbott, T.G., Abel, Z., Charlton, D., Demaine, E.D., Demaine, M.L., Kominers, S.: Hinged dissections exist. In: Proceedings of 24th Annual Symposium on Computational Geometry (SCG 2008), pp. 110–119 (2008) 7. Dehn, M.: Über den Rauminhalt. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, 345–354 (1900). Later published in Math. Ann. 55, 465–478 (1902) 8. Decomposite Congruent. http://www.kaynet.or.jp/~kay/misc/column/decompcongr.html. Accessed 28 Feb 2022. (in Japanese)

Perfect Circles, Amicable Triangles and Some of Their Properties – Angles Equality and About Two New Constants in the Triangle Michael Sejfried(B) ˙ Metal Union Company, ul. Zyzna 11 F, 42200 Cz˛estochowa, Poland [email protected]

Abstract. The family of perfect circles begins at the Fermat point, contains an incircle, and ends at the circumcircle of the reference triangle ABC. The relevant drawing shows this structure having many surprising properties. On the other hand, when looking for loci of vertices of bundles of equal angles (two or more), we find interesting curves. This, in conjunction with the perfect circles and amicable triangles, allows to determine two new constants in any triangle. One of them is the product of the three proportions s, t and u (like in Routh’s theorem), and the other - the angle occurring in each triangle at least 18 times. Keywords: Angles equality · Circle · Fermat · Constant · Triangle

1 About “Perfect Circles” and “Amicable Triangles” Perfect circles and amicable triangles are based on the 15-cevian structure in any triangle ABC and was already presented in different aspects at ICGG 2012 in Montreal [3] and then in Innsbruck [4], Beijing [5] and Sao Paulo [6] two years ago. Figure 1 illustrates this construction. What is it and how is it created? Let’s take a triangle ABC and a circle inside it and draw from each vertex lines tangent to this circle. The intersections of both upper tangents based on same side of the triangle ABC and similarly of both lower tangents determine the lines (A2 A3 , B2 B3 , C2 C3 ), which must become a cevian by adjusting the position of the circle. There is enough to do this for two cevians. So we get the perfect circle with given radius rx in the triangle ABC. Two possible cevianic triangles (their sides belong to cevians) KLM and K1 L1 M1 inscribed into the perfect circle become amicable triangles. Routh’s Theorem [1, 2] describes the dependence of the areas of KLM and K1 L1 M1 triangles on the area of the reference triangle ABC. The new properties of this structure are described in the following short study, which also includes a reference to the equality of the angles at which we observe the straight line segments from a specific point Z. In the triangle ABC we have three bundles of cevians, but only one of each bundle is the main cevian (marked pink) AA1 , BB1 or CC1 . So let’s take one of the main cevians – in this case BB1 - and let’s choose such a position for it that this cevian lies between Fermat point and Symmedian point. It is necessary and sufficient to determine the perfect circle in a completely unambiguous manner, both © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 209–218, 2023. https://doi.org/10.1007/978-3-031-13588-0_18

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in terms of its center Ox and radius rx . The lines originating from three vertices of the triangle ABC and tangent to perfect circle mark three pairs of intersections on the sides a, b and c of this triangle. Now, we need other three pairs of intersections formed by the extended sides (cevians) of amicable triangles. It is worth noting that both amicable triangles lie always inside the reference triangle ABC, and for rx = R they coincide with it.

B

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To simplify our considerations and without prejudice to their generality, let one of the sides (b) of the triangle ABC lie on the axis OX, and the vertex A of this triangle at the origin of coordinates. In the further part of the study, we will mainly deal with all sections lying on the side b (AT2 , T2 T, TB1 , B1 T1 , T1 T3 , T3 C).

2 Equality of Angles in the Bundle of Straight Lines 2.1 Equality of Two Angles Let’s choose the straight line segment AC and the point B on this segment. The proportion AB/BC is optional. Now, let’s take the point Z over this segment and consider, where should it lie to be the vertex of two equal angles: AZB and BZC (Fig. 2).

Z

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Also here let the points A, B and C lie on the axis OX, and the point A at the origin of coordinates. The locus of point Z is a circle with center OB and radius OB B1 = r0 . So, we get the following equations: r0 =

|AB| · |BC| |AB| − |BC|

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2.2 Equality of Three Angles Let’s choose now an additional point D on the line AC on the right side of C. The locus of the point Z for two equal angles m, then at least one linear parastichy contains more than one point. Therefore, a spiral pattern on a Voronoi diagram may be identiﬁed by recovering a sequence of generator points using the area of Voronoi cells in each linear parastichy. Firstly, we consider the distance between consecutive points in each linear parastichy. Assume that there are m linear parastichies. Let Si be i-th linear parastichy and pi,j be the generator point of i-th linear parastichy at position j-th , where j = 1, · · · , k, i.e., pi,1 , · · · , pi,k lie on Si . In the polar coordinates form, we note that, i.e. pi,k = (rk , θ + 2kπ). Since pi,1 , · · · , pi,k are on the same line, then θ = θ + 2kπ for some k and θ ∈ {1◦ , · · · , 180◦ }. Therefore, the distance between pi,k and pi,k−1 is 2 + r2 d(pi,k , pi,k−1 ) = ri,k i,k−1 − 2ri,k ri,k−1 cos(2π). This implies that the distance between consecutive points in each linear parastichy is rk − rk−1 , which depends on the same divergence angle θ. Hence, we get that in each linear parastichy, the Euclidean distances between generator points are equal.

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Fig. 2. Construction of a ray emanating from the origin

By the construction of a Voronoi diagram generated by phyllotaxis process, perpendicular bisectors of Voronoi generators exist between Si and Si−1 , Si and Si+1 where i = 1, · · · , m, and S0 := Sm , Sm+1 := S1 . Hence, the Voronoi vertices sets Vi,i−1 and Vi,i+1 contain points of Voronoi regions generated by generators on Si and Si−1 , and Si and Si+1 , respectively. Then, we can construct a ray emanating from the origin by considering Voronoi vertices which give the minimum angle and maximum angle among Vi,i−1 and Vi,i+1 . If lines that pass those points are constructed, triangles are obtained, including trapezoids, whose heights are equal, as shown in Fig. 2. Hence the area of the trapezoids arranged on the sequence of points on Si are compared using the properties of inscribed rectangles to conclude that the sequence of areas of Voronoi polygons on a linear parastichy is increasing. This comparison leads to the following theorem: Theorem 1. The sequence of areas of Voronoi polygons in each parastichy starting from its origin is increasing sequence. The proof of Theorem 1 follows from the construction of the Voronoi diagram and the convergence of Voronoi cells for Archimedean spiral. Then, the smallest area of the Voronoi diagram is likely would be expected to be considered as the generative spiral’s starting point. According to our observations, the smallest area of the Voronoi diagram holds when the divergence angle θ is between 36◦ and 180◦ . However, when the divergence angle θ is smaller than 36◦ , the procedure would be slightly more complicated. This characterization will be applied to our algorithm in Sect. 4. The observation is consistent with the results of Yamagishi et al. [16], which investigated the Voronoi diagram of an Archimedean spiral. When the points were generated by the phyllotaxis sequentially, the areas of Voronoi cells tended to increase. However, the precise property is still unknown.

4

Proposed Algorithm

Let V = {V (p1 ), . . . , V (pn )} be a given Voronoi diagram with respect to the generator set G = {p1 , . . . , pn } such that pi = (xi , yi ), i = 1, . . . , n. We also assume that A(V) = {Area(V (p1 )), . . . , Area(V (pn ))}. We would recover the Archimedean spiral using only information of points on the plane. Hence, we need

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to recover the following information: (1) the origin point of the Archimedean spiral; (2) information about the linear parastichies by detecting those points are on the line; (3) the sequence of points on subspiral, which are lines. This information is enough to recover the sequence of points on the main Archimedean spiral curve. Hence, the algorithms to recover the required information are provided as follows: 4.1

Finding the Beginning of Subspiral

Based on the results of Yamagishi et al. [16] and the current study’s observations, we consider the minimal area of the Spiral Voronoi diagram to be the starting point of the Archimedean spiral. This assumption does not remain true when the divergence angle θ is smaller than 36◦ , however. As a result, we examine the average angle of nearby the generating point. For the ﬁrst algorithm, we found the starting point of the spiral curve. In this algorithm, we ﬁrst considered the smallest area of Voronoi cells. However, it is necessary to verify whether the choice is correct. Remark that the pattern is generated radially with equal angles. If pi is the starting point, then the ray emanating from pi to the generators should divide a plane equally. Hence, the radial angle between those rays when we pick pi was checked. If pi is not the solution, we alternatively choose the next candidate by the larger area to the minimum area. Based on the observation, this trial and error will not repeat many times. The idea is represented as Algorithm 1. Algorithm 1. Detecting the starting point of the spiral. Input: A(V ) of the given Voronoi diagram V, G = {p1 , . . . , pn } a set of Voronoi generators Output: The starting point of the Archimedean spiral pi 1: Set P = ∅ 2: Compute bi := min A(V) 3: Choose the generator pi of bi 4: Choose neighboring generators of pi ; says pk and pk → P → 5: Choose an arbitrary pk ∈ P . Draw a ray − p− i pk − − → 6: Sweep the ray pi pk around the point pi clockwise |P | 7: Detect the point pl ∈ P and arrange it to the sequence {pl }l=1 8: Calculate ∠pk pi pl , ∠pl pi pi+1 , . . . , ∠p|P | pi pk 9: Compute m := min{∠pk pi pl , ∠pl pi pi+1 , . . . , ∠p|P | pi pk }; and 10: M := max{∠pk pi pl , ∠pl pi pi+1 , . . . , ∠p|P | pi pk } 11: if m | M then 12: pi is the origin of the Archimedean spiral 13: else 14: choose bi := min(A(V) − {bi }) 15: go to step 3 16: end if

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Detecting Linear Parastichies and Finding the Divergence Angle θ

After we obtained the beginning point of the Archimedean spiral using Algorithm 1, we use the information to detect the subspiral which are rays. However, we still need to recognize the number of rays to acquire the information of the divergence angle. This algorithm, has been divided into two main parts. We categorize the set of points in each linear parastichy by determining the slope. The number of linear parastichies is then used to determine the divergence angle θ. Due to the phyllotaxis processes, the point one considers raying on the radial line emanating from the spiral origin pi obtained from Algorithm 1. However, the number of slopes are not able to classify the diﬀerent radial line completely. So, we need to consider process into the following patterns: the even number of slopes that generate double radial lines, an odd number of slopes that generate odd radial lines, and an odd number of slopes that generate double radial lines. It is concluded in Algorithm 2. Theorem 1 conﬁrms that there exists− min{Area(V(pj,x ))} in Sj for some x and j = 1, . . . , t and the generators which are in the subspiral Si are arranged using an ascending sequence of Voronoi cells’ area. Moreover, by Algorithm 2, we obtain the sequence of Si for all i = 1, . . . , q. This implies that we can recover the Archimedean spiral curve C using the subspiral line. Algorithm 2. Recognition of the rays of Voronoi generators and divergence angle θ. Input: The initial point pi , Voronoi generator set G − {pi } Output: The set of subspiral S, the divergent angle θ, the number of radii q, The subscript of subspiral v Remark : The initial point pi is acquired from Algorithm 1 1: Set M = ∅ 2: for j = 1 to |G − {pi }| do 3: Calculate mj as the slope of pi pj and mj → M 4: end for 5: Cluster M to M1 , . . . , Mt sets with respect to t diﬀerent of slopes. 6: Set P1 , P2 , . . . , Pt = ∅ 7: if mj ∈ Mk then 8: pj → P k 9: end if 10: if 2—t then 11: q = 2t and θ = 360◦ /q 12: Set S1 , . . . , S2t = ∅ 13: for k = 1 to t do 14: Construct pm pn such that pm , pn ∈ Pt 15: Construct a perpendicular line lp⊥ to pm pn 16: Separate pα ∈ Pk to Sk and S2k with respect to lp⊥ 17: end for 18: Let pj,x be the point in subspiral Sj that gives the minimum area

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19: Compute aj = d(pj,x , pi ) for some x and j = 1, . . . , q 20: Sort by lowest to highest value of {a1 , . . . , aq } → {ab1 , . . . , abq } 21: Set v = bv for v ∈ {1, . . . , q} 22: else if 2 t and t = 5, 9, 15 then 23: Set Sk = Pk for all k 24: q=t 25: Construct Delaunay triangulation of D(G) 26: Choose pα ∈ Pk1 , for some k 27: Choose pβ ∈ Pk2 and pγ ∈ Pk3 such that k = k2 = k3 28: and pα , pβ , pγ are connected in the D(G) −−p→ · − −→ −−→ −−→ 29: Compute θ = cos−1 [p i β pi pα ]/|pi pβ ||pi pα | or − − → → −1 −−→ −−→ 30: θ = cos [pi pα · pi pγ ]/|pi pα ||− p− i pγ | ◦ 31: q = 360 /θ 32: Let pj,x be the point in Sj that give the minimum area 33: Compute aj = d(pj,x , pi ) for some x and j = 1, . . . , q 34: Sort by lowest to highest value of {a1 , . . . , aq } → {ab1 , . . . , abq } 35: Set v = bv for v ∈ {1, . . . , q} 36: else 37: Set Sk = Pk for all k 38: Let pj,x be the point in Sj that give the minimum area 39: Compute aj = d(pj,x , pi ) for some x and j = 1, . . . , q 40: Sort by lowest to highest value of {a1 , . . . , aq } → {ab1 , . . . , abq } 41: Set v = bv for v ∈ {1, . . . , q} 42: Choose pα ∈ Sk1 and pβ ∈ Sk1 +1 for some k1 −−p→ · − −→ −−→ −−→ 43: Compute θ = cos−1 [p i β pi pα ]/|pi pβ ||pi pα | ◦ 44: q = 360 /θ 45: end if

4.3

Recovering the Archimedean Spiral Curve C

We use this algorithm to verify our approach. The set of subspirals is then obtained S, which is already sorted the sequence and the divergence angle θ. As a consequence, we can recover the Archimedean spiral curve by phyllotaxis divergence process (generated points from the beginning point pi with a constant angle θ). Then we compare with the given set G by computing d(pi , pi ) for all i in order to know the error of our algorithm.

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Algorithm 3. Recovering the Archimedean spiral curve. Input: The set of subspiral S which is already labeled, The divergence angle θ, G = {p1 , . . . , pn } a set of Voronoi generators, the number of radii q Output: A sequence S of points which are on an Archimedean spiral Remark : q is acquired from Algorithm 2 1: Set t as the number of points of the Archimedean spiral which is generated from the divergence θ 2: Set S = ∅ 3: for j = 1 to q do 4: Compute mi := Area(V(pj,i )) for all i 5: Sort by the minimum to maximum value of {m1 , . . . , mk } → {mα1 , . . . , mαk } 6: Set k = αk 7: Set t = kj 8: pi,j → S 9: end for 10: Let G = {p l := (lθ cos(lθ), lθ sin(lθ)), l = 1, . . . , t} 11: Compute d(pi , pi ) for all i If d(pi , pi ) = 0, for all i, we can infer that the resulting set S and the generated generators G from the solution of θ in our algorithm are the same. This is illustrated in Algorithm 3. However, due to numerical errors during computation, the algorithm are probably due to a small tolerance , i.e., d(pi , pi ) < for all i.

5

Experiments

To verify the proposed algorithm, we apply it to all of the divergence angles θ in Table 1, although we will only show it for a few cases in this section. In this experiment, we ﬁrst started to generate spiral points using a ﬁxed angle θ. Then we discard the information of θ and apply Algorithms 1–3 to the generated data. Finally, we check whether the recovered results are the same or not. 5.1

Example for the Divergence Angle θ = 30◦

Firstly, we propose the spiral Voronoi diagram which is generated by 50 points with θ = 30◦ .

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Fig. 3. The result of the divergence angle θ = 30◦ (left) generated points; (middle) the first five ascending orders area of spiral Voronoi diagram; (right) recovering of the main spiral curve

By Algorithm 1, we compute the area of Voronoi diagrams. In this situation, we obtain that the minimum area is not given at the start of the spiral. By repeating this algorithm, we discover that the third area is the best starting point. Then, using Algorithm 2, we discover that t = 6, implying that there are 12 linear parastichies which are produced by 30◦ , as shown in Table 1. We receive an Archimedean spiral curve whenever Algorithm 3 is completed, as illustrated in the (right) Fig. 3. 5.2

Example for the Divergence Angle θ = 45◦

In this situation, the starting point has the smallest area. By using Algorithm 2, we obtain that t = 4, implying that 45◦ produces eight linear parastichies, as shown in Table 1. We receive an Archimedean spiral curve whenever Algorithm 3 is completed, as illustrated in the (right) Fig. 4.

Fig. 4. The result of the divergence angle θ = 45◦ (left) generated points; (middle) the first five ascending orders area of spiral Voronoi diagram; (right) recovering of the main spiral curve

5.3

Example for the Divergence Angle θ = 72◦ and 144◦

In this case, the starting point also has the smallest area. However, Table 1 shows that some of the groups have an equal number of linear parastichies. By

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comparing the distance between the beginning point and other points in each parastichy, we obtain that the sequence of linear parastichy is not the same. As a consequence, the ascending order areas of spiral Voronoi diagrams are not the same, as illustrated in the (middle) Figs. 5 and 6. We can characterize these situations and recover an Archimedean spiral in the (right) Figs. 5 and 6.

Fig. 5. The result of the divergence angle θ = 72◦ (left) generated points; (middle) the first five ascending orders area of spiral Voronoi diagram; (right) recovering of the main spiral curve

Fig. 6. The result of the divergence angle θ = 144◦ (left) generated points; (middle) the first five ascending orders area of spiral Voronoi diagram; (right) recovering of the main spiral curve

6

Concluding Remarks

This study investigated the properties of Voronoi diagrams in which the generators were on an Archimedean spiral curve with various divergence angles. Then, geometrical properties were applied to construct algorithms to allow for the recognition of whether the given Voronoi diagram was an Archimedean Voronoi diagram. This paper focused on the degenerate case of Voronoi diagrams, where the subspirals were radial lines. Spiral patterns in which the subspirals are curves were mainly observed. Based on our study, it is possible to consider those patterns using some transformations that transform a straight line into a curve. This provides an eﬃcient approach for recovering spiral curves, which may be applied more generally.

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Acknowledgement. The first author acknowledges the support of the DPST of IPST, Ministry of Education, Thailand. This research was supported by the Thailand Research Fund (TRF), Grant No. MRG6280164. The authors are grateful to Mr John Tucker, MA in Language Testing, University of Lancaster, for kind help in the English correction.

References 1. Adler, I., Barabe, D., Jean, R.V.: A history of the study of phyllotaxis. Ann. Bot. 80(3), 231–244 (1997) 2. Browne, C., Van Wamelen, P.: Spiral packing. Comput. Graph. 30(5), 834–842 (2006) 3. Erickson, R.O.: The geometry of phyllotaxis. In: The Growth and Functioning of Leaves, pp. 53–88 (1983) 4. Fortune, S.: Voronoi diagrams and Delaunay triangulations. Comput. Euclidean Geom. 225–265 (1995) 5. Fowler, D.R., Hanan, J., Prusinkiewicz, P.: Modelling spiral phyllotaxis. Comput. graph. 13(3), 291–296 (1989) 6. Godin, C., Gol´e, C., Douady, S.: Phyllotaxis: a remarkable example of developmental canalization in plants (2019) 7. Green, P.B., Baxter, D.R.: Phyllotactic patterns: characterization by geometrical activity at the formative region. J. Theor. Biol. 128(3), 387–395 (1987) 8. Gr¨ unbaum, B., Shephard, G.C.: Tilings by regular polygons. Math. Mag. 50(5), 227–247 (1977) 9. Gr¨ unbaum, B., Shephard, G.C.: Tilings and Patterns. Courier Dover Publications (1987) 10. Jean, R.V.: Introductory review: mathematical modeling in phyllotaxis: the state of the art. Math. Biosci. 64(1), 1–27 (1983) 11. Jean, R.V.: Phyllotaxis: A Systemic Study in Plant Morphogenesis. Cambridge University Press, Cambridge (2009) 12. Klaassen, B.: How to define a spiral tiling? Math. Mag. 90(1), 26–38 (2017) 13. Rivier, N.: A botanical quasicrystal. Le Journal de Physique Colloques 47(C3), C3-299 (1986) 14. Stock, D.L., Wichmann, B.A.: Odd spiral tilings. Math. Mag. 73(5), 339–346 (2000) 15. Voderberg, H.: Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente. Jahresber. Deutsch. Math.-Verein. 46, 229–231 (1936) 16. Yamagishi, Y., Sushida, T., Sadoc, J.F.: Area convergence of Voronoi cells on spiral lattices. Nonlinearity 34(5), 3163 (2021) 17. General Ontology. General ontology XXIXB. (n.d.). http://metafysica.nl/ ontology/general ontology 29b.html. Accessed 26 Dec 2021

Media Architecture as Innovative Method of Urban Environment Organizing Olga Semenyuk(B) , Assem Issina , Rakhima Chekaeva , Zhazira Bissenova , Timur Yensebayev , Askar Kalikhin , Bayan Ozganbayeva , Madi Zhussupov , Zhansaya Ashimova , and Aida Slyamkhanova L.N. Gumilyov Eurasian National University, Nur-Sultan 010000, Kazakhstan [email protected]

Abstract. The paper considers the aesthetic characteristics of architectural objects formed using media facades, features of the created artistic images on the examples of a number of implemented projects of the Nur-Sultan city. Another area of media design research is aimed at highly developed structures of cognition and their regularities: the tendency to completeness, harmony, symmetry, simplicity. The socio-psychological mechanism for the formation and assimilation of stable aesthetic stereotypes - “archetypes” can be interpreted as fixing social experience in the subconscious. This should include the formation of style, artistic image of media design, taste. The image of the city develops in the process of its everyday perception. The authors revealed the conditions for visual perception of the media design of the architectural and spatial environment of the city. Keywords: Media architecture · Graphic · Urban environment · Media facade

1 Introduction Media architecture is a type of art whose works are created and presented by modern information and communication technologies, mainly such as video, computer and multimedia technologies and the Internet. Media architecture includes various objects (buildings, structures) with any form of information, interactive and dynamic image technologies, which are designed for information exchange of the urban community. Digital data streams invade the space of a modern city, transform it, unite with it. Australian urbanist S. McQuire writes that the modern city is a media-architectural complex that arose as a result of the spread of spatial media platforms and the creation of hybrid spatial ensembles [1]. Thus, the widespread spread of digital networks in recent decades has led to the creation of media architecture as a new type of architectural objects.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 454–464, 2023. https://doi.org/10.1007/978-3-031-13588-0_39

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2 Prerequisites and Development of Media Architecture in the 21st Century 2.1 Historical Background of Media Architecture The rapid development of efficient lighting technologies at an affordable price has led to the creation of new architectural projects in which digital media are used to dynamically transform the external shell of the building. As the media facades became a global phenomenon, researchers from various disciplines began to explore its social, spatial and technological aspects under the general term “media architecture”. Initially focusing on the design of building-scale displays integrated into a built-up environment, the term has expanded to include urban multimedia applications that go beyond on-screen technologies and those that can take on different scales (see Fig. 1). The multimedia decoration of the facades of the Grand Alatau residential complex is a colorographic accompaniment of events and events held in the busiest recreational recreation area of citizens, equally perceived from anywhere in the central city park, embankment, city square in front of the government building (where official festive events are held.) Thus, the city authorities can convey information to more residents and guests of the capital. For example, the image of the flag, coat of arms for the day of state symbols of the Republic of Kazakhstan. At the same time, the color design is dynamic, unobtrusive. It amazes the scale and causes a sense of pride among the citizens of the Republic of Kazakhstan.

Fig. 1. Media architecture of Nur-Sultan city, Grand Alatau Residential Complex

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Buildings have always been used to demonstrate cultural psyche, norms and values. In ancient Greece, sculptures surrounding the facades conveyed stories of heroism. In Gothic religious architecture, elegant jewelry conveyed a theological message. And the grandeur of Baroque architecture was aimed at demonstrating strength, triumph and wealth. Over time, the facades separated from the structural structures. They began to act as independent skins, undergoing material, formal and technological experiments. Today, architecture often boasts dynamic lighting that can change appearance. This phenomenon is known as media architecture. Some architects even turned the facades into interactive canvases for creative expression. Others have made facades display environmental data, such as weather forecasts, internal activity, or energy consumption. The emergence and development of media architecture in the 21st century can be assessed as an attempt to return to the artistic process in architectural creativity. Media is a broad concept that includes means of communication and methods of transmitting information, as well as the medium they form, in other words, media space. Media architecture in the work refers to a type of art whose works are created and presented using modern information and communication (or media) technologies, mainly such as video, computer and multimedia technologies, the Internet. The work focuses on architectural objects with media facades. Media facade refers to a display built into the architectural appearance of a building of arbitrary size and shape on its surface, which is installed on the outer or inner (for transparent facades) part of the building. The creation of media facades was made possible due to the emergence of plastic LED screens, which are used both as advertising media and architectural coverage. The display of the media facade is intended for broadcasting media data - graphics, animation, photos and videos, as well as text messages. The proliferation of digital technologies and the growth of cultural practices based on media tools determine a new form of the urban environment, which is filled with various kinds of digital displays: plasma panels-screens, information displays in public transport systems, dynamic smart surfaces integrated into the facade design. The social housing areas can be “elevated” through art and cultural projects [2]. Architecture defines the worldview of society, its aesthetic principles and aesthetic ideals. In this process, the facade occupies the most important place, being the main means of architectural communication. The forms, proportions, decor of the facade over the centuries were determined by the purpose of the architectural structure, its structural features, and the stylistic solution of its architectural image. Practice shows that at present, “pulling” on the media facade, the architectural object changes these principles. Urbanist architect, professor at the Massachusetts Institute of Technology W. Mitchell emphasized: “Architecture is no longer a plastic game of mass in the light. It now includes playing digital information in space” [3]. 2.2 Aspects of Media Architecture Development In recent years, there has been a tendency towards the development of buildings with increasingly dynamic and spectacular external display shells. These digital media installations define the new conditions for the development of architecture, as a result of which digital media technology significantly affects the aesthetic qualities of architectural objects. Having tools for impact estimation allows for optimizing the product

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design parameters to increase the potential positive impact and reduce potential negative impact [4]. The need to study the aesthetic organization of media architecture is also associated with the fact that it is becoming more and more accessible and ubiquitous element of urban space. The names of several famous architects are associated with media architecture. Especially distinguished are such masters as Toyo Ito, Hadid, Eric van Egeraat, Hening Larsen, Heitz Neumann, Carlo Ratti, Ben van Berkel, Stuard Hiv, Andrew Melton, Carlos Ferre. Thus, the ability of the media object to affect the urban environment is increased due to the use of a wide range of visual and dynamic effects. In this regard, the study of the new visual language and aesthetic characteristics of media objects is an urgent task of the theory of modern architecture. The purpose of the study is to study the aesthetic characteristics of architectural objects formed using media facades, the features of the created artistic images using examples of several implemented projects. Despite the fact that the architecture of the 21st century has stepped into a new world information field and is developing in the conditions of coexistence and fusion of reality and the virtual world, created and supported by computers and network communications, the history of architecture makes it possible to objectively evaluate many processes of our time, which, it would seem, go beyond all known frames. Analysis of many media objects shows that many aesthetic characteristics of media architecture are like the artistic means of the Gothic style, separated from modern times for many centuries. During the study, the author attempts to substantiate this position by revealing it using the example of known objects of media architecture. 2.3 Development of Media Architecture Using the Example of the Nur-Sultan City The facades of the buildings of the Nur-Sultan city are increasingly revitalized by integrated light sources. Designers are increasingly focused on the visual perception of buildings. Projects of this category demonstrate creative media facade solutions. The flexibility of image/video display is based on an array of “pixels” (location, pitch, shape, color rendering). The problem is setting up creative “pixel” components (see Fig. 2). Color design should not be random. It is important to use deep knowledge in architectural coloristics, and the peculiarities of color perception on the volumetric forms of buildings and structures. Since, first of all, the facade of the building is part of the composition of the city ensemble and the volumetrically spatial form of the building, and only then the carrier of information. In this example, thanks to the array of pixels, the shape of the central part of the building was revealed. At the same time, the lateral parts are dissonant due to the use of a spectrum that does not apply in the architecture of buildings and structures. The main aesthetic means of the media center in the Nur-Sultan city were light and color, ephemeral substances that are prone to change and movement. The works of “electronic Gothic” amaze light-colored games of fantastic images. Unlike the architecture of modernism with its monochrome, poly color mysteries offer the viewer magical mystical transformations of color waves, a world that envelops, fascinates and gradually grows into the material reality of the city (see Fig. 3). In this case, our emotional experiences are perceived subjectively by each individual, but the entire multimedia structure of

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Fig. 2. Use of digital visual forms for aesthetic formation of volume space. Restaurant in NurSultan city.

the complex of buildings, thanks to the composition of colorographic elements (points, lines, planes), strengthens the shapes of buildings. Spaces become deep, due to horizontal lines on the facades, these lines lead to the main dominant - the ball. Correctly

Fig. 3. Media design on the example of Expo-2017 Nur Alem Hall.

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selected metrorhythmic repetitions, sinusoids of lines, not only enhance the shape of the ball, but, due to optical perception, set it in motion. Which at the subconscious level is compared with the rotation of the earth and so on (route of movement). The urban environment is a subject-spatial environment that ensures the vital activities of the urban community, participating in the formation of its behavior. You can specify the environment of an individual, a group of people, a people. Man not only adapts to the environment, but also transforms it, subjecting it to its goals, provides optimal conditions for his life and activity. The concept of the environment covers not only material elements, but also their spatial connections, and the person himself is included in the content of this concept. A person behaves in a certain way in the appropriate environment, actively experiences and evaluates this environment, developing an emotionally colored attitude to it, endows it with associative and symbolic meanings and images [5]. A person perceives his environment in the inseparability of the connections of the system. An urban environment is a complex object for adequate description and image, requiring special techniques. The complexity of the media design information is superimposed by the complexity of the perception structure. A new understanding of the term “environment” allows us to approach the formation of systems of an object-spatial environment and relies on all forms of space filling. Recognizing the objectivity of the aesthetic properties of city architecture, it is important to determine the ratio of rules and taste in the composition of the media design of the city environment. Excellent knowledge in aesthetic perception, which determines the subjective and social conditionality of evaluating the media design of architectural structures. Historical and social experience, psychological reaction, the role of the subjective factor are taken into account [6]. The art-historical approach to the study of media design of the environment of urban planning systems reveals the objective foundations laid down in the properties of urban planning objects: proportionality, proportionality of elements; in the rhythmic structure and drawing of plans, panoramas, prospects; in plastic and color wealth; in the harmony of the combination of artificial elements of the environment with the landscape. 2.4 Aspects of Urban Media Design Research Based on aesthetics and art science, one can form a methodological basis for studying the visual perception of the media design of the urban planning environment, a model for the development of the city’s artistic culture, using scientific methods for classifying various factors, extrapolating of trends. The scientific approach allows you to model the mechanism of aesthetic perception and artistic creativity, establishing the ratio of rules and taste in architectural creativity and in the perception of media design of the architectural environment. In a number of modern aesthetic directions of media design, the socio-psychological conditionality of aesthetic evaluation is investigated. Another area of media design research is aimed at highly developed structures of cognition and their regularities: the tendency to completeness, harmony, symmetry, simplicity. The socio-psychological mechanism for the formation and assimilation of stable aesthetic stereotypes - “archetypes” can be interpreted as fixing social experience

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in the subconscious. This should include the formation of style, artistic image of media design, taste. The image of the city develops in the process of its everyday perception [7]. Movement around the city determines the sequential change of objects of attention. During a special inspection, a person knowingly organizes visual information, choosing the most significant objects of media design of the environment. Associations are layered on visual information. The perception of the media design of the city environment can be imagined as a complex system organized in various planes [8]. Among the aspects of assessing the media design of the urban environment can be distinguished: • • • • •

Safety of life activities; Functional convenience; Comfort of work and leisure activities; Socio-aesthetic conditions; Historical and cultural values.

The perception of the characteristics of the media design of the architectural environment occurs at the psychophysiological, emotional and rational levels. The perception of the media design of the urban environment is diverse in nature and intensity. It is based on visual sensations that reflect the variety of shape, light, color. When perceiving the media design of the architectural and spatial environment, the role of other sensations besides visual ones increases: touch acts as an idea of various facets of surfaces, hearing perceives beeps of cars, crowd noise; sense of smell - various smells. In the process of studying media design in the architectural environment, the types of perception of the urban environment that are built are revealed: • when identifying the urban environment and taking into account the characteristics of perception; • when exploring the city at aesthetic and emotional-psychological levels; • at passive movement of population to any target passes with assessment of all characteristics of medium - for a short time. A person’s aesthetic attitude to the perceived medium of media architecture has an objective and subjective side. Objective - is determined by the subject environment and expresses its features, subjective - expresses thoughts, ideas, feelings, ideals, associations (see Fig. 4). Color causes associations. In many ways, cultural code works in perception. A person knowing the symbolism of color, comprehends the information. The combination of forms (horizontal, vertical, deep-spatial compositions) with the color and cultural code of the individual, evoke feelings, emotions, which in turn leads to the comprehension of visible space.

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Fig. 4. Media architecture. Association with the Egyptian Temple.

In the process of urban planning, it is necessary to provide for the presence and condition in the view frames of near, medium and long-range media plans of architectural design, the organization of visual connections of individual objects and parts of the city, the creation of visual accents and landmarks in the urban environment [9]. (see Fig. 5). In this image, the building grows as if from a hill. It is a continuation of the landscape, representing a single whole with urban space. The dome is a building, perceived as an artistic object, a point of attraction, representing an element of Land art. In turn, Land art is one of the types of multimedia design. The architect takes an active part in organizing the media design of the visual environment of the city, projecting the configurations of pedestrian streets and highways, setting the distance between the objects formulating the building street, the nature of the sequence of view frames. In the process of aesthetic perception of the media design of architecture and the formation of the image of the city, the observer captures not only highly artistic objects, but also disharmonic ones that cause a negative assessment. Associations, images, stereotypes that deepen the idea of the city are added to the visually perceived material. Thus, the aesthetic perception of the city is an active creative activity of human consciousness. The basis of this creativity is the daily accumulation of impressions, crystallization of images. Social psychology considers issues of aesthetic perception of the media design of the urban planning environment, revealing objective laws of visual

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Fig. 5. Media architecture of Nur-Sultan city, lighting design of urban spaces.

assimilation of information related to the peculiarities of the human psyche, psychological setting, and emotional assessment. The development of social and psychological research in the field of media design as artistic creativity and visual perception is aimed at creating rational guidelines. In the process of perceiving the media design of the architectural environment, there is a direct interaction between the novelty of aesthetic experience, psychological and intellectual potential [10]. In the image of the media design of the city, the content of urban life is expressed in many ways. The combination of diverse style and cultural meanings is expressed in the polyphonicity of the media design of the urban environment. The urban environment bears the imprint of historical layers, aesthetic qualities of nature and landscape [11]. The aesthetic perception of the city is manifested in the shift of artistic ideas and layouts of different eras. The structuring role of artistic perception of media design organizes visual information and compares in the mind of the observer with the image of the urban environment (see Fig. 6). The architecture of Nur-Sultan is specific. Many buildings, and sometimes recreational areas, carry historical cultural code, associated with myths and legends. Color combinations of backlight broadcast information that, in combination with landscape design, small architectural forms, creates a certain image of the urban environment.

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Fig. 6. Monument media design on Water-Green Boulevard, Nur-Sultan.

3 Conclusions The viewer perceives spatial objects holistically, as an environment with all its components. The formation of the media design of the urban environment should obey the laws of visual perception. The size and types of spaces, the forms of structures, the parameters of urban planning formations, the presence and types of small architectural forms determine the specifics of the architectural and artistic organization of the media design of the city environment, the application of all components of which should be consistent with the features of visual perception. The study revealed that the specifics of the formation of media design of the urban environment are that it is both a material environment and an art that takes into account the cultural, national and historical characteristics of the region; the social level and peculiarities of the national culture of the population determine the criteria for creating comfort in the urban planning environment. It was revealed that the conditions of visual perception of the media design of the architectural and spatial environment of the city form visual impressions, the quality of which is influenced by: – – – – –

Sequence of visible frames; Perception zone; Observation distance; Traffic route; Duration of perception.

Spatial characteristics of the media design of the Nur-Sultan city environment have been identified, which must be taken into account in the design process:

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– compositional connection of individual parts of the city, introduction of accents in the urban environment; – presence and ratio of near, medium and long-range plans of the urban environment; – differentiated ergonomic solution of media environment design based on different levels of perception and conditions for different population groups; – solution of problems of media design in urban environment by means of planning and means of architectural environment design.

References 1. McQuire, S.: Media City: Media, Architecture and Urban Space. Sage Publications, London (2010) 2. Eriksson, B., Sorensen, A.: Public art projects in exposed social housing areas in Denmark– dilemmas and potentials. J. Aesthetics Culture 1(13), 115–123 (2021) 3. Mitchell, W.: Me++: The Cyborg Self and the Networked City. Strelka Press, Moscow (2012) 4. Mabey, C., Armstrong, A., Mattson, C., Hatch, N., Dahlin, E.: A computational simulationbased framework for estimating potential product impact during product design. Des. Sci. 7(13), 1–23 (2021) 5. Dobritcina, I.: New Architecture Challenges in the Digital Culture Era. Academia 4, Moscow (2013) 6. Zabelshsky, G., Minervin, G., Rappaport, A., Somov, G.: Architecture and the emotional world of man. Stroyizdat, Moscow (2005) 7. Shimko, V.: Architectural design of the urban environment. Architecture -C, Moscow (2006) 8. Shimko, V.: Fundamentals of design and environmental design. Architecture -C, Moscow (2007) 9. Medvedev, V.: Essence of Design: Theoretical Basis of Design. SPUTD, Moscow (2009) 10. Filin, V.: Video ecology. What is good and what is bad for the eye. MC Video Ecology, Moscow (1997) 11. Nefedov, V.: Landscape design and environmental sustainability. Reconstruction, St. Petersburg (2002)

Le Corbusier’s Modulor and ‘le jeu des panneaux’: A Parametric Approach Wilson Florio(B) Mackenzie Presbyterian University, São Paulo, SP, Brazil [email protected]

Abstract. Historically, architecture gradually became controlled by modulation, proportion, rhythm, harmony, and orthogonality of regular forms. In his investigation, Le Corbusier renews this fact maintaining the precise orthogonality of the right angle and modulation as a means to achieve his mathematical ‘vérités réconfortantes’. In recent years parametric modelling allows us to combine a great number of parameters, conducting us to explore and discover new possibilities to plan architectural forms. In this paper we study corbusian window panels using PM. The methodological procedures were: i) drawing of the grid of proportions; ii) parametric redrawing of the red and blue series; iii) geometric study of the so-called “jeu des panneaux”, with the reconstruction of the 2.26 m square and its golden section; iv) development of algorithms based on the values of the red and blue series; v) generation of combinations among parameters; vi) discussion of results and analysis. Using Grasshopper, the aim of this article is to report the parametric study of the red and blue series of the corbusian Modulor. Keywords: Mathematics · Parametric study · Algorithm

1 Introduction Architects do not produce geometry; they consume it [1]. As applied knowledge, geometry is of fundamental importance in determining architectural forms. Geometry serves to guide the conception and interpretation of the architectural form. For Vitruvius, measuring by modules would allow for achieving the proportion and symmetry essential for harmony and compositional balance. The human body as a model of perfection to be followed. In Classical Architecture, the modulation given by the diameter of the column facilitated control of the composition. The proportion between parts and the whole has always been part of the geometric reasoning in construction. Geometry eliminated the problem of irrational quantities. So, we can infer that the use of proportions as a means of measurement to achieve the compositional balance, as well as the search to establish a parallel relationship between the parts and the whole of a building with the well-formed human body [2]. Especially in the Middle Ages, the templates served as models to ensure uniformity of modulation and repetition of stonework. This knowledge consisted mainly of proportion © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 465–476, 2023. https://doi.org/10.1007/978-3-031-13588-0_40

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systems. As John Harvey [3, p.30] stated, the simplest method of tracing a right angle on the ground was to trace a triangle with sides measuring 3, 4, and 5. During the Middle Ages, the main design secret consisted in applying the module [3]. The Roman foot (0.295 m) [3] remained a measurement standard in many parts of Europe, and many important buildings were established according to this standard. At the end of the 12th century, Leonardo Fibonacci defined the sequence in which every number is generated by adding together the two that precede it. The proportion is maintained at the ratio of approximately 1:1.1.6180339887498948482045868343656. On the other hand, according to Rudolf Wittkower [4, p.211], the Greeks created three types of proportion: the geometric – 1:2:4 arithmetic – 2:3:4 and the harmonic – 6:8:12 all related to music. Whether using a sequence of numbers, or simple geometric shapes (squares and rectangles), the proportion between numbers or forms was the means of expressing the relationship between parts and the whole. Consequently, architecture gradually became dominated by modulation, proportion, and orthogonality of regular forms. Historically, regular geometry provided the architect with the safety and accuracy necessary for a design and construction practice [2]. In this sense, Le Corbusier renews this interest in maintaining the orthogonality of the right angle and modulation as a means to achieve the desired proportion and harmony. Influential researchers [5–12] investigated Le Corbusier’s work. Papadaki [13], Cohen [7], Taboada [14], Mameli [15], have specifically researched the development trajectory of Corbusier’s research on the Modulor. The so-called game of panels (les jeu des panneaux) proposed by the architect aimed to demonstrate the possibility of creating constructive elements in series. In the book Modulor, the architect demonstrates some possibilities of panel combinations, highlighting that there were infinite possibilities for combinations. Currently, thousands of combinations can be performed exploring the resources of parametric modeling (PM). Using Grasshopper, the aim of this article is to report the parametric study of the red and blue series of the Corbusian Modulor, since the study of the human body, geometry and proportions have been one of the most valuable for establishing compositional principles. Consequently, we explored algorithms, based on the values of the red and blue series, to study multiple possibilities of combinations based on PM. We initially investigate the reasoning behind the Modulor formulation, and, subsequently, the parametric study of the red and blue series.

2 L’ Esprit Nouveau: Searching Mathematical Truths From early on, Corbusier defended the science resulting from industrialization as guiding principles for art and architecture. His texts published in the L’Esprit Nouveau Magazine, between 1920 and 1925, emphasize the machine and pure geometry to achieve the universal truth of Modern Architecture. Being attentive to industrial production, Corbusier notes that the module would be the rule to achieve scientific truth. In this sense, Corbusier declared: “To build well […] he measured, admitted a module, regulated his work, put it in order. […] By imposing his foot or arm’s order, he created a module that regulates all work; this work is in his scale […] It is on a human scale. It harmonizes with him […]” [16]. The lessons

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learned from classic architecture and the industrialization led Corbusier to conclude that the module was essential to achieve harmony and the relationship with the human scale. Thus, when presenting the designs of the regulating lines of the Villa to Garches (1927), Le Corbusier [5, p.144] stated: “La mathématique apporte ici vérités réconfortantes […] Un tracé régulateur est une assurance contre l’arbitraire.” [16]. For that reason, mathematics would be the right way to get to the truth. In fact, Corbusier’s aim was to produce elements using serial production: “[…] serial house will impose the unity of elements, windows, doors, construction processes, materials […] [17]. L’architecture est chose de plastique. La plastique, c’est ce qu’on voit et ce qu’on mesure par les yeux” [18]. In 1925, Corbusier declared: “There is no good human work without geometry. Geometry is the very essence of Architecture. To introduce the series into the city’s construction, the building must be industrialized” [19]. His intention was to achieve industrial mass production of building elements, particularly windows.

3 Le Corbusier’s Modulor: Golden Ratio to Serial Production In 1950, Le Corbusier released the book titled Le Modulor [20] (Fig. 1). In his book, he asks: “What is the rule that orders and unifies all things?” [21]. His interest in regulating lines emerged from reading the Auguste Choisy’s book [21]. In addition, the mathematician Matila Ghyka, in his book [22], contributed to Corbusier’s research. However, during World War II, a committee for standards was formed to implement modulation [7]. But the Assemblée des constructeurs pour la révolution architecturale, created by Corbusier in 1942, that impelled the architect to examine deeper his studies on the Modulor. Consequently, Corbusier encourages his collaborator Gérald Hanning to investigate the normalization of construction and industrialization based on a system of measurement. In 1944, Hanning’s drawings were based on two diagonals of the initial square, while Elisa Maillard’s ones had the ratio of the golden section (Fig. 1, upper left). But on March 30, 1945, Corbusier together his collaborators [20] created the grid of proportions of this geometric construction, based on a human figure of 1.75 m and the golden section. During his travel to New York on December 9, 1945 (Fig. 1, right), Corbusier measured spaces in his cabin and the ship, finding the measure 2.16 m. In the merchant ship, Corbusier sketches the answer: “First, I classified as red series […] Next, I classified as blue series, the series based on the double unit 216. Next, I drew a man of a height of 1.7 m engaged at four numbers: 0, 108, 175, 216.” [20, p.70]. Thus, the red series sequence, based on Fibonacci, has three measures, a, b, and c, proportional to each other, being the smallest number (a), the intermediate number (b), and the largest number (c), in a mathematical ratio a:b = b:c, where a + b = c. To obtain the idealization of the human body, he initially establishes the height of 1.75 m, and later on the height of 1.83 m, reconciling two systems: meters and inches. Finally, the architect establishes relationships with the golden section and the Fibonacci sequence. Back in Paris, in February 1946, Corbusier choose the name of his invention: Modulor [20].

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Fig. 1. Grid of proportions Hanning-Maillard (1944); Corbusier’ sketch of modulor at Vernon (1946); cover of modulor books (1950–1955). Source: Fondation Le Corbusier.

Therefore, the word Modulor is a combination of the words “module” and “gold” (or). The sequence of numbers in the series is regulated by the golden ratio. The grid of proportions, taken to New York and drawn in his office, generated the following statement: “Le Modulor est un outil de mesure issu de la stature humaine et de la mathèmatique.” [20, p. 75]. Corbusier declared: “The combinations resulting from the application of Modulor are unlimited” [21]. Unfortunately, Corbusier wrote, “almost all of these values in meters were untranslatable to integers in feet and inches” [20, p. 76]. The height of 1.75 m referred to a French man, but British police officers were six feet tall. Thus, to contemplate the dimensions of the British man, the dimensions became the height of 1.83 m, with 1.13 m at the center, and 2.26 m with arm upraised. From then on, the Modulor was six feet high: 183 cm. The golden ratio is maintained both in the red series and in the blue series. Both hold values closer to 1.618 than Fibonacci’s.

4 Conception of the “le jeu des panneaux” Algorithm In order to explore the richness of “combinations” provided by Modulor, algorithms were developed to generate hundreds of alternatives for windows, similar to those designed and called the “le jeu des panneaux” in volume 1 of Modulor. The research steps were as follows: a) study of the Fibonacci sequence; b) drawing of the proportions grid originated by the Maillard tracing reconciled with the Hanning tracing; c) drawing of the red series; d) drawing of the blue series; e) study of proportions based on the height of 1.75, and then 1.83 m; f) drawing of the red grid; g) drawing of

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the blue grid; h) superposition of the two grids; i) geometric study of the so-called “jeu”, with the reconstruction of the 2.26 m square and its half to study the combinations, as well as the 2.26 m square and its golden Sect. 1.397; j) development of algorithms based on the values of the red and blue series; k) generation of combinations among parameters. By redrawing the Hanning-Maillard draft, it was possible to understand the geometry, proportions, and relationship linked with the human body. The drawing of the red series provided the understanding of the Fibonacci series resulting from the ø ratio, while the drawing of the blue series, its double (Fig. 2). In addition, when redrawing the series, the fractional numbers were noticed, and the real architect’s need to round off the values for practical application. The conciliation between the unit in meters and inches forced the establishment of successive approximations of values. Doubling, adding the section of another, and subtracting the gold section was understood by redoing the numerical values of the red and blue series. The ratio between the rounded values of the red series results in: 10/6 = 1.66; 16/10 = 1.6; 27/16 = 1.68; 43/27 = 1.59; 70/43 = 1.62; 113/70 = 1.61; 183/113 = 1.61. So, the arithmetic and geometry contained in the Modulor serve to guide the proportions.

Fig. 2. Modulor parametric modeling steps.

The conception of the algorithm required the reconstruction of the trajectory of the steps declared in the 1st volume of the Modulor book. Based initially on squares,

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the “panel game” (or le jeux des panneaux) starts with simple combinations of golden rectangles, proportional to each other, derived from the values established by Modulor. The steps in the making of the algorithm were the following: a) definition of the parameters, based on the values of the red and blue series; b) definition of minimum and maximum values; c) study of different sequences of insertion of rectangles derived from the series values; d) study of opaque and transparent panels; e) generation of panels. Five algorithms were developed to progressively investigate the different ways of combining the frame panels, but only two we will explain in this paper. Trial and error are a very common procedure to learn parametric modeling. From the three initials algorithms we learned that it is natural that the number of combinations will result from: a) the number of parameters; b) the number of values of each one of them; c) restrictions (such as square); d) alignments or not; and not necessarily the number of panels. Instead of using 3 or 4 panels (used in the three algorithms) we decide to increase the complexity, using 6 or 7 panels subdivisions. In algorithm, openings were generated. Combinations between six panels were generated (Fig. 3). Our intention was investigating the Modulor’s eyes level. There are six parameters: Two parameters allow us to control the width and length of rectangle r2; Two parameters control rectangle r3; only one parameter to control the width of the rectangle r4; and only one parameter to control the rotation of all panels. The parameters for the dimensions of the panels were previously defined from the red series – 0.16/0.27/0.43/0.70/1.13/1.83, and from blue series – 0.86/1.40. The rotation parameter contains 4 variation possibilities: 0, 90, 180, and 270°. The order of the definition sequence goes from r1 to r7.

Fig. 3. Eight algorithm steps controlled by parameters.

Figure 3 shows the 8 steps of algorithm. In step 1, rectangle r1 measuring 2.26 × 2.26 m was created, connecting points a and b. Rectangle r2 was created from vertices a-c-d-e. In step 3, the rectangle r3 was created from the definition of vertices d–f–g–h. Rectangle r4 was defined by the vertices on opposite sides, g–j. Therefore, the successive rectangles

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r2, r3, and r4 are interdependent on the dimensions (width and height) established in the creation order definition. In steps 5 to 8 (Fig. 3), rectangles r5 to r7 are the result of the dimensions and vertices established for rectangles r1 to r4. Rectangles r1 to r4 are derived from the dimensions of the red series, while rectangles r5 to r7 are derived from the red or blue series. For example, in Fig. 3–3 r3 is 1.40 × 0.70, and r7 is 0.86 × 1.83 [Fig. 3, 4, 5, 6 and 7], both combine dimensions of the red and blue series.

Fig. 4. Panels combination from algorithm 3.

We thought to better study the apertures and the relationship with the Modulor proportions. It was interesting to note that the dimensions of the red series, related to the parts of the human body, provide unexpected and interesting combinations of modulations and subdivisions. In Fig. 4 we can visualize some of thousands of panels combinations which are possible based on red and blue series is possible. One notice that the proportions of the panels, derived from golden rectangles, provide, in most of the options, an adequate design in functional and aesthetic terms.

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Algorithm 4 consists of parameters that control the modulations of the red and blue series in a useful way. A square measuring 2.26 × 2.26 m was defined. Then it was subdivided into four equal squares. Each square contains a set of possibilities for manipulating the internal subdivisions according to the values of the established parameters. Figure 5 shows these steps.

Fig. 5. Algorithm 4: steps of subdivision of each quadrant.

While the values of the parameters established for each parameter are changed, different modulations can be obtained following the dimensions established by the red and blue series (Fig. 5). Three sliders were defined, containing 4 variations of the red series 0.16/0.27/0.43/0.70 and 1 of the blue series 0.86, totaling 5 variations each. Therefore, each quadrant can contain 5 × 5 × 5 = 125 variations resulting from the combinations between these 3 sliders. The same procedure was performed for the other three quadrants. Sixteen different panel configuration options were generated for each quadrant (Fig. 6, left). Each of the 16 options in each quadrant can fluctuate according to the three sliders, like those in option 8 (Fig. 6, right). As a result, there are 16 layout options for quadrant

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1, 16 for quadrant two, 16 for quadrant three and 16 for quadrant four. As each slider has 5 variations, and there are three sliders containing the red and blue series, this is 5 × 5 × 5 = 125 possible combinations of dimensions multiplied by 16 × 16 × 16 × 16 = 65556. Therefore, there are 125 (series dimension combinations) × 65556 = 8,194,500 combination options between the four panels.

Fig. 6. Sixteen panel options generated for the quadrant 1 (left). For example, some variations of panel 8 (right).

In Fig. 7 it can be seen that the lines drawn at 45°, corresponding to the diagonals of each quadrant, demonstrate, through the regulating tracing (trace régulateur), the proportion between the modules between themselves and the whole. As these are dimensions resulting from the red and blue series, the sequence of golden rectangles guarantees the proportion of the rectangles among themselves. The algorithm was organized with the possibility of generating 1, 2 or 3 openings in each quadrant, in order to differentiate opaque and glass panels, i.e., to study the variations in the openings of the modular windows (Fig. 8). Following Le Corbusier’s idea, the joinery panels (currently probably metallic) and the glass windows panels would be produced in series, in order to provide a combination between opaque and transparent panels. In this sense, form, function, and technique are achieved simultaneously according to the project’s principles.

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Fig. 7. Tracé régulateur and golden rectangles: proportion between parts and the whole.

Fig. 8. Window openings and opaque panels.

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5 Discussion For Le Corbusier, mathematics and its branch, geometry, were the means for achieving success in his projects from the vérités réconfortantes. The study that started with the regulating lines in the 1920s was finalized in the 1940s with the writings of the two volumes of the Modulor book, published in 1950, but applied systematically for the first time in the Unité d’Habitacion in Marseille in 1947. Corbusier’s innovation was establishing a connection of this ancient knowledge with other studies, creating his own modules. The basic sequence of 16, 23, 47, 70, 113, 183, close to the Fibonacci Series, would henceforth be present in the geometric definitions of his projects in a systemic way. In this research, parametric modeling allowed us to investigate profoundly the relationship among the two series applied to panels. The algorithm created allows the generation of hundreds of combinations, interleaving openings with opaque panels. The algorithm makes it possible to generate unexpected and surprising combinations, since they were defined by random combinations. In his book Corbusier selected a few panel configuration options and combined them in a selective way, as there are hundreds of possible combinations. One can imagine the difficulty of drawing, with restricted precision, the dimensions of the red and blue series of dozens of panels, with the help of some designers in his office. But in our investigation the problem was the contrary: too many possibilities of parameters combinations. The emphasis on the study and application of regulating lines, proportions, and the human figure as the basis of the geometric thinking of his designs is remarkable. The algorithm created allowed us to investigate how efficient is the measure system created by red and blue series, and how convenient was to apply to create serial windows. In fact, one’s can imagine that the industrialized window modules would be, in the architect’s view, a real opportunity for the industry to introduce a series production of building elements. Corbusier achieved his dream with the elaboration of the construction grid, designed to fit the man placed within it, thus reconciling “human stature (man with arm upraised) and mathematics…” It was from the horizontal and vertical lines, and the right angle praised in the geometric studies published in the L’Esprit Nouveau Magazine, up to the investigations that lead to the dimensions of the Modulor’s Grid that Corbusier would achieve the desired scientific, mathematical proof that would guide, with certainty, his spatial and aesthetic proposal in the field of architecture. If “for the eyes, everything is geometric,” one would assume that, for Corbusier, the proportions would be the basis to achieve the desired sense of beauty and plastic emotion in the design of shapes and spaces in architecture and the arts. The “visual” event requires that the composition - the arrangement, organization, and dimensioning of spaces - be carried out to provoke the desired sensations, affected bodily, but above all by sight. The human eye level, at 1.60 m, allows the panels at the top to frame the landscape and allow for indoor-outdoor integration, while the panels at the bottom allow the eye of a seated person to enjoy the view. Therefore, there is always a panel suitable for different heights of the human eye. The play of numbers, the mathematics of the “gods,”

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was unveiled by establishing the red and blue series, inspired by Fibonacci and many other important mathematicians but implemented by Corbusier in Modern Architecture.

References 1. Evans, R.: The Projective Cast: Architecture and its Three Geometries. MIT Press, Cambridge (1995) 2. Florio, W.: O uso de ferramentas de modelagem vetorial na concepção de uma arquitetura de formas complexas. PhD in Technology of Architecture. Programa de Pós-Graduação da Faculdade de Arquitetura da Universidade de São Paulo. FAUUSP, São Paulo (2005) 3. Harvey, J.: The Master Builder: Architecture in the Middle Ages. McGraw-Hill Book Company, New York (1971) 4. Wittkower, R.: Los fundamentos de la arquitectura en la edad del humanismo. Original Title: Architectural Principles in the Age of Humanism. Alianza Editorial, Madrid (1995) 5. Boesiger, W.: Le Corbusier et Pierre Jeanneret : œuvre complète. Volume 1 to 8. Les Editions d’Architecture Zurich/Edition Girsberger, Zurich (1964) 6. Cohen, J-L. : Le Corbusier et la mystique de L’URSS : théories et projets pour Moscou 1928–1936. Pierre Mardaga Editeur, Liège (1987) 7. Cohen, J-L.: Le Corbusier’s Modulor and the debate on proportion in France. Archit. Histories, 2(1), 23, 1–14 (2014). https://doi.org/10.5334/ah.by 8. Boyer, M.C.: Le Corbusier : homme de lettres. Princeton Architectural Press, New York (2011) 9. Lucan, J.: Le Corbusier une encyclopedie. Éditions du Centre Georges Pompidou, Paris (1987) 10. von Moos, S.: Le Corbusier. Elements of a Synthesis. The MIT Press, Cambridge (1988) 11. Curtis, W.J.R.: Le Corbusier: Ideas & Forms. Phaidon Press, London (2015) 12. Samuel, F.: Le Corbusier and the Architectural Promenade. Birkhauser, Basel (2010) 13. Papadaki, S.: Le Corbusier: The Gradations of Modulor. Durisol (1947) 14. Taboada, J.M.F.: The Modulor by Le Corbusier 1943–54. Independently published. (2018) 15. Mameli, M.: Le Corbusier and the American modulor. In: Jorge Torres Cueco (Ed.) International Congress Le Corbusier 50 years later, pp.1282–1294, Universitá Politécnica, Valencia (2015) 16. Corbusier, L., Saugnier, A.: Les Tracés Régulateurs. L’Esprit Nouveau 5, 563–572 (1921) 17. Corbusier, L., Saugnier, A.: Les maisons en série. L’Esprit Nouveau 13, 1525–1542 (1921) 18. Corbusier, L., Saugnier, A.: Architecture III. Pure création de L’esprit. L’Esprit Nouveau 16, 1903–1920 (1922) 19. Corbusier, L.: Une ville contemporaine. L’Esprit Nouveau 28, 2392–2409 (1925) 20. Corbusier, L.: O Modulor. Original Title: Le Modulor. 1950. Orfeu Negro, Lisboa (2010) 21. Choisy, A.: Historie de L’Architecture. Tome I. Gauthier-Villars, Paris (1899) 22. Ghyka, M.: Le nombre d’or. Rites et Rythmes Pythagoriciens dans le développement de la civilisation occidentale, Gallimard, Paris (1931)

Grid as Memory in City and Architecture Yuji Katagiri1(B) , Taizo Iwashita2 , Hirotoshi Takeuchi3 , Takahiro Ohmura4 , Ikko Yokoyama5 , and Tatsuo Iwaoka5 1 Tokyo City University, Setagaya City, Tokyo 1588557, Japan

[email protected]

2 Musashino Art University, Tokyo, Kodaira City 1870032, Japan

[email protected]

3 Nippon Institute of Technology, Minamisaitama District, Saitama 3450826, Japan

[email protected]

4 Window Research Institute, Chiyoda City, Tokyo 1010024, Japan

[email protected]

5 Tokyo University of Science, Noda-city Chiba 2788510, Japan

[email protected]

Abstract. The architects had been attracted by grid systems and applied grids into realized architectures. We published Grid on Architecture in 2018 and discussed the relationship between the geometry of the grid and architectural works. In this study, we reckon the meaning of the grid in city and architecture by comparing game and toys with the grid. First, on reconsidering contemporary architectural theories, the grid, regarded as the “sea”, is associated with architectural monumentality, isolated but inside the city, regarded as an “island”, which makes us recognize the city and keep in the memory. We discuss the gameboard of the grid and brick toy. We can say that the game board grid is a “closed grid”, for its isolated world inside from contexts outside. On the other hand, brick toy like Lego is regarded as an “open grid”, for its multiplying square or cube system. Then, we discuss that these two ideas can also be seen in the idea of the city and architecture. In this sense, closed grids are seen in the ideal city of India and China, and open grids in an expansion of urban infrastructure, for example, Cardo and Decumanus, and Barcelona plan of Cerdà. After that, it treats how contemporary architects translated urban grid to their works on the real map. Keywords: Grid · Memory · Design · Architecture · City

1 Grid in Design Human beings have designed the geometry of the grid, from small toys to architecture, to urbanism. A grid is a simple geometric shape of a repeating pattern formed by intersecting a plurality of lines at equal intervals at right angles or diagonally. Because of its homogeneity, it is sometimes perceived as a homogeneous space or a uniform and boring space. However, if you look closely at an object made of a grid, you will see a variety of interesting details such as the shape and appearance of the squares separated by lines, the size and extent, and the center and edges of the area. Some of them were © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 477–488, 2023. https://doi.org/10.1007/978-3-031-13588-0_41

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intentionally created, while others were inevitably or accidentally created about parts other than the grid. Like Periodic Table, Calendars, and address books, the Cartesian grid helps us understand an amount of information in one sight. From virtual stuff like table games in Japan, toy bricks, and Mandalas, we use the grids, and the lattice systems to recognize the world. (Fig. 1) As far as design, “Grid” includes not only Cartesian lattice as an orthogonal coordinate system but also several patterns of the repetition of lines that have the same distance as a polar coordinate system.

Fig. 1. Example of grid designs by human being

However, why do we apply the geometry of the grid to designed works? In the study, we focus on the meaning of the grid as memory in architecture and cities. For the designs made by human beings in quotidian things, we remember that human beings used grid from small things to huge planning. In this research, we will discuss the grid-like spaces that exist on various scales in various environments created by humans, from virtual game-boards to real ones such as cities.

2 Grid in Contemporary Architectural Theory Pier Vittorio Aureli, an Italian architect-theorist, discussed that the grid as “sea” in architecture is associated with politics of land possession. (Aureli 2018, 139) Aureli pointed out that grid at the urban scale has been used as a composition of appropriation of land and people have been obliged to obey the law of asymmetry political balance with subdivision of grid in the city as land possession. Expounding Aureli’s critical view, in terms of architecture and city, people could not escape the composition of appropriation with the geometry of the grid. When we design architecture, we can communicate with each other with a comprehensive schéma of the grid, using standard lines, plan of ramen frame, and square windows aligned in the grid. For Aureli, grid system such as Cardo and Decumanus is regarded as urbanization, and he put importance on architecture as the island inside the expanding grid.

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On the contrary, Andrew Witt reported his chronological view in terms of the relationship between mathematical formulation and modern architectural design. (Witt 2021, 275) According to Witt’s view, from the 1970s to the 1980s, the cubic grid had imported economic patterns of thought into architecture as a unit of measurement and simulation, and even as a game like framework for economic or political choice. In contrast of Aureli’s view of grid as land possession at urban scale, Witt insisted that cubic matrix only stared to show importance in 1930’s and became tacit social system or a mathematical equivalent of an idealized economy (Witt 2021, 276). The difference of political position between the two, although it might be derived from European Marxist position of Aureli and American Non-Marxist position of Witt, come from the difference of architectural scale of their views. Aureli focused on the large range of chronological culture of human habitation in the terms of urban scale and dwelling, while Witt focused on how human being design with formulation of cubic matrix in terms of architectural scale and idealized form. In Aureli’s viewpoint, it is strongly premised that the primitive form in the collective memory – on which Aldo Rossi had put importance in the architectural design – should have been observed in any cultural traces of dwelling. However, in Witt s viewpoint, after the appearance of Modern geometrical culture for design, more and more human beings have applied mathematics into architectural design for investigating new architectural forms, molecular structurelike architectures of Louis Kahn’s and Metabolist Movement.

3 Grid of Architectural Design and Exception in the City Architects have struggled to design inside of the existing city inside the politics or power, because of large scale of material, size, and costs. Aldo Rossi, who was one of the most influential architect-theorists in the twentieth century, designed architectural projects in his youth with Gianugo Polesello and Luca Meda. Polesello, Rossi, and Meda made a project for Centro Direzionale di Torino in 1962. According to Aureli, the project of the building of a 320 m square ring with a 280 m huge courtyard represents not only extrusions of the chessboard of grid outside of Turin, but also a form of exception, for being located on the periphery of the city. (Aureli 2008, 67.) Their project for a new office and directional center, about which Aureli indicated that the three architects had revealed the concealed power of the dominant class in those days with rigorous and simple geometry, was derived from the existing building inside the grid of Turin, Mole Antonelliana (Fig. 2). The high building with brick over 100 m height grid represents the state of an exception inside the city, which causes the inversion of negative and the positive of our recognition of the city. In other words, Mole Antonelliana is only a monument in the grid of the historical city, but acts as the essential reference for the whole grid of the Turin. They understood the effect caused by the relationship between monumental existence and the city grid, and applied it to their non-realized city-center outside the city, associating the concept of New Centro Direzionale with the Mole Antenelliana as the exception inside the Turin grid. (Fig. 3).

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Fig. 2. (left). Mole Antonelliana. Image: author

Fig. 3. (right). Rossi’s drawing of the City grid of Turin with concept of Centro Direzionale Image cited from https://mitpress. mit.edu/books/architectures-desire, last accessed 28 February 2022.

4 Defining Its Monumentality with Grid Aureli is not only influenced by Rossi but also by Gianugo Polesello. Polesello, who was one of his most important collaborators of Rossi, put importance on the city grid for architectural design. In the project for Centro Direzionale di Firenze in 1977, Polesello aligned nine towers in 3 × 3 square. (Fig. 4) Also, Polesello had tried to apply 3D gridlattice of cubes in his architectural design (Mase 1994: 17–18) Mase Masahiko reported that Polesello had used a 15m grid with these standard lines (15 m = 3.75 m × 4 or 3 m × 5 = 15 m) both in the plans and the sections. (Mase 1994: 17, 27) In the Project of Centro Direzionale di Firenze (1977), Polesello and his collaborators designed nine towers with 15m grid.

Fig. 4. Project for centro Direzionale di Firenze (1977). Image cited from Polesello (1982), p.53.

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Fig. 5. Isometric projection of a 3D model of the structural frame of Centro Direzionale di Firenze. Image: author

In the project for the new city center of Florence, each tower is rigorously aligned in 3D-lattice of cubes of 15 m × 15 m × 15 m, supported by the same structure-frame made from steel and concrete of the square section of 1 m × 1 m. (Fig. 5) In some sense, 3D-lattice of Polesello was amplified with squares and cubes into the structure-module of the building, which was engraved in the imaginary city grid. In the work for Florence, Polesello tried to connect structural frame and architectural design in as a purist way as possible, by associating the 3D lattice of the cube with a ramen frame of steel and concrete. The grid of the cube towers is geometrically and rigorously defined and then would operate the existing periphery around Centro Direzionale to prevent over-development of the land for its monumentality. Thus, what Polesello discussed is “their sense (of the already-given) of monuments, to regard/see isolated or approached, grouped” (Polesello 2000, 105) means that the architectural monumentality of simple geometry acts as “a fact” that exists for a long time ever before inside the city. Moreover, Aureli and Tattara, who were taught by Polesello and Rossi at IUAV University in Venice, co-founded the office called DOGMA and designed the City Wall with Office Kersten Geers Devid Van Severen. (Fig. 6) The cruciforms of buildings make huge square spaces and restrict the development of the space. This urban plan shows that rigid and rigorous geometry urges us to keep the monumentality even though it represents the open grid system.

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Fig. 6. City Wall. Image traced by authors from Dogma’s project collaborated with KGDVS.

5 Grid of Game Board: Closed Grid Why are architects attracted by the grid for the design? Inspired by Cristian NorbergSchulz, Kishida (2012) redefined the idea of schéma for architectural design. Kishida points out that schéma cannot be invented but be held in human existence as a simple scheme (not in an iconic diagram) and points out that the grid is one of the most communicable schéma for architects to design forms. Not only for architects but also non-architectural people, the schéma of the grid is effectively communicable. The board game is a comprehensive example. The game boards with planner grids make human beings understand the fictional system with Cartesian coordinates to define where the agents are located. For example, Wordle, the web game which is updated as a daily quiz, uses a 5 × 6 grid on which the player can put the five-letters-word in one time and try six times at maximum. For example, the upper word “WIRED” is incorrect but the letter “R” (yellow square) is included somewhere in the correct answer except for the third letter. (Fig. 7) Next, the word “ROAST” is also incorrect but the letters “R”, “O”, and “T” are included in the different positions. If we guess a correct position with the correct alphabet in one word, the squares highlight green, as the “T” and “O” in the third row. Finally, the player can find the correct answer as “THORN” with all five letters with green squares in the day.

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Fig. 7. Web game in contemporary. Image: author on the Wordle

The game Wordle shows a simple connection between the player and the interface of the grid, where people put each trial on the one grid of 5 × 6. The correct answer is changed every day and one player had only one chance to guess in one day. Even though we had known the correct five-letter word, it is possible to guess the answer. Indeed, we can seek the five-letter word in the internet or web dictionary. In this sense, Wordle is an open-resource-game, because we solve the game by investigating other resources. The grid of Wordle represents a rigid chance to enter the five-letter word six times maximum but the time to access information is premised on each chance to enter the word. The grid in the game represents the stronger rigidness of the rule if the other player joins the game of the grid. Senet, in the New Kingdom of Ancient Egypt, has a gameboard of a rectangular grid of 3 × 10. (Fig. 8) In the game, the two players compete on the game board like snakes and ladder. The original rule is unknown for no records have ever been discovered. However, some have attempted to reconstruct the rules and it can be explained through Kendall’s rule as below. As with numbered and gridded squares, both players have five pieces in the first row of 1–10 by turn. (Fig. 9) Then, one player put each piece on 1, 3, 5, 7, 9, and the other on 2, 4, 6, 8, and 10. All the pieces run toward the goal of square 30 and when a piece arrives at the goal or particular squares, the player can remove it from the grid board. Finally, the player who removes all the five pieces will win.

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Fig. 8. Table game in ancient Egypt, Senet. Image: author

Fig. 9. Grid order of Senet, Image: Public domain

It is different to regard grid systems as a mass of squares or textile of vertical and horizontal lines. For example, the game of shogi in East-Asian culture has a 9 x 9 grid in which the pieces are put on each square, while the pieces in the game called Go are put in the cross points 18 × 18 grid. (Fig. 10). The grid game board of each game can be said as a “closed grid”, for they make a world system inside the game board, isolated and separated from the other system. As if they are “islands”, the grid game boards urge us to act according to the system inside.

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Fig. 10. Grid board of Shogi (left) and Go (right). Image: author

6 Grid of Brick Toy: Open Grid

Fig. 11. The base plate of Lego (left) and Diplo (right). Image: author

The block of Lego is a systematic toy that can be made into various things by assembling several types of inset blocks. People put the studs on one side of the basic rectangle at an 8 mm pitch and fit the studs in the gaps on the other side to assemble firmly. (Fig. 11, left) There is also a series called Diplo that uses blocks larger than those of Lego. (Fig. 11, right) The block system of Diplo is composed of 16 mm studs, which is adapted to the system of 8mm studs of Lego. This is why Lego blocks with Duplo blocks can be mixed. In this case, although they are limited as base plate, the grid of bricks can be said as an “open grid”, which enables us to compose things by extracting the system of 3D-lattice of bricks, which is defined in 2D-Grid in the campus, the standards of grid urge us to understand staffs as the composition of divided cubes and expand the composition inside the grid system.

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7 City Grid: Closed Grid and Open Grid Considering these games with grids, we can say that the two types, of closed and open, grids, enable us to recognize the world. The closed grid would work as if it were the boundary of the city wall and the open grid as a system of expansion in the land organization and occupation.

Fig. 12. City grid in ancient India (Left) and China (Right) Image: author

The two types of the grid can also be seen in the architectural ideal city. Since ancient times, closed grids (square design) have been used in urban design. In ancient India, the book Alta shastra states that the city should be surrounded by the city walls, and divided by three main streets vertical and three horizontal. The city was sectored in a 4 × 4 grid. (Fig. 12, left) This grid is supposed to be derived from Mandalas, the painting of grid composition with the Hindu Gods. (Fig. 13) Besides the ancient Indian concept of ), the Record of Trades, Records of Examination of the city, The Kaogong Ji ( Craftsman, Book of Diverse Crafts or Artificers’ Record, during the Spring and Autumn period in China, states the grid of 4 × 4, similar to the grid in Ancient India. In ancient China during the period, between main streets or main streets and the one side of the wall, the street appeared and the grid was rendered from 4 × 4 to 8 × 8 grid. (Fig. 12, right) The concept was integrated into East Asian countries, for example, in ancient Japanese cities of the 8th century, Nara (old name: Heijo-Kyo) and Kyoto (old name: Heian-Kyo). These ideal cities with a grid for the concept of geometrical reference of the powerful city with an orthogonal coordinate system. Also, there are examples of the ideal city with a polar coordinate system, for example, the Plan of Sforzinda made by Filarete and Palmanova City in Italy.

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Fig. 13. Mandala in grid.Image: author

The ideal city with the grid is defined by the walls and makes the boundary as if they were an island. On the contrary, the grid system of Cardo and Decumanus in the Ancient Roman empire works as an expandable system with enlargement of the state power, as if it were “sea”. The expanding system of the grid is shown in Ildenfons Cerdà’s grid of Barcelona and Colonial cities, to which Aureli indicated urbanization (Aureli 2011, page.x.) The open grid works as a system in the expansion of infrastructure and circulation, for example, Hausmann’s reconstruction of Paris as a polar coordinate system.

8 Grid as Memory in Architectural Design Finally, we can say that the grid system has two faces; one is a resource of the standard line, defined as an open grid; the other rigorously defining razor of the stuff or things with boundaries, defined as a closed grid. These two faces mix each other, then compose our cognition of the quotidian stuff from the small to the huge. At the stake is that the grid helps us keep the stuff in the mind or recognize things to recompose in the memory. Although the grid strictly works as a system in the city, as if it were the “sea”, the geometrical boundary of architecture can be defined and work as an “island.” In this meaning, the grid acts not only to integrate architecture as an “open grid” but also to separate itself from the others and underline architecture itself. That might be why we keep using the grid and memorize things with the grid, even though it is boring, too much familiar, mediocre, or commonplace.

References Aureli, P.V.: Appropriation, subdivision, abstraction: a political history of the urban grid. Log 44, 139–167 (2018) Aureli, P.V.: The difficult whole. Grey Room 9, 39–61 (2007) Aureli, P.V.: The Project of Autonomy. Princeton Architectural Press, New York (2008)

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Aureli, P.V.: More and more about less and less. Log 16, 7–18 (2009) Aureli, P.V.: The possibility of an absolute architecture. The MIT Press, Cambridge, Mass (2011) Elderfield, J.: Grids. first published in Artforum (1972). https://www.artforum.com/print/197205/ grids-36215. Accessed 26 Feb 2022 Group, Z: Grid on Architecture, Tokai Education Research Institute, Tokyo (2020) Katagiri, Y.: Gianugo polesello’s architectural ideology and column debate with aldo rossi collaboration in architect adolescence of polesello-rossi for theoretical contribution to la tendenza. J. Archit. Plann. (Trans. AIJ) 777, 2437–2445 (2020) Katagiri, Y.: Grid on the cube of new san cataldo cemetery. In: Proceedings of The 13th Asian Forum on Graphic Science (AFGS2021), pp. 14–19, Springer, Hong Kong (2021) Kishida, S.: Kenchiku isho Ron. Architectural Design Theory), Maruzen Publishing, Tokyo (2012) Make and play Senet - Royal Albert Memorial Museum. https://rammuseum.org.uk/wp-content/ uploads/2020/05/Make-and-play-Senet.pdf. Accessed 28 Feb 2022 Lucan, J.: Composition, Non-composition: Architecture and Theory in the Nineteenth and Twentieth Centuries. Routledge, London (2012) Polesello, G.: Gianugo Polesello, Progetti di architettura. Pierluigi Grandinetti(ed.), Edizioni Kappa, Rome (1983) Polesello, G.: Città, monumenti antichi e nuovi monumenti: il progetto per la Camera dei Deputati a Montecitorio, Il Progetto Del Monumento - Tra Memoria E Invenzione, 101–107. Edizioni Gabriele Mazzotta, Milan (2000) Proto, F.: Abject objects: perversion and the modernist grid. Archit. Cult. 8(3–4), 564–582 (2020) Stoppani, T.: Paradigm Islands Manhattan and Venice Discourses on Architecture and the City. Routledge, London (2011) Takeuchi, H., Iwaoka, T.: Study on the module of framework in Japanese contemporary houses. J. Archit. Plann. (Trans. AIJ), 591, 233–238 (2005) Takeuchi, H., Iwaoka, T., Hanyu, S.: Study on the meaning of scale in architecture. J. Archit. Plann. (Trans. AIJ),594, 231–236 (2005) Witt, A.: Formulations. The MIT Press, Cambridge, Mass (2021) Yasuoka. Y.: A new interpretation of the grid system reform in the late period. J. Egypt. Archaeol. 107, 1–2, 265–280 (2021)

Stencils Are like Pencils. On the Ambiguous Visuality of Laser-Cutting Templates from Model Making – Substance Versus Cutout as Constructive Vagueness Niels-Christian Fritsche(B) Technische Universität, Dresden, Germany [email protected]

Abstract. Stencils focus on what ought to be painted, printed or sprayed on other surfaces by omitting it. Hence, a plus-minus ambivalence is introduced where a negative (cutout) allows for a positive (message). This yin and yang resonates in areas such as information design with airbrushed words and numbers as well as stencil art, ranging from political messages to abstract graffiti. With the advent of cost-efficient laser cutting for model making, cutout form templates – in contrast to stencil carriers – emerge in great numbers in dumpsters that can be considered illegible leftover waste (the cutout substance missing) or as off-cuts that spur imagination. We see visible and therefore substantial contour forms that are fundamentally empty in terms of the immateriality of the enclosed areas, oxymorons of ‘substantial’ omissions in 2D. Likewise 3D. 3D plastic printing of sophisticated objects requires interim support structures. Once exposed in the cleaning process, these support structures reveal impressive 3D balances of geometric reasonability and visual arbitrariness. Such 2D and 3D templates inspire to view geometry and graphics differently, to flip through the pages of art history, and – and most intriguingly – consider artistic reuse. Keywords: Stencil Residue · Template Ambiguity · Geometry and Graphics · Projective Test · Design Inspiration

1 Introduction: Ambiguous Obviousness The story so far: Fascinated by low-threshold 3D plastic printing and laser cutting, I overlooked the leftovers, with a clear focus on achievement in the long tradition of Western thinking in hierarchies and oppositions, thereby masking the yin-and-yang appeal of geometric yet negative 2D stencil recess and incomprehensible yet positive support structures in 3D plastic printing.

2 Ambiguity of Modern Art Pattern recognition defines a bracket in perceptual psychology. We recognize patterns that are important to us either as opportunities for progression or endangerment thereof. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 489–496, 2023. https://doi.org/10.1007/978-3-031-13588-0_42

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Our brains compute that harmlessness equals insignificance – be it every day routine or unthreatening strangeness – to save us the effort of unnecessary thinking. Hence, the arts and art history have entertained themselves with recognizable representation and figurative sculpture. Focus on substance as such suppresses considerations about side effects like waste. Modern art has trained us to accept abstract paintings and non-figurative sculpture. In 2019, Peter Weibel curated an extensive exhibition about “trajectories of sculpture in the 20th and 21st centuries”, called “Negative Space”. Apart from abstraction, Weibel recounts new oppositions such as “void instead of volume” and “weightless instead of heavy” [1]. A beautiful moment of ambiguity close to, or equal to, art experience struck me when I passed the dumpster of our makerspace. Layers of cardboard and plywood stencils – hang on, these are anonymous templates (or, in engineering: templets) with multitudes of irregular recesses and randomly piled up – constituted imagery in itself. Suddenly, there it was. Deliberate outlines of purpose, ambiguous obviousness of templates, geometrically appealing yet incomprehensible and ripe for informal reuse, the oxymoron of ‘substantial’ omission. I was enchanted (see Fig. 1).

Fig. 1. Left: Dumpster diving for anonymous laser-cutting residue for model making that has outlived its purpose – model-making components long since removed. Right: Garbage dumpster image window – ambiguous stencils form layers of bits of information out of focus, almost like a look into a well. Images by the author (2018).

Subsequently, I became a vivid dumpster diver, rescuing stencil arrays from the elements and the garbage collection company.

3 Yin and Yang of Laser-Cutting Residue from Model Making The art of stenciling and use of material stencils, patterns, jigs, screens and molds highlights the eternal duality of substance in the shape of stencil and color as well as accident. We brush or spray paint onto stencils as daub with fuzzy edges, take the stencil away, and what is supposed to appear are near perfect colored surfaces with crisp edges on the surfaces we initially put the stencils on. Stencils are supposed to be physically firm whereas surfaces to be colored shall appear two-dimensional, even if – and sometimes despite – color being applied to rough surfaces. Primarily, stencils are transmitters of information, not an end in itself. They bear physical substance but it is their voids that define purpose (why) and message (what).

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The ambiguity of both the laser-cutting residue from model making, the templates, and the cutout immateriality appears to be striking in terms of the defined geometry of the forms – based on mathematical rules – and their fuzzy, ambiguous graphic appeal. We see ambivalent designs with constructive vagueness, voided voids and freed compulsions, at the same time. As forensic laymen, the missing parts make us question what they were made for. Psychologically, the gaps remind us of the Rorschach test. Common stencils allow the transfer of messages onto surfaces full size, 1:1. Templates of architectural models to scale facilitate mostly rectangular cutouts of baseplates and elevations, three dimensions in one plane. Decorative, flowery templates trigger the imagination to an even greater extent. In landscape architecture, models consist of piles of curvy yet scaled cutouts, representing contour lines (elevations) in the field. Neither template serves a purpose any longer. Yet, precisely this nature of waste makes a perfect setup of vector geometry prone to pixel color application beyond stenciling purposes. A geometry-and-graphics template yin and yang appears, freely quotable according to Laozi’s Tao Te Ching: You can mold clay into a vessel; yet, it is its emptiness that makes it useful. How do we cope with emptiness? Images shown in this paper possess work-inprogress character to support the didactics of this paper. 3.1 Laser-Cutting Residue from Model Making as Positive Material Substance with Voids and Isolatable In-Between Forms First, the templates, residue of laser-cut stencils for model making, act as objects in themselves. They share rectangular bases with regular and irregular voids in differing degrees of tightness. What You See Is What You Get. A parallelism of comprehensibility and take it as what you like it to be opens up. I exhibited the ambiguous laser-cutting residue. Some voids began to talk like speech bubbles; others remained silent (see Fig. 2).

Fig. 2. Left: An array of templates (2018). Right: Selection of plywood stability bridges (4 × 37 cm, 14 × 25 cm) (2019). Images by the author.

Templates with recognizable voids such as the “RUSH” sample (Fig. 2, left) ask the viewer to reach for the spray can although the purpose of that template, the isolated letters R, U, S and H, obviously was not stenciling (otherwise, the template wouldn’t have ended up as garbage, unused). Generic and curvy compositions were so rich in form that image-stitching software would face problems with pattern recognition later. A first hesitant approach to action happened by accident: Fragile stability bridges, thin pieces of a stencil collection, fell out due to material overstress. Oops? No! Another

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beautiful layer of consideration! Why not destroy the templates intentionally? Templates with multiple patterns contain a message layer with each single void and a concrete – if coincidental – layer of in-between forms that – if broken out – may become an even richer if more ambiguous inventory of unbiased geometry (see Fig. 3).

Fig. 3. Left: An anonymous plywood template (30 × 15 cm). Center: Detail (15 × 15 cm) and detail broken out with the look of a face looking to the left. Right: Reversed image with new associations as if spray-painted. Images by the author (2021).

Other thoughts weren’t pursued further. Coincidental dumpster arrangements possess artistic random generator value. Stencil templates remind us of 3D collages in modern art such as Kurt Schwitter’s “Merzbau” from the 1920s. By heritage, residue from model making represents feedstock for new sculpturality. 3.2 Laser-Cut Templates as Negative Image Stock of Omissions Merge Picture Frame and Painting Surface First, templates define picture carriers already. With their noncontinuous surfaces, templates operate like images with lost or censored portions of their content, vaguely resembling iconoclasms, the destruction of icons, images or monuments. The application of paint comes next, not only because the laser-cut residue from model making resembles stencils. Consequently, templates apply as fragmentary surfaces for projection. Sprayed and painted templates merge intermediate object, the application of color, and permanence (the paint is there to stay) (see Fig. 5, left and center). Second comes printing in two fashions: Freshly painted templates with their noncontinuous surfaces function like printing plates in relief printing techniques. At some point, stencils may become paint-laden like a painter’s spatula. Paint-splattered surfaces turn into reliefs, enriched potential for stamping. Whereas soiled edges – paint creeping between plate and paper – ruin woodblock printing and etching, here, in informal template printing, surplus paint adds to the abundant balance of geometric predestination and quaintness. The material is the medium that is – freely associated with Marshall McLuhan – a biunique carrier of information unlike message, a filing without a linguistically describable statement (see Fig. 4, 5). In contrast to other art forms, we have two applicable sides – front and back – at our disposal. We mirror back and forth, turn patterns upside down, bend the templates, and so forth… Eventually, painting by brush, spraying in stenciling fashion and multiple applications of various templates with fresh paint on either side onto surfaces at will form a magical box of possibilities. Even simple single-form letter and character template stencils lose the predetermined nature of their content if half raised on one side, used in reverse and by spraying one color on top of another still-wet one and mixing them (see Fig. 4, right).

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Fig. 4. Left: Floppy cardboard template with traces of spray color (90 × 60 cm) and spray-paint on paper monotype (unique print) with ambiguous images and reversible figures (100 × 70 cm) (2019). Center: Work-in-progress abstract graffiti, sprayed with various templates on paper (77 × 150 cm) (2019). Right: Spraying from different angles, intentionally out of focus. Sketch (2022). Images by the author.

Fig. 5. Left: Template collection (2019). Center: Spray-painted plywood template (55 × 100 cm) as object in itself and informal window view frame (2022). Right: Work-in-progress surface with hugging pattern, defying the logic of stacked layers (detail, 46 × 58 cm) (2019). Images by the author.

Combining spray paint jet effect, brush stroke appeal and the printing of paint-stained templates transforms rigid stenciling into broad expressionism that keeps carrying the geometrically clear outlines according to Theo van Doesburg’s line, plane and color as impersonal expression (concrete art) and adds wildstyle graffiti attitude to it. Far from algorithmically generated “multi-layer stencil creation from images” [2], the focus is on the encounter of garbage with and as projective potential. Sharp laser-cut outlines with procedurally burned edges, traces of spray perfect at the center and blurred towards the edges as well as brushstrokes and thinner application transform serial stenciling with recognition value to unique prints (monotyping). The visible, and therefore positive, contour as negative blind spot of incomprehensible form is positive in terms of conceivability. We see something that is evidently designed on purpose, not arbitrary, but not self-explanatory either. Yet, it does appear coincidental. Sometimes I encounter templates that appear much richer in form than their supposed cutouts (see Fig. 6). Stencil ambiguity and the range of color applications with imperfect, unspecific traces of work appear valuable for actively teaching self-referred visibility – what you see is what you get; please feel free to take it from here – as a fundamentally unexplored component of our wide-range perception that intrigues me the most.

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Fig. 6. Reverse engineering (2019). Left: Laser-cut stencil residue (65 × 72 cm) spray-painted on paper (77 × 150 cm). Center: Ornamental plywood-bridge detail (30 × 40 cm). Right: Inverted image. The ornamental motif conceals the mass production of same-size water-drop shape plywood patterns (17 × 16 cm). Images by the author.

4 Playing with 3D Printing Vertical Support Structures 4.1 Miraculous Support Structures in Plastic 3D Printing Low-threshold 3D printing (additive manufacturing) focuses on polymeric materials. Buildups with voids and overhanging parts need technology-determined support structures. Whereas sophisticated 3D plaster prints are fragile but easily vacuumed after completion, sophisticated 3D plastic prints require rather careful freeing from support by hand. During my “transitions” curriculum addressing the structural and design routines of modelling geometry in computer-aided design with the slogan “wild combination of irregular transitions” [3] I experienced another garbage déjà vu with plastic print residue equating to 3D templates (see Fig. 7). What great characters they are!

Fig. 7. Left: Scan of support structures of plastic 3D printing (29 × 43 cm). Center: Photo of polymeric support structures. Right: Polymeric support structure (13 × 13 cm). Images by the author (2021).

Initially, the joy of liberating 3D plastic prints from their support structures overshadowed the richness of the carrying structures themselves. Again, as with the initial dumpster impression of 2D laser cut templates, the leftover nature of disenfranchised 3D support structures masks their visual richness and their potential for imagination. How do we arrange geometry and graphics this time?

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4.2 Footprints, Slipstreams, Shadow Residue and Idling Cycles of Basic Design Laws Some support structures appear to be pieces of art in itself. Other structures leave remarkable footprints via flatbed scans and spray paint. They cast shadows and spray-paint footprints with new associations in the slipstreams around and in-between geometry and graphics. Bits and pieces start new rounds of stimulation, from the look-and-feel of children’s play boxes all the way to pareidolia, our intent to impose meaningful interpretations on nebulous stimuli, preferably faces. Let’s allow ourselves to recognize nonsense (see Fig. 8).

Fig. 8. Left: Generic support structure footprint that resembles a Rorschach test image. Spraypaint on paper (30 × 43 cm). Center: Photo of a spray-painted support structure with the appeal of an extruded island on a globe. Right: Ambiguous photo image – color, forms and image carrier form a 3D camouflage picture. Images by the author (2022).

The basic design laws from the beginning of the 20th century come to mind, but with a twist. In the template jumble, we try to pick out circles, squares, and triangles according to the law of the simple form only to realize that all these now missing shapes were meant to build cardboard and plywood models, i.e. rather sophisticated 3D designs. Similarly, we focus on elements that are close to each other according to the law of proximity as well as similar elements (the law of equality), both to no avail. Someone placed forms onto the laser cut template not with an intended aesthetic or design, but rather with a space and cost-saving agenda. The same happens to our observance of the laws of closure, experience and constancy. Finally, and most prominently, we struggle with the law of figure-ground separation since the cutouts – absent 2D figures on present 2D templates for prospective 3D model making – are missing.

5 Synthesis: Let’s Talk About “Concrete Stenciling” The view of the cutouts as two-dimensional negatives, i.e. what the laser cutting is missing, but what it was made for in terms of three-dimensional model making, suggests a preliminary mental stroll through art categories. The template jumble reminds us of the dazzle painting to camouflage ships in World War I and since. Color, applied in zebra stripe patterns, was supposed to break form, the silhouettes of ships on the horizon seen from submarine periscopes. Modern artists used stenciling in figurative and abstract fashion as a staple of their work. Thereby, stencils were intrinsic to the compositions. This piece of art, with the help of stencils, ought to start looking like that. Yet, the recycling of laser-cut templates

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begs to differ. The strange, anonymous template, made for unknown purposes, precedes the signature composition. Look, here is something. What are we going to do with it in its second life? Even though playing with laser-cut residue resembles stencil graffiti, pochoir (French for stenciling as visual art) doesn’t apply to the reuse of laser-cut templates. Pochoirs aim at recognizability and brand recognition via transfer pictures (decals), in opposition to monotype transfer printing like klecksography and imprint technique (“Abklatschtechnik”) according to Max Ernst’s series of graphic inventions such as frottage, grattage and decalcomania. Next to destructive iconoclasm, already mentioned, templates bring to mind glitch art as artistic iconoclasm, advancing through errors. More research is need to determine the span of creative destruction via unprejudiced trial-and-error-habit to rather nihilistic punk ethos. The aesthetics of waste certainly apply. Dumpster diving defines a form of recycling that fosters bricolage (French for do-it-yourself projects), the use of materials at hand in contrast to costly high-end art supplies. Let’s relate the reuse of residue to databending and datamoshing in the digital realm as well as to wabi-sabi, the traditional Japanese aesthetics with emphasis on imperfection, asymmetry and austerity among others in the physical world. Whereas the term “abstract stenciling” refers to decorative design in the tradition of home decoration cutouts, perhaps the concept of “concrete stenciling” or “templating” – as a portmanteau word made of template and contemplating – starts to sound reasonable. Concrete art focuses on geometrical abstraction, as is the case with laser-cutting residue. The missing parts of the templates were designed for model-making purposes. With the purpose gone, the templates with leftover-cutouts remain geometrically concrete, productive, yet oxymoronic omissions of substance. Let’s contemplate the departed content of the templates in front of us.

References 1. Weibel, P. (ed.): Negativer Raum. Trajectories of Sculpture in the 20th and 21st Centuries. The MIT Press, Cambridge, Mass, London, England (2021) 2. Jain, A., Chen, C., Thormählen, T., Metaxas, D., Seidel, H.-P.: Multi-layer stencil creation from images. Comput. Graph. 48, 11–22 (2015). https://doi.org/10.1016/j.cag.2015.02.003 3. Fritsche, N.-C.: Modelling exceptions. sense-free transitions by guerrilla design: cheat on habits and software. In: Nexus 2021 – Architecture and Mathematics. Conference Book, pp. 69–74. Kim Williams Books, Turin (2021)

Gems Geometry: From Raw Structure to Precious Stone Nicola Pisacane(B)

, Pasquale Argenziano , and Alessandra Avella

Department of Architecture and Industrial Design, Università della Campania Luigi Vanvitelli, Monastero di San Lorenzo ad septimum, 81031 Aversa, CE, Italy {nicola.pisacane,pasquale.argenziano, alessandra.avella}@unicampania.it

Abstract. This paper introduces some considerations regarding the geometric principles that lead the configuration of precious gems. Geometry crosses and characterises different aspects of precious gems: from the atomic composition of its raw structure, to the classification of the crystalline system, up to defining shapes and configurations of possible cuts, as well as to control the functioning of the instruments that allow its execution. From age to age, geometry is leading the lapidary art that bases its foundations and rules on it, mainly in order to cut and manufacture the stones, and also to identify the rules that could enhance the brightness or minimize the wastefulness of precious material during the processing. Starting from research still in progress by the authors, this paper will outline the role of geometry in the fields of crystallographic classification, the cuts classification, the methods for design and execution, also through the conspicuous iconography and the digital modelling. Keywords: Crystallography · Gemstone cutting · Gemstone digital modeling

1 Introduction If the disciplines that study the manufacturing of isotropic materials are based on Geometry, and its declinations in order to the type of material processed and the aim of application are numerous. In the context of building constructions, the importance of Stereotomy, developed in the broader of Descriptive Geometry, cannot be overlooked. If Stereotomy – as its etymology indicates – is generally aimed at cutting isotropic materials (stones, wood, metals), the widest application concerns stone materials and its applications to architecture. In this context, the Stereotomy bases its principles on rules that combine geometric principles with principles of stability of buildings, as well as a knowledge of rocky materials and their characteristics. It is no coincidence that this discipline, which had wide diffusion in Europe, in France found a fertile background due to the extensive use of calcareous and sedimentary rocks for the construction of buildings and infrastructures [1]. Otherwise, when isotropic materials are natural mineral substances and the field of application refers to the manufacturing of precious stones, the discipline is “lapidary art”, and still recognises its rules in geometry [2]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 497–508, 2023. https://doi.org/10.1007/978-3-031-13588-0_43

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In lapidary art, the minerals are classified into precious stone, semi-precious stone or gem, and pietra dura according characteristics of decreasing transparency (precious stone has high levels of transparency and hard stone has no prevailing transparency) and different morphological, physical and chemical properties. The lapidary art in the manufacturing of minerals for ornamental purposes dates back, according to historians, to the times of the river civilisations of the Mesopotamian area (V century of BC), where the polishing of the genuine faces of crystals and of pebbles belonging to alluvial deposits was made. Only later, the invention of cabochon manufacturing introduced the polishing of dome minerals without cuts, with smooth and rounded edges. The first lapidary models cut into facets to obtain prismatic shapes that enhanced the known qualities of brilliance, started from the XIII century AC. The most complex processing systems, such as carving and engraving, and refining originated from simple polishing and crystal faceting processing systems. All these processes belong to the artistic technique called “glyptic” (from the Greek “carving”) and were transmitted almost unchanged until the introduction of mechanisation processes in the XVIII century [3]. Since the Twentieth century, in the field of lapidary techniques applied to precious stones, specific studies on the determination of input parameters for the design of models, including digital models, have been undertaken and are still ongoing. These parameters are able to menage the cut in order to maximise the optical characteristics of the precious stones and the yield obtained. Despite the undoubted interest of the topic and the substantial volume of business of the gemological business, the research in progress is still not widespread and the areas involved are limited, as well as there are few economic resources invested in the advancement of knowledge. This research is part of a wider study carried out by the authors into the Research Group “Gems and Jewels: History and Design” [4] at the Department of Architecture and Industrial Design [5]. In this paper, the authors outline the ordering principles of the research, based on statistical, mathematical and digital studies on the topic, tracing in the latter the recurring geometric basis that will be declined in the study of the crystallographic systems models, in the analysis of cutting configurations, and in the aspects properly connected to dimensional survey and gem manufacturing, as better argued in the following paragraphs.

2 Gems Geometric Principles in its Raw State. Crystallographic Models and Proportionality Rules The geometric fundamentals that regulate the gem cutting find their origins in the configuration of the crystallographic structures. In fact, the approach to the study of gemstone shapes and cuts starts from the analysis of the different atomic structures and the geometric principles that steer their organization. In this paper, far from deepen the crystallographic aspects of mineralogy, the mainly geometric aspects that characterize crystalline structures will be developed. The geometric organization of crystals is already introduced at the end of the 18th century, a period in which the observation of mineral species led to a scientific approach in the study of the crystallographic and physical properties of minerals. In this period, starting from the work of the Abbot René Just

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Haüy, the classification of crystalline forms and symmetries began to be systematized [6–8] (Fig. 1).

Fig. 1. Planche 6 from Traité de cristallographie by René Just Haüy published in Paris in 1822.

Starting from his studies the discipline takes on a scientific character and in the following decades researchs and theories will follow transferring principles of spherical trigonometry and geometry of polyhedrons to crystallography, determining a new approach to the study of minerals [9]. This approach finds its foundation in the study of the arrangement in 3D space of atoms according to ordered, regular, constant and repetitive series, following the geometric rules of tessellation of space. On this spatial organization are classified the crystalline systems and their geometric lattices. Specifically, in crystallography, it is due to the French physicist Auguste Bravais the systematization in 1848 of the models of geometric arrangement of the molecular entities that define the structure of crystals [10, 11]. Bravais’s work is significant not only because it confirms the geometric approach to the study of crystalline structure but marks the transition from solid spatial systems to discrete point systems. Crystallography, in fact, first identified seven possible crystalline systems, subsequently refining this classification through the introduction of spatial

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lattices and fourteen possible models of 3D organization based on the arrangement in space of the particles that make up the crystal itself according sets of points through infinitely repeatable patterns in space. Both geometric models (solid and point) find their origin from the physical and atomic structure underlying the mineral itself and whose knowledge is fundamental for the definition of the possible best cuts and processes to which they can be subjected. The organization of the seven possible crystalline systems is based on the length of the faces and the angular ratio between them and, therefore, on the type of symmetry that is created between the faces themselves. These dimensional and morphological relationships refer to the elementary cell that by reiterating in space gives life to the crystal and its 3D configuration that can also differ from the geometric shape of the constituent cell. The declination of the different crystallographic systems is based on the spatial arrangement of the crystallographic axes that find their common origin in the center of the crystal and from this are oriented according to well-defined angles and directions. Already Haüy in 1782 stated that the reference axes of a crystal are three oriented lines, parallel to three converging and non-coplanar edges. Depending on the length of the axes and the angles between them the cubic system (axes of equal length and three right angles), tetragonal (two equal sides and one unequal but three equal and right angles), orthorhombic (three unequal sides and three right angles), trigonal (three equal sides to each other and three angles equal to each other but differently inclined with respect to the right angle), hexagonal (two equal sides that form an angle of 120° between them, while the other two angles are straight and the third side of different length than the previous ones), monoclinic (three unequal sides and only two right angles) and triclinic (three sides and three unequal angles) are determined (Fig. 2). These arrangements in 3D space, in addition to the seven aforementioned models, also identify different symmetry ratios between the geometrically and physically homologous portions of the crystal [12]. Also in case of symmetries three possible types are defined: symmetry to a plane or reflection in relation to its surfaces, according to an axis or rotation in relation to it and according to a point or inversion in relation to the same point. The symmetry of a crystalline structure, however, should not be understood and referred only to the external geometric shape of the crystal but also to that recognizable after a chemicalphysical analysis of the crystal to allow it to trace its minimum geometric structure. This symmetry, called real or true symmetry, can be inventoried according to 32 classes, each of them corresponds to one or more typical minerals. Each class also corresponds to a crystalline system based on geometric shapes also born from the aggregation of regular polyhedrons or from solid volumes in general that also define the properties of individual minerals and their predisposition to be cut according to particular shapes or geometries. The developments of studies in mineralogy led to the transition from solid to punctual geometric models for the classification of crystal structures. These models still effective provide the space organization according to 3D lattices in whose nodes are located the atoms constituting the minerals. These nodes also represent the elements of conjunction with further possible lattices according to a geometric structure theoretically repeatable to infinity. August Bravais, who introduced such models, also hypothesized the existence of only 14 possible fundamental schemes according to which the particles that form the crystal can be distributed. These schemes base their geometric structure on the already

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discussed crystalline systems, identifying in these the possible positions assumed by the atoms and therefore the symmetry relationships between the parts. For example, the cubic crystalline system can correspond to three possible lattice structures depending on whether the atoms are located only in the vertices of the same cube, or even in its center of gravity or in the centroid of the six faces and so also for the tetragonal, trigonal, hexagonal, orthorhombic, monoclinic and triclinic systems different declinations of lattice structures are possible, each characterized by one or more forms of symmetry.

Fig. 2. Crystallographic axes system (top left) with the identification of axes a (crystal length), b (crystal width) and c (crystal height), the origin O of the axes and the angles α, β and γ shaped by the aforementioned axes. The seven crystallographic systems defined in relation to the angular and dimensional relationships between the parts. (Drawing by Francesca Fabozzi)

Knowledge of the atomic-scale organization of the crystal structure is fundamental for the influence it has in determining the characteristics of each precious gemstone. Each of these, in fact, is characterized by a specific crystalline system that defines, in theory and according to models that exclude the presence of impurities and imperfections typical of a natural mineral, the properties of resistance and response to light also in relation to the geometric and dimensional configurations of the molecular structure. It will be these geometric characteristics that will partly orient the cutting of the gem itself according to principles and cutting shapes that will find their foundation in geometry.

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3 Gemstones Cutting: Geometric Principles and Morphological Configurations The geometry regulates the conformation of precious gems from the raw state to the cut stone. In the raw state, as can be seen in the previous paragraph, geometry governs the atomic crystalline structure of the stone; in the cutting process, as described in this paragraph, geometry leads the possible best cuts to obtain a specific shape of the gemstone. The raw stone used in ornamentation and artistic processing is a naturally crystallized mineral that appears like regular solid geometries, with generally opaque surfaces and intrinsic structural characteristics of color, transparency and brightness highlighted by the cutting process, from which the precious characteristics or semi-precious depend [13]. If in ancient times, the stones were used as they appeared in nature, applying rudimentary polishing and cutting actions to enhance their appearance, starting from the Sixteenth Century there are the first examples of stone cutting that can to be considered the early cutting processes as still used. According to this processing, the cut is applied not only in the top of the pyramid of the crystal along a plane perpendicular to main direction of the crystal, known as “table”, but also in the different edges of the crystal generating multiple facets on which to refract the light enhancing the stone brightness. This cut, known as “faceting”, and the “cabochon” one, are the main types of gemstones cutting. The “cabochon” cut is still widely used in the East because it alters the raw state of the gem as little as possible. It is mainly applied to opaque, semi-opaque or transparent and coloured stones - such as opal, turquoise, moonstone, cat’s eye and the like - which take on a convex and semi-ovoid shape. When it is made in the complete ovoid shape, rarely, it is known as “double cabochon”. The faceting, on the other hand, is applied to transparent or semi-transparent stones exclusively such as diamonds, emeralds and rubies, because it enhances their reflective properties. While the “cabochon” cut slightly alters the original nature of the stone, on the other it does not allow to enhance its dichroism, nor to highlight its intrinsic dispersive and reflective characteristics, as happens with the faceting [3]. According to the studies of Arthur Herbert Church, the different gemstone cutting forms can be grouped into “flat” and “curved” surface; mixed facets and curved surfaces is rarely into the same specimen. The “surface” group includes Brilliant-cut, Step or trap-cut, Table-cut, Rose-cut and various declinations of these cuts; the “curved” one, on the other hand, includes Single cabochon, Doable cabochon, Hollowed cabochon, Tallow top [14]. Although Arthur Herbert Church recognises two main cutting forms, “flat” surface cuts and their geometric principles are explored below. The quality of a specific cut is evaluated about the improvement of the intrinsic optical performance and weight maintenance of the stone, from which the degree of preciousness depends. The preciousness, in fact, is estimated on the basis of the parameters of brilliance and of material dispersion. By brilliance it is meant the response of the precious stone in terms of its ability to refract light. By material dispersion it is meant the weight ratio between rough raw stone and faceted precious stone.

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The geometry leads the choice of angular inclination of faces both to maximise the characteristics of the crystal to refract light and obtain a better brightness, and to obtain the biggest faceted gemstone, called “yield”, starting from the raw stone. Over the centuries, the developments of studies in optical mineralogy has allowed to better control the results of the different faceting methods applied, thus overcoming the limitations of the past technologies. The ancient cutting methods, in fact, did not always highlight the optical characteristics of the crystal, nor they did consider its physical characteristics, and the results were almost always obtained empirically. Taking into account the physical characteristics of the crystal means faceting the stone by adapting to the geometric structure of its crystalline system by operating along the cleavage planes, where the bonds of the molecular structures are weaker and therefore more suitable for cutting, with the further advantage of reducing the possibilities of breakage and waste of the material. During the cutting design step, therefore, it is necessary to calculate the critical angle of the specific material to obtain the greatest possible brilliance from the processing in addition to evaluate the cleavage planes, the class and the degree of symmetry of the stone [15]. The brilliance is defined as the fraction of the incident light that returns to an external observer positioned in front of the crown, after successive internal reflections. When a ray of incident light reaches the surface of the gem, part of it is reflected and the other part is transmitted or refracted through the air-gem interface. The amount of reflected light is proportional to the refractive index of the specific material, a constant number that indicates the gem’s characteristic to slow down light. Furthermore, the angle of reflection is equal to the angle of incidence, they both lie on the same plane and are measured with respect to the normal (always perpendicular) to the surface of the gem at the point where the ray of light hits the surface. The refracted ray, on the other hand, deviates from the incident ray towards the normal and slows down according to the refractive index of the specific material. The refracted fraction of the ray travels inside the gem changing direction and repeating this phenomenon when it reaches another gem-air interface, this time on an internal surface [16]. In the gemological field, ideally, the faceting of a gem must be made according to specific inclined cutting planes intersecting each other by angles that allow the entry of natural light and its exit after it has bounced numerous times on the inner faces of the solid. In order for the light ray entering the gem to reflect inwards (internal reflection) ricocheting an indefinite number of times on the surfaces of the gem, the angles that are formed by hitting these surfaces must be greater than that of the limit angle of the itself gem. When this ray hits one of the internal surfaces at an angle smaller than the limit angle, it manages to come out of the material [17]. Although the gemological literature recognizes an ever-increasing number of cutting models based on specific geometric and proportional principles, assuming the cut of the colored gem as a generalization of the diamond cut, in this discussion we will refer to the “brilliant cut” according to the ideal model designed by Marcel Tolkowsky in 1919. The different parts that make up the diamond, and by generalization a faceted gemstone, are: the crown (the upper part), the table (a large flat facet in the center of the crown), the girdle (the outer edge of the stone that divides the crown from the pavilion), the pavilion (the lower part of the faceted stone and extends from the girdle to the culet) and the culet

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(a small facet at the base of the pavilion, useful to prevent the stone from splintering or damaging). The proportions that play an important role in determining the brilliance of a diamond are the percentage of the table, the angle of the crown, the angle of the pavilion and the thickness of the girdle. These proportions, in a variety of combinations, can produce high and low levels of brilliance [18]. In the Tolkowsky ideal model, in which the brilliance is maximum, the dimensions of such parts are regulated by rigorous proportional relationships calculated on the basis of the diameter of the girdle, according to which: the total depth of the gem should be 59.3%, of which 43.1% should be the depth of the pavilion and 16.2% the height of the crown, the table should not measure more than 53%, the angle of the crown should be 34.5°, while the angle of the pavilion it should be 40.75° [19]. It is possible to state that the geometry of the faceted model is parametric: given a specific faceted model, the angular values of the facets remain constant as the lengths of the edges vary. This perfectly follows the first fundamental rule of crystallography enunciated by Niels Stensen in 1667 [12]. Consequently, we can deduce that man tries to reproduce in gemology what is observed in crystallography. Given the geometric principles that lead the different cuts, each cut can have different shapes, depending on the plane figure that envelops the crown and the physical properties of the atomic structure of the specific crystal, and can generate multiple spatial configurations. The complexity of the morphology obtained is directly proportional over time to the technological progress of cutting tools and to the development of new abrasives (Fig. 3). In particular, more articulated shapes are obtained starting from the early 1800s with the application of the French jambpeg technology which, not being equipped with systems for calibrating the angles, does not yet allow full control of the orientation of the facet planes, which only the subsequent technologies will perfect [20]. This jambpeg technology allows to overcome the limits of the first handheld equipments, which with still rudimentary techniques carried out only the smoothing of the rough stone, and of the first mechanical tools used from 1400 to 1600 which did not allow to orient the stone to be cut, to which a single horizontal cut was applied. The improvement of cutting technology is recorded starting from 1900 with the use also in the gemological field of the first machines equipped with a vertical shaft and goniometer, which through a graduated crown allows you to control the rotation of the stone by setting the orientation angles of the cutting plans, in relation to the atomic structure of the rough stone and obviously of the desired shape [17, 21–25]. This cutting technology is still in use today for the colored stones, while only to diamond cutting, in particular for the “Tolkowsky brilliant cut”, many processing steps have been automated through the use of laser cutting machines, then digitized through closerange photogrammetric technology, which is still being tested also in the gemological field, as better described in the following paragraph.

4 Precious Stones: 3D Shape and Digital Modeling In the fields of lapidary and glyptics – as can be seen in the previous paragraphs – a known theoretical aspect is confirmed: survey and design activities are mutually linked through geometry and measurement, in relation to the material characteristics of the

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Fig. 3. Time-line framework: cutting classification, gemstones cutting shapes, faceting and polishing equipments. (Drawing by Francesca Fabozzi)

stone. Stone material determines its shape; geometry presides over its conformation and it is fundamental for its study; the measurement defines the dimensional scalability of the stone – in relation to the International System of Units – within standard values of tolerance and accuracy. In glyptics – restricting my thought to the subject of this paper – detecting a raw stone or a cut stone has a remarkable importance. The characterisation of the raw stone (mineralogical classification, size, weight, for example) is necessary for the technician to have a full knowledge of the object, and therefore to design the best faceting, looking out for the balance among the values of brilliance and the material wastefulness. The deep characterization of the raw stone clarifies what are the planes of cleavage, the class and the degree of symmetry of that stone. In other words, the cutting geometry that can be applied to a raw stone must correspond to its internal geometry; if there were no such balance, a decay of the brilliance would occur after faceting, or even the cutting process would not be successful. For this reason, the faceting tools are basically goniometers. The raw stone – put in the center of this tool – rotates in order to the “gear index” that in turn depends on the periodicity of the cutting angles and of the planes of symmetry, as we read in the previous paragraphs. The gear index is a numerical value corresponding to the equal division of the turn angle, which is countable in: 32, 64, 72, 77, 80, 84, 86, 96 and 120. Among these, the toothed

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crowns #64 and #96 allow the greatest number of cuts in application to the greatest number of stones; on the other hand, the toothed crown #77 is used only for three types of cutting. The number and angle of the cutting planes are the basis of the parametric modelling software, made for the digital design of precious stones. The most popular software in this field are: “Gem Cad” and “GemRay” (designed by Robert W. Strickland in 2002) [26], and “Gem Cut Studio” [27]. The digital approach follows traditional one: 3D modelling starts from a primitive solid (cylinder or cube) [28] in whose center of gravity a threeorthogonal triplet is placed, and the parametric values of the cutting angle, of the planes of symmetry, and of the gear index are set. After the cut-stone modelling in a digital environment, the aforementioned software allow the simulation of the brilliance. While Robert W. Strickland developed two separate software that interchange files, Gem Cut Studio allows real-time faceting, photorealistic simulation, and analytical verification of brilliance with advantages in the project workflow. The geometric and structural survey of a cut-stone are necessary to determine its quality – the diamond, for example, has a dimensional standard tolerance of ±0.01 mm – and in some cases, they are even necessary to identify a specific precious stone among numerous similar ones [29]. In parallel with the development of two-circle goniometers for faceting, in the Nineteenth century two-circle reflecting goniometer were developed to measure inter-facet angles of the stones. These tools were used until the invention of new ones, based on the X-ray diffraction technique; in some cases, the former still attest to a high degree of accuracy of measurements [30]. In the geometric survey of the cut-stone, on the other hand, the caliber is still the main tool for measurements, given its high standard of metric accuracy in relation to the instrumental type. The most used calibers are three with accuracies of ±0.05 mm, ±0.01 mm and ±0.001 mm. Digitisation has also been applied to this field, and therefore close-range digital photogrammetry and 3D laser scanner are being tested for the capturing and modelling of cut-stone; the photogrammetric model becomes the “digital twin” of the real cut stone, and so geometric and metric evaluations can be carried out into the digital environment.

5 Conclusions The research highlights a common geometric aspect in all phases, starting from the crystalline structure of the raw stone to the cut shape. It is interesting to note that the geometry of the crystal lattice influences the form of the crystal regardless of its size and consequently orients the possible cuts of the stone, taking up the first statement of Niels Stensen (1667) [12]. The geometric relationships that bind the different phases are exemplified in the diagram of Fig. 4 in which they are highlighted: the cubic mesh octahedral geometric matrix of the diamond in the rough state, the possible symmetry relations of each minimum cubic-shaped unit, the orientation of the cutting planes with a position consistent with the aforementioned planes of symmetry, and finally the symmetry relationships of the cut stone from which derive the only possible configurations of the precious form. The progress of the research will decline the analyses conducted so far to the different crystals, deepening the most suitable cuts according to the different crystallographic

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Fig. 4. Comparative diagram of the raw diamond, its cubic crystalline structure matrix and the model of the precious stone worked according to the standard round brillant cut (above). Symmetry of the octahedral structure of the raw gem, the minimum cubic unit also based on its atomic structure, the cut stone concerning to the same mineral (below). (Drawing by Authors)

structures with reference to the geometric characteristics and in particular to the tessellation of the space, to the symmetry ratios, to the angles of position of the symmetry and cutting planes, to the proportionality relationships between the parts of the precious stone. Acknowledgments. The authors designed the research and shared its methodology and contents. In particular, the paragraph entitled Gems geometric principles in its raw state. Crystallographic models and proportionality rules is edited by Nicola Pisacane, the paragraph entitled Gemstones cutting: geometric principles and morphological configurations is edited by Alessandra Avella, while the paragraph entitled Precious stones: 3d shape and digital modeling is edited by Pasquale Argenziano. Introduction and Conclusions are edited by all the authors.

References 1. Perouse De Montclos, J.M.: L’architecture à la française, Picard, Paris (2001) 2. Nicols, T.: A Lapidary or, the History of pretious stones. Thomas Buck, Cambridge (1652) 3. Sborgi, F.: Glittica e lavorazioni affini. In: Baccheschi, E., et al. (eds.) Le Tecniche Artistiche, pp. 63-82. Mursia, Milano (1973) 4. https://www.architettura.unicampania.it/images/ricerca/gruppi/Jacazzi_Gemme_e_Gioielli_ HIDEeG2_2021_ITA.pdf, Accessed 05 Jan 2022 5. Pisacane, N., Argenziano, P., Avella, A. Dalla stereotomia, spunti per il disegno delle gemme. Insights into the gems’ drawing from stereotomy. In: Jacazzi, D., Morelli, M.D. (eds.) (a cura di) Gemme e Gioielli: Storia e Design, pp. 68–69. DADI Press (2021) 6. Haüy, R.J.: Essai d’une théorie sur la structure des crystaux, appliquée à plusieurs genres de substances crystallisées. Paris (1784)

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N. Pisacane et al. Haüy, R.J.: Traité des pierres précieuses. Paris (1817) Haüy, R.J.: Traité de cristallographie. Paris (1822) Borchardt-Ott, W.: Crystallography: An Introduction. Springer, Berlin (2011) Bravais, M.A.: Mémoire sur les polyèdres de forme symétrique. Imprimerie de Bachelier, Paris (1849) Bravais, M.A.: Études Cristallographiques. Gauthier-Villars, Paris (1866) Mottana, A., Crespi, R., Liborio, G.: Minerali e rocce. Mondadori Editore, Milano (1977) Farrington, O.C.: Gems and Gem minerals. A. W. Mumford, Chicago (1903) Church, A.H.: Precious Stones considered in Their Scientific and Artistic Relations. Wyman and Sons, London (1905) da Silva, J.T., Lazaretti Zanatta, A.: A real-valued genetic algorithm for gemstone cutting. In: XXXVIII Conferencia Latinoamericana En Informatica (CLEI), pp. 1–8 (2012) Mol, A.A., Martins-Filho, L.S., da Silva, J.D.S., Rocha, R.: Efficiency parameters estimation in gemstones cut design using artificial neural networks. Comput. Mater. Sci. 38(4), 727–736 (2007) Smith, G.F.H.: Gem-Stones and Their Distinctive Characters. Methuen & Co. LTD, London (1912) Sasia´n, J.M., Yantzer, P., Tivol, T.: Optics Photon. News 14(4), 24–31 (2003) Tolkowsky, M.: Diamond Design. Spon & Chamberlain, New York (1919) Brard, C.P.: Mineralogie appliquée aux arts. Paris (1821) Schmetzer, K.: A 15th-century polishing machine for gemstones attributed to henri arnaut. J. Gemmol. 6(36), 544–550 (2019) Toll, A.: Gemmarum et lapidum historia quam olim edidit Anselmus Boetius de Boot Brugensis. Joannes Maire (1636) Holtzapffel, C.: Turning and Manipulation. Holtzapffel & Co, London (1864) Prim, J.K., Lapidary Technology Through the Ages: Laps and Polish. https://medium.com/jus tin-k-prim/lapidary-technology-through-the-ages-laps-andpolish-59c29f05a11a, Accessed 05 Jan 2022 Latin 7295 codex, attributed to H. Arnaut and others, first half of the 15th century. Bibliothèque Nationale de France, Paris. https://gallica.bnf.fr/ark:/12148/btv1b90725989, Accessed 05 Jan 2022 https://www.gemcad.com. Accessed 05 Jan 2022 https://gemcutstudio.com. Accessed 05 Jan 2022 Sangveraphunsiri, V., Kankriangkrai, S., Prachya, P.: Development of a 3-D solid modeling system based on the parasolid kernel for gem stones faceting. In: The 22nd Conference of Mechanical Engineering Network of Thailand (2018) GIAResearch team, Measurement Tollerances. Accurancy and Precision in the Gem Industry, Gemology, pp. 183–185 (2005) Shen, A.H., Bassett, W.A., Skalwold, E.A., Fan, N.J., Tao, Y.: Precision measurement of inter-facet angles on faceted gems using a goniometer, gems & gemology, pp. 32–38 (2012)

A Multi-scale Investigation of Visual Interactions in the Built Environment via the Generation of Parametric Procedures Matteo Cavaglià(B) Department of Architecture, Built Environment and Construction Engineering (DABC), Politecnico di Milano, Via Giuseppe Ponzio 31 Bld. 15 - Bonardi Campus, 20133 Milan, Italy [email protected]

Abstract. This contribution aims at presenting the current progress on the definition of automated procedures able to test different types of basic visual interactions within the built environment. Following the example of past thinkers in the field, the set of proposed operations aims at collecting different kinds of data to better support multi-scale analysis about the impact of transformations processes in different types of contexts. The proposed work will also serve as a base for further elaboration on the topic. Keywords: Visual analysis · Multi-scale analysis · Data driven design · Parametric design

1 Introduction The following contribution describes a series of hypotheses about testing simple visual interactions between different elements, developing a work-flow able to highlight various characteristics of how an environment can be observed. As the sense of sight is usually our primary access to the space we live in, its related perceptions hold great importance in the formulation of architectural designs, being acts of manipulating the space around us. As such, properly assessing the visual relationships that people establish within a certain space, is a key factor for almost any undertaking. Once, this kind of inquiries were generally answered via graphical representation, until the scientific progress allowed the use of more compact and objective calculations. However, as the digitalization of many professions is consolidating, different expertise can now more easily intertwine their different points of view widening their possibility.

2 Visual Interactions as Tools to Control Parametric Design Processes Over the years, parametric tools of representation have contributed to defining noteworthy structural changes in many professional sectors with strong links to the discipline © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 509–518, 2023. https://doi.org/10.1007/978-3-031-13588-0_44

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of graphic representation [1]. The main point of interest of these instruments is the possibility to shift the subject of a design process, from the realization of an object to the generative process itself. Processes defined in this way become dynamic entities, capable of being implemented not only for the realization of a single solution but also multiple variations, via the manipulation of some basic parameters. An algorithmic framework that was already widely rooted both in human behavior and mentality [2], but that has become more versatile in our contemporaneity after the introduction of digital computing systems [3]. Also, thanks to the application of visual programming systems, coupled with libraries of commands closer to the mindset of graphic representation, more and more actors originating from “artistic” disciplines were allowed to easily access these tools [4]. One of the most interesting consequences of the increasing exchange of different professional fields is identified in the production of systems, specialized in the composition of various objects. It is a profound conceptual transformation, where the final product derived from the participation of a professional within a project is no longer a finished and well-defined instance but rather an open and dynamic algorithm [5]. A system capable of integrating a specific work process, potentially customizable by anyone through the implementation of their own and personal application parameters, in such a way as to produce a unique and unrepeatable solution, sometimes even without the direct intervention of the author of the original work process. The replacement of a physical entity, with a specific editor able to generate it, has been analyzed in different ways as an innovative way to include more actors within the traditional design processes, such as the end-users of the generated designs, in what is now sometimes defined as “Participatory Design” [6]. Precisely on these scenarios, for example, Professors Branko Kolarevic and José Pinto Duarte strongly contributed to the debate on the subject, favoring through their work the dialogue of various voices on the potential consequences of the application of what has been defined as “Mass Customization and Design Democratization” [7]. Even without reaching these extreme consequences, imagining such applications can still pose several open questions, in particular on how to verify the operational results extracted from the use of these processes, in such a way as to prevent the implementation of incorrect or inconvenient solutions [8]. We can also deal with this problem in a broader sense, questioning the very relationship between traditional professional figures and these innovative generative processes. Potential answers to these doubts can already be found in the past. Indeed, even before technology made the adequate tools available, several figures had already had the sensitivity to interpret with surreal awareness the opportunities that would have been available in our present [9]. Among these, the architect Luigi Moretti can be mentioned with particular attention for having disseminated, between the 50s and the 80s, numerous studies regarding a future algorithmic turn of design [10]. On this topic, Moretti envisions a highly symbiotic and collaborative vision of the relationship between the human and the algorithmic dimensions involved in the design process. In his opinion, “parametric architecture” would have allowed professionals to increase the awareness of their choices, and their consequences [11]. In explaining his position, Moretti does not seem to show serious concerns about the reliability of the results of an algorithmic design, probably because it is always

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considered a tool used to strictly support the work of the designer, and not an easy way to substitute his authority or responsibility. The problem initially posed is however only partially solved by this information, since although we can now consider the participation of a human parameter within the process to be very important, the next question is to understand how, and by what means, this participation should take place, to be effective. This deeper question is definitely too complex to find a synthetic solution, but once again approaching Moretti’s writings can become particularly useful. On numerous occasions, Moretti places a strong emphasis on the characteristics of the “parameters” as the base of the mental processes that lead to the generation of the final form [12]. The “parameters” are, therefore, the starting point of each subsequent operation and can strongly influence any outcome that may subsequently be reached through the execution of the tools built on their examination. Furthermore, the definition of basic parameters can take advantage, in the same way as other phases of the work, of both the intuition and the capacity of imagination of the designer. These parameters can, in fact, become an object of study for more intimate and personal evaluations, for example, by defining aggregate syntheses of multiple factors, optimized in a compact form by personal interpretation to better support the operations required in the future. Once again, Moretti’s lesson on the subject is precise and effective, as in the architecture manifest of his ideas about parametric architecture, where he bases the generative work process on a development parameter based on a personal evaluation system. This system originated from the composite aggregation of different data in optical, geometric, and compositional nature. The so-called Luigi Moretti Soccer Stadium, presented at the Milan Triennale in 1960, in collaboration with the mathematician Bruno de Finetti, is configured as a formally complex structure, derived from the desire to provide each spectator with an optimal view of the surface of the playing area. To achieve this goal, the previously mentioned evaluation parameter is defined to measure the visual completeness of each seat, and through an optimization process, the shape of the stands is defined by the points where this value offers more appropriate measurements [13]. Even today this lesson may prove useful in guiding greater awareness of the use of parametric tools. In particular, the attention given to the parameter processing phase, and the freedom in the management and combination of known data, constitute, in my opinion, a very important practice, often not sufficiently valued. Following this thought, the next paragraph will present some preliminary results of a sort of brainstorming on this theme.

3 A Parametric Setup to Perform Multi-scale Visibility Analysis The results shown below represent a group of comments and tests, in the context of the application of parametric scripts useful to implement visual analyzes within different categories of architectural and urban spaces. I propose that the use of parametric workflows within this context can provide different kinds of advantages over the comprehension of a given environment. In particular, through the implementation of visual programming systems, such as the Grasshopper [14] application, capable of easily applying logical processes derived from the context

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of the geometrical representation of space. In this regard, the fundamental concepts of projective geometry (projection and section [15]) were used here as the basis to perform visual analysis, through the reconstruction of simple relationships between points in space, in the form of projecting rays. Subsequent evaluations of interference, between the obtained rays, and the solid shapes placed in a given environment, can allow the acquisition of a wide range of data which, albeit in an elementary way, can become the basis for subsequent and more refined evaluations. By examining the simple basic relationship, between a point of view connected to a second point observed in space over a certain distance, various non-trivial information can be extrapolated, such as the possibility of observing that point, as well as the possibility of not seeing it (see Fig. 1).

Fig. 1. The figure summarizes different possible evaluations that may be implemented with automatized visual analysis scripts: starting with basic outputs as the possibility to see something (A), or its inability (B). In addition, more detailed data may be computed, like the visual angle covered by all the relevant elements visible from a certain point (C), or the raw number of visible points (D).

In addition to this, if we connect the point of view to more points, one could also determine the overall viewing angle within which the different visible points are observed from that particular point of view. Finally, one may also introduce the possibility of examining multiple points of view at the same time, allowing the system to simultaneously determine the visual relationship of a large sample of positions, generating analyzes of greater detail and complexity. The procedures created in this way may also be potentially applied to any context, allowing multi-scale investigation able to seamlessly connect different scales of analysis of the same space.

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Simple automation may also be applied to the generation of the points of view to test, allowing the production of rapid scene setup, for early-stage planning. Also, the placement of viewpoints may consider different types of spatial arrangements over the target surfaces in relation to their uses. For example, to perform a view analysis of entire buildings, the points of view may be placed over the building envelope, to simulate openings and windows (see Fig. 2). The output of such procedures comes in the form of data linked to each specific point of view. Therefore, it would be possible to unpack specific portion of this stream of information to implement it elsewhere, enlarging the scope of design algorithm with new input parameters.

Fig. 2. The figure displays the process used to place points of view to test over the envelope of a test building. Each surface is subdivided to simulate openings and windows potentially located in each areas, and the center of each obtained part is used as a reference to locate a point of view (A). The process can also account for analyzing surfaces with specific spatial orientations (B) and can also control the final quality of the analysis by editing the density of the initial subdivision (C).

In the following examples, different data were used to color the analyzed envelope of a test building in relation to the ability to see specific points in the environment. Also, by representing the obtained measurements via a color gradient, highly readable images can be generated, to efficiently share and evaluate the environment visual conditions (see Fig. 3, 4, 5, 6).

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Fig. 3. The following graphic shows the output of a simple application of the proposed visual analysis script. The central test volume is used to generate on its surfaces several viewpoints to check and see the environment around it. Each viewpoint is used to tests if specific notable points of the territory are visible, while also taking into accounts the presence of possible visual obstacles. The results are used to color via a gradient, the surfaces linked to each viewpoint.

Fig. 4. The following graphic shows the output analysis derived from different environmental setup.

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Fig. 5. Focus on a filter showing the total number of notable points of the territory visible from each test point of view located on the surfaces of the test volume.

Fig. 6. Focus on a filter showing the maximum width of the visual angle that envelops all the notable points of the territory visible from each test view.

At the present stage, the procedures compiled can only account for these basic types of visual relationships, but still they can already be implemented in larger generative design framework, allowing the interaction of “visual relationships” with other kinds of data. For example, different types of optimization procedures, may include in their fitness goals, also the scores resulting from these operations. Their implementation certainly cannot be applied without constraints but, for example, via the implementation of different weights for all the various parameters involved, a more aware compounded synthesis may be reached, following the spirit already displayed by the previously cited best practices.

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Fig. 7. Focus on a possible use of the presented methodology. All the different parameters extracted from the script can be used to evaluate possible spatial configurations of a transformation project. In this example an open green area is analyzed to measure the interactions of the placements of trees (here represented with a simplified shape comprehensive of a general volume) on the possible visibility of valuable points around the landscape.

Fig. 8. Focus on the application of the presented script in a larger urban context. Like the previous example. This test displays the possibility to easily switch the context of reference of the analysis with scenario characterized by different scales. In the picture on the left, it can be observed the whole context of reference, with the notable points to test for visibility marked with red dots. On the right instead, we can see an enlarged representation of the old town previously linked to each point in the environment, and the output of the visibility analysis linked to each building.

Finally, the resulting experience has been successfully applied to more complex test environments where it was possible to evaluate the derived outcomes in relation to real spatial configurations. Sites of different scales and intricacy were carefully selected to test the capabilities of the presented system to elaborate meaningful information across different scales of analysis. Alongside the conceptual tests at the district scale, two major

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analyses were performed with the aim to link the human scale across large distances, even reaching to the territorial scale of analysis in relation to the landscape configuration (see Fig. 7, 8). The preliminary results show good potential, and future developments of the research will aim to improve all the fundamental features of the current system, like the number of obtainable data, the reliability of the information, and the generation of a database easily implementable in wider workflows.

4 The Possible Positive Interaction of Techniques of Artistic Representation The results presented here do not have yet strong links with similar, but more advanced practices, pertaining to the field of Space Syntax. Nonetheless, this work was developed following the very same interest about the relation between human beings and their inhabited spaces [16] that define the core of Space Syntax itself. While future developments will examine more analytically the vast background of best practices that were developed within the context, it is interesting to note how the field of graphical representation, and its contemporary developments, may contribute greatly to expanding the possible outputs of established visual analysis frameworks. This is something that was already asserted before. In the last few decades various fields developed significant theoretical advances, in synergy with the innovation of digital graphical representation, and this can be applied with particular emphasis to Space Syntax [17]. However, while nurturing important leaps across multiples disciplines, graphical representation itself is extending towards complex and engaging horizons. One of the wider frontlines of this advancement is the quest for more accessible photorealistic representation tools, able to achieve real-time photorealism on par with real life [18]. This aim may seem out of place compared to the frame outlined by the current text, but this may not be the case. At the end of the day, the realistic representation of a virtual scene, is just another way of approaching the description of space. Substantial differences con be found in the background of the generation of a rendering, starting from the tools, its goals or the amount and quality of information packed in the final output. Nevertheless, what is obtained in the end, can be thought as a description of space. As a different path looking towards the same subject, which is the description of space, maybe this expertise may be implemented into more analytical setup and contribute to supplement design processes with complete interpretation of the human space. Photorealism may not be an end per se, but just the necessary condition to allow further uses of the medium to be born [19].

5 Conclusions The outputs presented here followed a path towards the formulation of automated procedures to better support transformation processes of the built environment by translating the visual relationships existing within it, in a parametric language. While at the present, the possible results that may be extracted from these processes are quite limited, the acquired data can already be implemented, with the proper weights, in different kinds of basic optimization procedures.

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However, the aim of this work is not the generation of entirely autonomous entities, on the contrary, it follows the inspiration of past thinkers in the field and aims at collecting useful data to perform more sophisticated analysis of the selected environments by crossing different fields and visions. Future investigations on the topic will probe further, established best practices from the field of Space Syntax and a possible link between those parametric processes and more artistical inputs from different fields.

References 1. Tedeschi, A.: AAD Algorithms-Aided Design: Parametric Strategies using Grasshopper. 1st ed. Le Penseur, Potenza (2014) 2. Moretti, L.: Spazio, Gli editoriali e altri scritti. 1st ed. Christian Marotti Edizioni, Milano (2019) 3. Laura, L.: Breve e universale storia degli algoritmi, 1st edn. Luiss University Press, Milano (2019) 4. Casey, R.: Form+Code in Design, Art, and Architecture. Princeton Architectural Press, New York (2010) 5. Bohnacker, H., Gross, B., Laub, J., Lazzeroni, C.: Generative Design: Visualize, Program, and Create with Processing. 1st ed. Princeton Architectural Press (2012) 6. Tommasino, T.: Come se Giancarlo De Carlo giocasse a Doom. 5. Domus Website. https://www.domusweb.it/it/architettura/gallery/2020/03/23/doom-come-buona-praticadi-progettazione-partecipata-.html. Accessed 22 Jan 2022 7. Kolarevic, B., Duarte, J.P.: Preface. In: Mass Customization and Design Democratization, Preface. 1st ed. Taylor & Francis, New York (2018) 8. Kolarevic, B., Duarte, J.P.: From Massive to Mass Customization and Design Democratization. In: Mass Customization and Design Democratization, Chapter 1. 1st ed. Taylor & Francis, New York (2018) 9. Frazer, J.H.: Parametric computation: history and future. Archit. Des. 86, 18–23 (2016) 10. Moretti, L.: Spazio, Gli editoriali e altri scritti. 1st edition. Christian Marotti Edizioni, Milano (2019) 11. Moretti, L.: Ricerca Matematica in Architettura e Urbanistica. In: Moebius Unità Della Cultura Architettura Urbanistica Arte. Orsa Maggiore, Roma (1972) 12. Moretti, L.: Eclettismo e unità di linguaggi. In: Spazio, Gli editoriali e altri scritti. 1st edition. Christian Marotti Edizioni, Milano (2019) 13. Bianconi, F., Filippucci, M., Buffi, A., Vitali, L.: Morphological and visual optimization in stadium design: a digital reinterpretation of Luigi Moretti’s stadiums. Archit. Sci. Rev. 63(2), 194–209 (2019) 14. Grasshopper Homepage. https://www.grasshopper3d.com/. Accessed 26 Jan 2022 15. Betti, R.: Geometria leggera: Introduzione all’idea di spazio matematico. 1st ed. Franco Angeli Edizioni, Milano (2015) 16. Dursun, P.: Space syntax in architectural design. In: Proceedings of the 6th International Space Syntax Symposium (2007) 17. Hillier, B.: Space is the Machine: A Configurational Theory of Architecture, 6th edn. Cambridge University Press, Cambridge (1996) 18. A first look at Unreal Engine 5. unrealengine Website. https://www.unrealengine.com/en-US/ blog/a-first-look-at-unreal-engine-5. Accessed 9 Feb 2022 19. Beyond Photorealism: Conveying Emotion and Sense of Place Through Rendering. ArchDaily Website. https://www.archdaily.com/948857/beyond-photorealism-conveying-emotion-andsense-of-place-through-rendering. Accessed 9 Feb 2022

Generating Spatial Configurations of Floating Settlement Branch Structures for Urban Atoll Islands Jovana Stankovi´c(B) , Branislava Stoiljkovi´c, Sonja Krasi´c, and Nastasija Koci´c Faculty of Civil Engineering and Architecture, University of Niš, Aleksandra Medvedeva 14, 18106 Niš, Serbia [email protected]

Abstract. Urban atoll islands are most susceptible to rising sea levels due to low altitudes. The constant immigration of the local population to the water surface affects the growth of the existing spatial configurations of floating settlements and calls into question their stability. The aim of the research in this paper is to find out which spatial configuration would correspond to the unhindered expansion and dynamic geography of a floating settlement for an unlimited number of inhabitants. Hexagonal modular forms, branch structures and principles of generative design will be helpful. By defining 100 inhabitants as a sample and representing 3 different types of floating houses in percentages, a generative design process can be started. The aim of the research is to find a solution with the least number of different types of modular platforms and to show the possibility of unhindered expansion after generating possible combinations of floating houses on one platform and combinations of platforms necessary for a floating settlement. Keywords: Generative design · Floating settlement · Spatial configuration · Combinatorics

1 Introduction Urban atoll islands are most susceptible to rising sea levels due to low altitude [1]. The problem of land loss is largely solved by occupying the water surface and building floating settlements with low-budget modular houses. The spatial configurations of these settlements are changing because a certain number of households emigrate daily from the mainland to the water surface in accordance with the trend of rising sea levels. The question is how to ensure the unhindered expansion of spatial configurations, i.e. dynamic geography with minimal impact on the stability of existing floating structures. According to Czapiewska et al. [2], the system of floating houses can be formed in the form of islands, branch, composite structure and a single large structure. The possibility of dynamic geography is realized by islands and branch structures. In the structure of the island, each floating house is located on one platform and connected to the others by hinged joints, while in the branch option, several floating houses are located on one platform and connected to modular platforms with hinged or rigid joints. Considering © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 519–527, 2023. https://doi.org/10.1007/978-3-031-13588-0_45

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the stability, the branch option proved to be more suitable. In previous works [3, 4] the author team Stankovi´c et al. investigated these structures and found that the advantage of islands structure is the possibility of removing/adding floating house modules, which is very important in case of failure of a module or family expansion, and the disadvantage is the inability to add new floating houses within the already adopted scheme. It is not the case with branch structure for which it was found that an unlimited number of houses could be added. The growth of the spatial configuration and the creation of ordered formations is possible by applying modular forms [5] and branching growth pattern [6], where modular platforms aggregate on the water surface according to the principles of a growing branch. Previous research has shown that the hexagon is a suitable shape for this process while being a structurally stable structure with a maximum surface area and a minimum perimeter [3, 4]. The size of modular platforms depends on social needs, functionality, location characteristics, financial and structural constraints [5], and above all on the number, type and connection of floating houses located on one platform, while the size of floating settlements depends on changes in the number of inhabitants. It can be concluded that there are a large number of possible combinations of floating houses on one platform and a combination of platforms necessary for a floating settlement of an unlimited number of inhabitants. Therefore, it is necessary to take a sample of several different types of the floating house and a certain number of inhabitants, which reduces that number to the final, and later increases exponentially. After generating possible spatial configurations of the floating settlement of the adopted sample, it is necessary to choose the most favorable solution with the least number of different types of platforms. Generative design (GD) is often used nowadays in architectural design and is defined as a design approach that uses algorithms to generate solutions [7] and combines parametric design and artificial intelligence with the limitations and data included by the designer [8]. If the branch structure and principles of generative design are connected, the subject problem of unhindered expansion of spatial configurations can be solved. In this paper, the subject of research will be the generating of spatial configurations of floating settlements branch structures inside of hexagonal grid on a sample of 100 inhabitants and 3 different types of the floating house given in percentages. One platform will contain 3 floating houses of any type. The aim of the research is to find a solution with the least number of different types of platforms and to show the possibility of unhindered expansion after generating possible combinations of floating houses on one platform and combinations of platforms necessary for a floating settlement.

2 Generative Design of Floating Settlement Floating settlement planning is not easy, so GD will make the whole process much easier. The key stages in this design are [9]: 1. 2. 3. 4.

Defining the problem; Gathering data; Setting evaluation criteria; Generating the model;

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5. Evaluating the results; 6. Evolving the design and 7. Selection and refinement. 2.1 Defining the Problem The problem is production of the spatial configuration of the floating settlements of the branch structure for urban atoll islands for an unlimited number of people because the number of people immigrating to the water surface in these areas is increasing every day. 2.2 Gathering Data Due to the impossibility of designing a floating settlement for an unlimited number of people, the sample that is considered relevant is 100 inhabitants. A study of the literature established that households with 6 family members are the most represented in urban atoll islands [4]. Therefore, in this settlement, 50% of the population will be accommodated in houses for a family of six, 25% in houses for a family of four and 25% in houses for a family of eight, so there will be 9 houses for six members, 6 houses for four members and 3 houses for eight members. For easier understanding in the continuation of the research, the house for a family of four is marked with type 1, for a family of six with type 2 and for a family of eight with type 3 (see Fig. 1).

Fig. 1. Types of floating houses for a family of 4, 6 or 8

The adopted number of houses is 18 per 100 inhabitants, so the number of floating houses per platform will be taken as 3. In the end, the floating settlement for 100 inhabitants will have 6 platform structures. The research continues in the Mathematica software package, where the input data are processed and the result is all types of platforms with three floating houses of any type, i.e. 10 of them different. The result is presented in the form of a set of three numbers where the first number shows how many type 1 houses are on the platform, the second number type 2 houses and the third number type 3 houses (see Fig. 2).

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Fig. 2. All possible platform types in the branch structure

2.3 Setting Evaluation Criteria Before starting to generate a combination of platforms for solving a floating settlement, it is necessary to define the evaluation criteria, i.e. determine the goal of this process. In this case, the goal is to create a floating island with the least number of different platforms, because a smaller number of different platforms will reduce costs and enable dynamic spatial growth compatible with urban needs. In order to find out that number, it is necessary to get all possible combinations of platforms. 2.4 Generating the Model The research continues in the Mathematica software package, in order to find out how many possible combinations the floating island can have. The iterations we set in this program are: • • • • •

list of all combinations of sixes; selection of sixes where the sum in the first place is equal to 3; selection of sixes where the sum in the second place is equal to 6; selection of sixes where the sum in third place is equal to 9 and delete duplicates. The number of possible combinations is 54 (see Fig. 3).

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Fig. 3. All possible combinations of platforms for floating settlement branch structures

2.5 Evaluating the Results After generating possible solutions of the floating settlement (see Fig. 3) of the branch structure, it can be concluded that there are 3 solutions with 2 different platform types, 7 with 3, 27 with 4, 15 with 5 and 2 solutions with 6 different platform types. The set goal was a solution with the least number of different types of platforms so that 3 solutions with 2 different types of platforms are further considered. 2.6 Evolving the Design After the evaluation, it is necessary to develop three solutions for the floating settlement (see Fig. 4). In the first solution, there are 2 houses of type 1 and 1 house of type 3 on the first type of platform, and 3 houses of type 2 on the second. In the second solution, there are 2 houses of type 2 and 1 house of type 3 on the first type of platform, and 2

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houses of type 1 and 1 house of type 2 on the second. Finally, in the third solution there are 1 house of type 1 and 2 houses of type 2 on the first type of platform, and 1 house of type 1, 1 house of type 2 and 1 house of type 3 on the second.

Fig. 4. Evolving the design of selected platform types

The connection of these floating houses is realized under the following conditions: • the axes of symmetry of all floating houses on one platform intersect at one point; • animal shelter module positioned towards the outside of the platform and • there must be one distance module between each floating house on the platform to achieve physical and architectural comfort. 2.7 Selection and Refinement At the end of the research, it remains to connect the platforms and visualize the floating settlement with the least number of different platforms for each of the solutions from the previous step (see Fig. 5, 6 and 7). There are many combinations when connecting platforms, so the choice of a representative example, in this case, was influenced by the human factor. The pictures show how the settlement can be expanded with the same types of platforms.

Generating Spatial Configurations of Floating Settlement

Fig. 5. An example of a floating settlement with the first platform solution

Fig. 6. An example of a floating settlement with the second platform solution

Fig. 7. An example of a floating settlement with the third platform solution

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3 Results and Discussion Based on the analysis of three floating settlement solutions with the least number of different platforms obtained by generative design, it can be concluded that the third solution is not suitable for expanding the spatial configuration due to the unfavorable shape of the platform. This shape of the platform is caused by the presence of all three types of floating houses. The first and second solutions are characterized by platforms with 2 different types of houses, which means that in terms of expanding the spatial configuration, both can be considered in the same way.

4 Future Research Generative design is a good tool for forming settlements with different types of platforms in different ways. In this paper, one way of merging platforms is presented. Future research would be focused on the development of floating settlements with all possible combinations of connecting an infinite number of platforms of 2 different types based on the first and second solution from this research. This would increase the number of conditions in the design and thus get a more optimal and rational solution.

5 Conclusion Generative design is current in the modern world. With its potential, it can help solve world problems such as the consequences of rising sea levels faced by urban atoll islands and increasing the spatial configuration in line with population migrations. By applying modular hexagonal shapes, generative design principles and branch growning pattern, all types of platforms can be obtained, solutions with the least number of different platforms can be selected and finally a floating settlement can be designed for an unlimited number of people.

References 1. Oppenheimer, M., et al.: Sea level rise and implications for low lying Islands, coasts and communities. In: Pörtner, H.-O., et al. (eds), IPCC Special Report on the Ocean and Cryosphere in a Changing Climate, Cambridge University Press, Cambridge, UK (2019) 2. Czapiewska, K., Roeffen, B., Zanon, B.D.B., de Graaf, R.: Seasteading Implementation Plan, Final report. Delft, The Netherlands (2013) 3. Stankovic, J., Krasic, S., Mitkovic, P., Nikolic, M., Kocic, N., Mitkovic, M.: Floating modular houses as solution for rising sea levels-a case study in Kiribati island. In: Stojakovic, V., Tepavcevic, B. (eds.) Towards a new, configurable architecture - Proceedings of the 39th eCAADe Conference, vol. 1, pp. 161–170. eCAADe (Education and Research in Computer Aided Architectural Design in Europe) Brussels, Belgium/ FTN (Faculty of Technical Sciences, University of Novi Sad), Novi Sad (2021). http://papers.cumincad.org/data/works/att/ ecaade2021_264.pdf

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4. Stankovic, J., Krasic, S., Nikolic, M., Kocic, N.: Spatial configurations of floating settlementsparametric method. In: Jeli, Z., Popkonstantinovi´c, B., Miši´c, S., Obradovi´c, R. (eds.), MONGEOMETRIJA 2021, Proceedings of The 8th International Scientific Conference on Geometry and Graphics, pp. 201–210, Serbian Society for Geometry and Graphics (SUGIG)/ Faculty of Mechanical Engineering, University of Belgrade, Belgrade, Serbia (2021) 5. El-Shihy, A.A., Ezquiaga, J.M.: Architectural design concept and guidelines for floating structures for tackling sea level rise impacts on Abu-Qir. Alex. Eng. J. 58(2), 507–518 (2019) 6. Kizilova, S.: Form and functional features of modular floating structures. In: E3S Web of Conferences, Topical Problems of Architecture, Civil Engineering and Environmental Economics (TPACEE 2018), EDP Sciences, vol. 91, p. 05013 (2019) 7. Caetano, I., Santos, L., Leitão, A.: Computational design in architecture: defining parametric, generative, and algorithmic design. Front. Arch. Res. 9(2), 287–300 (2020) 8. Archdaily, How Will Generative Design Impact Architecture. https://www.archdaily.com/937 772/how-will-generative-design-impact-architecture. Accessed 06 Feb 2022 9. Autodesk University, Generative Design for Architectural Space Planning. https://medium. com/autodesk-university/generative-design-for-architectural-space-planning-9f82cf5dcdc0. Accessed 21 Feb 2022 10. Obradovi´c, M.: Tiling the lateral surface of the concave cupolae of the second sort. Nexus Netw. J. 21(1), 59–77 (2018). https://doi.org/10.1007/s00004-018-0417-5 11. Stavric, M., Wiltsche, A.: Geometrical elaboration of auxetic structures. Nexus Netw. J. 21(1), 79–90 (2019). https://doi.org/10.1007/s00004-019-00428-5

Planning the Infill Patterns and the Resulting Density Percentage Error in Additive Manufacturing Yasaman Farahnak Majd1 , Marcos de Sales Guerra Tsuzuki2 , and Ahmad Barari1(B) 1 Advanced Digital Design, Manufacturing, and Metrology Laboratories (AD2MLabs),

Ontaria Tech University of Technology, Oshawa, Canada [email protected] 2 University of São Paulo, São Paulo, Brazil

Abstract. This paper aims to introduce the implementation parameters of the layered infill patterns in additive manufacturing to evaluate their effects on the actual density in the fabricated parts. To achieve this goal, parameters involved in shaping different infill patterns are presented and the significance of different input parameters on the output (actual) infill density is evaluated. The reference models for infill implementation are generic cubic and cylindrical models as their specific geometries lead to fewer geometric parameters affecting infill density error and provide informative case studies. Keywords: Additive manufacturing · Layer-based AM · Infill patterns · Infill density error · Infill patterns’ parameters

1 Introduction Additive Manufacturing (AM) is a wide range of technologies that have drawn significant attention in recent decades as it provides the opportunity to build complex parts by adding material instead of subtractive manufacturing. Layer-based AM which is the focus of this paper is a branch of AM that creates a 3D model by putting 2 ½ dimensional layers on top of each other. A layer-based 3D printing starts with a stereolithography (STL) file format that represents a 3D model with triangles. These triangles are cut with a group of z-planes to extract the information of each layer. After slicing, each layer will consist of one or more polygonal shapes named contour. Contour is the most external perimeter that shapes the boundary of the part. After this stage, inner wall(s) are generated with an offset (depending on contour and inner wall width) from the contour. Inside the inner wall boundary, there is an empty polygonal area that should be filled. Infill patterns are groups of line combinations that are used to fill the empty area inside each layer. After these three steps of path generation (contour, inner wall, and infill), machine instruction (G-Code) will be produced to specify printer head movement in each layer.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 528–540, 2023. https://doi.org/10.1007/978-3-031-13588-0_46

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There are two general types of infill patterns, 2D and 3D. In 2D infill patterns, the same pattern will be repeated in each layer but in 3D ones, the pattern changes over layers making a 3D shape inside the part. The focus of this paper is on different types of 2D infill patterns. Although there are slicing software tools that work with different kinds of infill patterns, the parameters involved in the implementation of the patterns are not clearly specified, whole knowing the parameters is the key to investigating the accuracy of infill density percentage. One of the leading parameters that has effects on the stiffness of the 3D printed part is infill density percentage as the mechanical properties of a part are directly related to its geometrical properties. The goal of this paper is to specify 2D infill patterns’ input parameters and investigate the accuracy of the actual infill density percentage in comparison with the infill density defined by the user as the input. Input infill density is named nominal density in this paper. The result shows which input parameters have significant effects on the accuracy of infill density and how changing these parameters can lead to a smaller error involved.

2 Background Additive Manufacturing has a significant role in digitalization which is the main component of Industry 4.0. Therefore, a lot of researchers are working on the improvement of AM processes to be adapted to be used in various industries [1]. AM is growing so fast in different directions, and sometimes this high-speed development happens while some basic details are skipped. For instance, uncertainties and their sources in AM are one of those topics that are less addressed in the literature [2]. In order to have control over the manufacturing results, considering accuracy in different aspects of AM is necessary. This section aims to introduce some previous efforts for accuracy improvement in AM and find the gap to be furtherly discussed. Many researchers have followed AM accuracy improvement by addressing issues such as cusp geometry (volume and height) and also surface roughness [3–8]. Gohari et al. worked on the volumetric deviation minimization in AM presenting a methodology for intelligent process planning based on adaptive slicing [9]. Later, their group worked on minimizing the geometric deviations between a 3D model designed in CAD software and the resultant 3D printed part [10, 11]. Some researchers put efforts into finding manufacturing errors of 3D printed parts by developing inspection and modelling systems that predict the final product details [12–14]. Considering manufacturing error reduction in process planning and also design modification with the goal of reducing sensitivity to the sources of the errors are also the ideas behind some methodologies to reach the final product with the desired features [15–19].

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Infill patterns are one of the components of AM that has effects on the mechanical properties of the final 3D printed part [20]. Therefore, accuracy in infill path generation is a key step in reaching the desired strength. Although finding some infill patterns’ parameters is straightforward, having the parameters in hand helps to reach a common language in defining infill patterns. That’s why this paper aims to present infill patterns’ parameters as they are not provided in the literature clearly. On the other hand, there are some efforts done for infill pattern modification to reach the desired mechanical properties in the final 3D printed part [21–23], but the density accuracy of the generated infill (as the most significant parameter in the effect of infill pattern on the strength) is not discussed previously, which is the second focus of this paper.

3 Methodology The focus of this paper is on the geometry specifications of various types of 2D infill patterns, and the calculation of infill density percentage error after 3D printing with different geometrical setups. To follow this goal, a slicer program is developed in python to make different infill patterns implementation possible. Six different infill patterns are implemented to be used for density error evaluation. Although the program is developed to work for any 3D model, all the results in this paper are based on generic cubical and cylindrical models to reach the minimum geometrical complexity in the model and see the pure effects of infill parameters on the infill density error in the absence of model parameters’ effects. Following, infill patterns are classified, the required input parameters are presented, and the procedure for density error evaluation is provided. 3.1 Infill Patterns Classification This research focuses on six types of infill patterns that are the most used patterns in different commercial software tools. These six types include grid, triangle, tri-hexagon, line, honeycomb, and wiggle which are introduced in Table 1.

Planning the Infill Patterns and the Resulting Density Percentage Error Table 1. A short description and sample images for six groups of infill patterns Infill Name

Description

Grid

• Two groups of parallel straight lines cross • Angle difference between two groups of lines is 90ᵒ • Material accumulates in the crossing points of two lines

Triangle

• Three groups of parallel straight lines cross • There is 60ᵒ angle difference between each two consecutive groups of lines • Lines shape triangle structures together in each layer • Material accumulates in the crossing points of three lines

Tri-hexagon

• The same as triangle, three groups of lines cross with 60ᵒ angle difference • The only difference that makes the shape different from triangle is shifting the third group of lines.

Line

• There is only one group of parallel lines in each layer. • The direction of the parallel lines in consecutive layers can be different. Therefore, a group of layers makes a repeatable pattern together. • (a), (b) and (c) show three consecutive layers with different line angles. The pattern of this set of layers repeats every three-layer in this case. • No material accumulation in one layer

Image

a) 1st layer with 30ᵒ lines

b) 2nd layer with 45ᵒ lines

c) 3rd layer with 60ᵒ lines

Honeycomb

• Honeycomb pattern shapes by repeating two same pattern lines next to each other in the whole layer. • The lines do not cross in the layer, therefore no material accumulation happens • The same pattern repeats in all layers

Wiggle

• Wiggle pattern shapes by repeating one pattern line all over the layer in the form of wiggly lines. • The distance between each two pattern lines is calculated based on the pattern line parameters and the required infill density percentage.

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3.2 Infill Patterns Parameters The most important step to generate an infill path algorithm is specifying the input variables. Some of the input variables are common among all infill patterns and some are related to the specific properties of each infill pattern. Common input parameters include infill density percentage and infill raster width. The other parameters are related to the specific geometry of each infill pattern. Grid The first group of parallel lines makes α angle with +x. The angle of the second group of lines has 90° difference in direction, therefore it is considered a dependent parameter. Input parameters of the grid pattern include infill density percentage, infill width, angle of the first group of lines and the intercept of the starting infill line for each group of parallel lines. Figure 1 shows the position of the first infill line. Lines in red have a distance equal to the infill lines’ distance and specify the boundary that the first infill line can move in. The red point represents the intercept of the infill line.

Fig. 1. First infill line position based on the input intercept

Line As there is one group of parallel lines in each layer using line pattern, generating this pattern is the same as the grid with some differences. Input parameters include infill density percentage, infill width, a list of angles showing the angles of infill lines of repetitive layers in order, and the intercept of starting infill line in each angle. For example, the image of the line pattern in Table 1 shows three repetitive layers with 30°, 45°, and 60°, therefore the input angle list would become (30, 45, 60). Triangle Input parameters of the Triangle pattern are the same as Grid pattern but the concept behind them is a little bit different. There are three groups of lines with 60° angle difference (adding two dependent parameters). The third group of lines should pass the intersected points of the first and second groups of lines. Therefore, only first and second groups of lines need an input for the intercept of their first line. This way, lines make a triangle shape next to each other. Figure 2 shows how the triangle pattern will appear by intersecting three groups of lines.

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Fig. 2. Three sets of straight lines making triangle pattern

Tri-Hexagon The general shape of this pattern is the same as the triangle infill pattern, so the same input parameters can be used for generating tri-hexagon infill lines. The only important difference is in the condition of the third group of lines. To make a tri-hexagon pattern, the third group of lines, unlike the triangle pattern, should pass the middle of the perpendicular distance of two consecutive intersected points. Figure 3 shows the position of the third group of lines to shape a tri-hexagon pattern.

Fig. 3. Three groups of lines (0°, 60°, 120°) making Tri-hexagon pattern

Honeycomb The first step to generate a honeycomb pattern is determining the parameter(s) of the repetitive element based on the inputs. Figure 4 shows an element of the honeycomb pattern, its geometric parameters and material distribution in an element. The shape of this element is created based on the form of pattern lines next to each other. Each honeycomb element is a hexagon. Therefore, each side length is the only unknown parameter that should be calculated based on the infill density percentage and infill raster width.

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(a)

(b)

Fig. 4. Honeycomb element, a) parameters that specify the shape of the element, b) material distribution in a repetitive element

One other input to generate the honeycomb pattern is the start point of the line pattern. Four possible starting points are considered on the pattern line as shown in Fig. 5. This way, input parameters of the honeycomb pattern include infill density percentage, infill width (w), start point number on the pattern line, and starting point coordination. Figure 6 shows the pattern difference with different starting points. In this figure, the coordination of the selected starting point is set to the left corner of the largest rectangle.

Fig. 5. Possible starting points on the honeycomb pattern line

Fig. 6. Honeycomb pattern on the largest rectangle with different start points

Wiggle To generate a wiggle infill in a layer, the repetitive element and its parameters should be specified. Figure 7 shows the effective geometric parameters of a wiggle element. The edge length (a) and the angle between every two pieces of lines (α) are the inputs and by having infill density and width(w), the distance between each two consecutive pattern lines will be calculated.

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(a)

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(b)

Fig. 7. Wiggle element, a) parameters that specify the shape of the element, b) material distribution in a repetitive element

The same as the honeycomb pattern, the starting point on the pattern line is also a parameter to generate wiggle infill. Two possible starting points are considered as it is shown in Fig. 8. By having this information, the wiggle and honeycomb algorithms are almost the same. The input parameters to generate the wiggle pattern include the angle between every two pieces of lines (α), edge length (a), infill width (w), starting point number, starting point coordination on the largest rectangle, and infill density percentage.

Fig. 8. Position of two starting points on the wiggly line pattern

3.3 Volumetric Percentage Error The error that shows the difference between the nominal and actual infill density is named Volumetric Percentage Error (VPE) which is the investigated error in this paper. Equation 1 represents VPE; where Dn is the nominal infill density (user input) and Da is the actual infill density. Dn is the main input parameter for infill generation and Da is a good approximation of the actual infill density after printing which is calculated using Eq. 2; where Vinfill is the infill volume represented in Eq. 3, and VTotal is the total volume of the boundary that is filled with infill lines discussed in Eq. 4. The reason Da is called an approximation of the actual infill density is that it is not determined by the actual printed part measurement. Instead, by considering the details of printing behavior and the infill information, infill volume is calculated, and actual infill density is determined. In Eq. 3 and Eq. 4, n is the total number of layers, m is the total number of straight lines in case of grid, triangle, tri-hexagon and line patterns, and represents total number of pattern lines in case of honeycomb and wiggle patterns, len shows length calculation, w is the

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infill raster width, h is the layer height, Acontour is the total area filled with contour lines in each layer, Ainnerwalls is the area filled with inner walls, and Aeffective is the effective area in infill generation which is the boundary filled with infill in each layer. VPE = Da = Vinfill =

Dn − Da × 100 Dn

(1)

Vinfill × 100 VTotal

(2)

m n π w2 )×h (((len(i) × w) + 4

(3)

l=1 i=1

VTotal =

n

Aeffective (l) × h

(4)

l=1

Aeffective (l) = ATotal (l) − Acontour (l) − Ainnerwalls (l) Based on Eq. 1, VPE changes by nominal and actual infill density, which makes it dependent on the infill parameters setup. Therefore, the goal is to show the value of VPE and the pattern of its changes in different infill setups for the introduced infill patterns. For the 3D model, Cubic and cylindrical shapes are selected as their specific geometry helps to minimize the effect of the model’s properties on the output. VPE occurrence is related to two incidents during infill generation. First is the possibility of gap creation at the end of each group of parallel lines, and second is the round shape that is created at the end of straight lines or pattern lines while the extruder retracts the filament to move to a new point. The second term in Eq. 3 is the added volume due to the round shape effect.

4 Results As it is discussed previously, each infill pattern has its specific input parameters and changing each of them has effects on the actual infill density value. Although all those parameters are worth discussing, this paper focuses on two input parameters; infill lines’ angle and infill density percentage as they are the most significant ones. Six introduced infill patterns with 0.1 mm width are generated in different setups inside two models: a 10 × 10 × 10 cm3 cube, and a cylinder with 10 cm height and 10 cm diameter. Changing infill width makes a change in VPE as both possibility of gap creation and the number of round shapes at the end of straight lines are dependent to infill width. Comparing different infill width results (not presented here) has shown that smaller infill width will lead to smaller VPE. Therefore, 0.1 mm is chosen for infill width as a small value to evaluate the best-case scenario. Contour and inner wall thickness is set to 0.1 mm, but it doesn’t have much effect on the result as VPE is calculated based on the effective area which is the boundary of infill region surrounded by inner walls.

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4.1 Straight-Line Based Infill Patterns Four groups of patterns, grid, triangle, tri-hexagon and line consist of straight parallel lines crossing in different conditions. This common format of generation makes the comparison of these four patterns easier. Figure 9 shows the change of VPE by the change of infill patterns’ angles and density percentage. Discrete lines represent the data related to the cubical model, while continuous lines show the cylindrical model results. In all cases, the overall amount of error generated in the cylindrical model is more than the error generated in the cubical model. This difference shows how the geometry of the model can be effective on the amount of VPE involved.

(a)

(b)

(c)

(d)

Fig. 9. VPE based on first group of parallel lines angle in different infill density percentage generated on a cubic model (represented with continuous lines in the graphs) and a cylindrical model (represented with discrete lines in the graphs); a) Grid, b) Triangle, c) Tri-hexagon, d) Line

The result shows that on average VPE reduces while the infill density percentage increases. The reason lies on the fact that the probability of unfilled gap creation increases in lower densities, as the infill lines’ distance is larger. In the cylindrical model, the amount of VPE changes by a small value when the angle of infill lines changes. The reason is that the cross-sectional area of a cylindrical model is a polygon in shape of a circle and the orientation doesn’t mean in a circle. Figure 9 can be used as a guide to decide on choosing infill pattern type based on the minimum VPE occurrence in a specific setup of infill parameters. Also, in each infill density, choosing infill orientation can happen based on the VPE amount in different angles to find the angle that gives the minimum error. This shows that these kinds of comparisons can lead to best infill setup (type and parameters). Although the results

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here are based on two models with specific geometries (best case scenario), the same methodology can be used for any freeform model. 4.2 Pattern-Based Infill Types Honeycomb and wiggle patterns are generated using a pattern line like in Fig. 5 and Fig. 8. Therefore, the type of input parameters in pattern-based infill types is a little bit different from infill patterns that are generated using groups of straight parallel lines. Based on the honeycomb input parameters, to see the behaviour of the honeycomb pattern, besides infill width, starting point number is the only parameter that can change while infill density is changing. However, the starting point number is found less significant based on the outputs (that are not presented here). Figure 10(a) represents VPE in different infill density percentages generated in the two models. The result shows that VPE increases by the increase of infill density percentage. This behaviour originates from the nature of hexagons which are the honeycomb’s element. The unfilled gap inside the hexagon prevents it from reaching the nominal infill density, especially in higher density percentages.

(a)

(b)

Fig. 10. Volumetric Percentage Error; a) in different infill density percentages for the honeycomb pattern, b) in different infill density percentages for the wiggle pattern while the angle between each two consecutive lines changes

Among different parameters involved in shaping the wiggle pattern, edge length and starting point number are found less effective (results not presented), therefore, VPE is calculated by the change of the angle between every two consecutive lines in the pattern in different infill density percentages. The result is presented in Fig. 10(b) and shows the same behaviour as straight-line based infill patterns approximately. The amount of error decreases by an increase in infill density. The reason is that pattern line distance is adjustable while the input parameters change and this prevents unfilled gap creation in the infill structure, unlike the honeycomb pattern.

5 Conclusion Commonly used layered infill patterns for additive manufacturing parts are presented and their input parameters are introduced. In the next step, the Volumetric Percentage

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Error (VPE) is discussed to show the difference between actual (output) and nominal (desired user input) infill density error for the introduced patterns in different setups. The results are presented for two models with specific geometries: a cube and a cylinder. The observed results show that VPE amount is in average smaller in the cubic model, and also VPE decreases when infill density increases, except for the honeycomb pattern when VPE increases from 5% in 10% nominal infill density to 35% in 100% nominal infill density. This makes the honeycomb pattern a relatively inaccurate option for the higher infill densities. Although other infill patterns produce relatively smaller amounts of VPE, the comparison of cubic and cylindrical models as case studies shows that VPE depends on the model’s geometry and might increase by the model’s complexity. To be more specific, VPE increases when two conditions happen; the number of cut straight or pattern lines increases (increasing round shape effect that occurs when narrow features become more in cross-sectional areas of a model), or infill geometry or parameter setup be in a way that the number of unfilled gaps increases. Results show that, generally, finding a behavioral pattern for different parameters’ setup is not accurate enough to rely on, therefore, to continue this study on finding the best infill parameter setup based on VPE minimization, an optimization algorithm should be presented to consider all scenarios. The optimization problem can change based on the requirement. For instance, it can be designed to calculate the amount of nominal infill density to reach the desired actual infill density in a specific setup. Also, it can be designed to find the best infill type and its input parameters for a desired nominal infill density.

References 1. Barari, A., Tsuzuki, M.S.G., Cohen, Y. and Macchi, M.: Intelligent manufacturing systems towards Industry 4.0 era. In: Canadian Society for Mechanical Engineering International Congress, vol. 3 (2021) 2. Barari, A., Jamiolahmadi, S.: Convergence of a finite difference approach for detailed deviation zone estimation in coordinate metrology. Acta Imeko 4(4), 20–25 (2015) 3. Dolenc, A., Mäkelä, I.: Slicing procedures for layered manufacturing techniques. Comput. Aided Des. 26, 119–126 (1994) 4. Zhou, M.Y., Xi, J.T., Yan, J.Q.: Adaptive direct slicing with non-uniform cusp heights for rapid prototyping. Int. J. Adv. Manuf. Technol. 23, 20–27 (2004) 5. Rianmora, S., Koomsap, P.: Recommended slicing positions for adaptive direct slicing by image processing technique. Int. J. Adv. Manuf. Technol. 46, 1021–1033 (2010) 6. Kumar, C., Choudhury, A.R.: Volume deviation in direct slicing. Rapid Prototyp. J. 11, 174– 184 (2005) 7. Jain, P.K., Taufik, M.: Surface roughness improvement using volumetric error control through adaptive slicing. Int. J. Rapid Manuf. 6, 279–302 (2017) 8. Pandey, P.M., Reddy, N.V., Dhande, S.G.S.: Real time adaptive slicing for fused deposition modelling. Int. J. Mach. Tools Manuf. 43, 61–71 (2003) 9. Gohari, H., Barari, A., Kishawy, H.: Using multistep methods in slicing 2 ½ dimensional parametric surfaces for additive manufacturing applications. IFAC-PapersOnLine 49(31), 67–72 (2016)

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10. Gohari, H., Bender, B., Barari, A.: 3D printable pneumatic valves for rapidly manufacturable mechanical ventilator amid Covid-19 outbreak. In: 2020 IEEE 10th International Conference Nanomaterials: Applications & Properties (NAP), vol. 2, pp. 02SAMA23-1–02SAMA23-4 (2020) 11. Gohari, H., Tsuzuki, M.S.G., Barari, A.: Improving geometric tolerances in 3D printable pneumatic valves designed for respiratory mechanical ventilators amid Covid-19 pandemic. In: 17th IFAC Symposium on Information Control Problems in Manufacturing, vol. 54, pp. 1053–1058 (2021) 12. Barari, A., ElMaraghy, H.A., Orban, P.: NURBS representation of actual machined surfaces. Int. J. Comput. Integr. Manuf. 22, 395–410 (2009) 13. Ueda, EK., Tsuzuki, M.S.G., Barari, A.: Piecewise Bézier curve fitting of a point cloud boundary by simulated annealing. In: 2018 13th IEEE International Conference on Industry Applications (INDUSCON), pp. 1335–1340 (2018) 14. Barari, A.: Automotive body inspection uncertainty associated with computational processes. Int. J. Veh. Des. 57, 230–241 (2011) 15. Barari, A., ElMaraghy, H.A., Knopf, G.K.: Evaluation of geometric deviations in sculptured surfaces using probability density estimation. In: Davidson, J.K. (ed.) Models for Computer Aided Tolerancing in Design and Manufacturing, pp. 135–146. Springer, Dordrecht (2007). https://doi.org/10.1007/1-4020-5438-6_15 16. Askari, H., Esmailzadeh, E., Barari, A.: A unified approach for nonlinear vibration analysis of curved structures using non-uniform rational B-spline representation. J. Sound Vib. 353, 292–307 (2014) 17. Berry, C., Barari, A.: Cyber-physical system utilizing work-piece memory in digital manufacturing. IFAC-PapersOnLine 52(10), 201–206 (2019) 18. Martins, TC., et al.: Algorithmic iterative sampling in coordinate metrology plan for coordinate metrology using dynamic uncertainty analysis. In: 2014 12th IEEE International Conference on Industrial Informatics (INDIN), pp. 316–319. IEEE (2014) 19. Barari, A.: CAM-based inspection of machined surfaces. In: 5th International Conference on Advances in Production Engineering - APE2010, pp. 17–19 (2010) 20. Parab, S., Zaveri, N.: Investigating the influence of infill pattern on the compressive strength of fused deposition modelled PLA parts. In: Vasudevan, H., Kottur, V.K.N., Raina, A.A. (eds.) Proceedings of International Conference on Intelligent Manufacturing and Automation. LNME, pp. 239–247. Springer, Singapore (2020). https://doi.org/10.1007/978-981-15-44859_25 21. Xia, L., Lin, S., Ma, G.: Stress-based tool-path planning methodology for fused filament fabrication. Addit. Manuf. 32, 101020 (2020) 22. Clausen, A., Aage, N., Sigmund, O.: Exploiting additive manufacturing infill in topology optimization for improved buckling load. Engineering 2, 250–257 (2016) 23. Kain, S., Ecker, J.V., Haider, A., Mussi, M., Petutschinigg, A.: Effects of the infill pattern on mechanical properties of fused layer modelling (FLM) 3D printed wood/polylactic acid (PLA) composites. Eur. J. Wood Wood Prod. 78, 65–74 (2020)

Equirectangular Pictures and Surrounding Visual Experience. Spherical Immersive Photographic Projections at: Boito Architetto Archivio Digitale, Historical Exhibition at Politecnico di Milano Federico Alberto Brunetti(B) Design Department, Politecnico di Milano, Milan, Italy [email protected]

Abstract. The inter-institutional university research between the Politecnico di Milano and the Accademia di Brera (2021–2022), to which this survey has participated, has collected, classified and shared in digital formats the various archival and documentary sources about Camillo Boito: one of the most significant architects in the historical and cultural period of the Unification of Italy. Some of my previous publications on spherical photo-cameras, and about the devices for displaying these images through AR/VR viewers, are here contextualised in particular around the case of research on the meticulous spatiality conceived and realized by the architect Camillo Boito (1836–1914) in some significant projects. This further experience on the field made it possible to verify the functionality and effectiveness of the visualisation of spherical views, not only in technological terms, but in a comparative manner with respect to the usual methods of representing architecture, implying some methodological deductions. In this short essay I therefore briefly propose some methods and results from the survey on two case studies of 19th century interior spaces, through the graphic aspects that can be visualised by the geometric algorithms for their treatment as spherical images, and some considerations deduced from these experiments in the process of geometric visualisation of the designed space. Keywords: Equirectangular projection · Spherical view · Architectural survey · 19th century Italian architecture · Camillo Boito · Experience design

1 Introduction The instrumental availability and technological research being carried out on spherical images requires some clarification in order to better understand the whole and the origin of these - apparently disruptive - new possibilities of perception and representation of three-dimensional space. At the basis of these innovations is the geometric definition of equirectangular projections - also known as cylindrical equidistant - whose conception, dating back to Marinus of Tyre (AD 70–130), was also the basis for the oldest terrestrial © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 541–553, 2023. https://doi.org/10.1007/978-3-031-13588-0_47

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cartographies. The problem of planar unwinding of spherical surfaces was subsequently reinterpreted using different geometric protocols, depending on specific purposes and related accuracy and deformation issues. The current computational possibility of interpolating, and digitally translating, between the geometries adopted in the various spherical representations has been made relatively manageable by the now widespread and proven conversion protocols of current digital cartographic softwares (Yang et al. 2000). Similarly these geometries, already applied in the representation of the spherical convex shape of the earth, have also been adopted for the description and astronomical restitution of the configuration of celestial space, which surrounds the earth as an ideal spherical concave configuration of the universe from the point of view of our planet. Obviously, recent discoveries about the far position and relative distances of the stars in the space of the universe have made it possible to evolve these original polar and angular views, arranged on a spherical map, into computational models of three-dimensional and dynamic representations. However, verified the antecedent and relevance of a cultural and mathematical heritage experienced in studies generated in the cartographic and astronomical fields, this paper focuses on the use of spherical views in the representation of perceived space at an architectural and environmental scale, as an extension and simultaneous re-presentation of the observer’s entire empirical visual field. Some of my previous publications on spherical photo-cameras, and on the devices for displaying these images through AR/VR viewers (Brunetti 2021), are here contextualised in particular around the case of research on the meticulous spatiality conceived and experimented by the architect Camillo Boito (1836–1914) in some significant projects. This further experience on the field made it possible to verify the functionality and effectiveness of the visualisation of spherical views not only in technological terms, but in a comparative manner with respect to the usual methods of representing architecture, implying some methodological deductions. In this short essay I therefore briefly propose the methods and results of the survey - in the meantime - from two case studies of 19th century interior spaces, through the graphic aspects that can be visualised by the geometric algorithms for their treatment as spherical images, and finally some considerations deduced from these experiments in the process of geometric visualisation of the designed space.

2 Boito Architetto Archivio Digitale The inter-institutional university research between the Politecnico di Milano and the Accademia di Brera (2021–2022)1 to which this survey participated, has collected, classified and shared in digital formats the various archival and documentary sources about this architect. Camillo Boito was one of the most significant and authoritative architects in the historical and cultural period of the Unification of Italy (1871); his work carried on a lively debate in the culture of design the new civil necessities and the foundation of a stylistic contemporary language, heir to a millenary historical tradition to be preserved 1 https://www.eventi.polimi.it/events/boito-architetto-archivio-digitale/?lang=en

https://www. auic.polimi.it/en/school/projects/galleria-del-progetto-exhibition-space/boito-architetto-arc hivio-digitale.

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and revived, using well-accepted and refined methods of representation. This 19th century author expressed himself in a decidedly interdisciplinary way between tradition, innovation, identity, applied arts and theoretical debate, thanks to a solid foundation in geometric design, material knowledge of building and consolidated skills in the craft resources then available. In his architectural, theoretical and didactic work, the art of drawing (Boito 1882) was at the foundation of design knowledge: both in the relief of existing monuments and in the building of new ones, based on geometric skills in two and three dimensions - as can be seen in his and his students’ drawing templates - with orthogonal projections of plans, sections, facades and perspectives. These general rules are enhanced by the expertise also derived from ornamental traditions and archaeological studies on the origin of geometry as a regulatory and symbolic tool of Romanesque cosmatesque stone inlay. In this era, the frequent use of stone in construction had still maintained an intense contribution from the study of stereometry, conceived as a precise forecast of the solids geometry in the designing role of each individual construction element, compared to the three-dimensional drawing of the whole: from the working of stone for the static realization - and initially on the ground - of architectural works, as well as a numbered repertoire of forms for use by the competent workers for the realisation of artefacts. The spatial, formal and material attention of Camillo Boito was at the same time directed towards the valorisation and integration of the now consolidated serial procedures - and no longer only artisan/pre-industrial - for the realization of constructive components: he managed a national level specialized publishing, and an indepth cultural debate also for the development of technical training schools. These skills, founded on the study of compositional and geometric languages of the past (Zucconi 1997), were based in geometry and drawing, almost like as an “algorithm” of comprehension and application of a common language. Such methods in Architectural debate can be compared to different but similar references in the Arts and Craft movement by William Morris in Great Britain, or even in the Deutscher Werkbund in Germany founded by H. Muthesius and in Austria in the Wiener Werkstätte, founded by Koloman Moser and Josef Hoffmann. While maintaining a rare executive and direct competence in architectural drawing - as documented by the numerous dossiers and design drawings collected and digitised for the exhibition - it is interesting to note that, from a certain threshold onwards, the focus on innovations in architectural representation techniques is highlighted by the new role assumed by photography. Shortly after the invention of photography, the distinguished professor C. Boito began also to collect the first available prints and to use this new technique for the visual representation of the various phases in the study of architecture: in the systematic survey of historical buildings, in the comparative documentation of the phases of construction sites between the original state and completion in restoration works, in the comparative cataloguing of samples of elements produced serially by the industrial arts for the decoration of buildings (Brunetti 2018). The foundation of academic institutional photolibraries were conceived, in consequence, as iconographic repertoire for the ‘remote’ study of the monumental and artistic heritage.

3 The Global Frame of Equirectangular Picture In order to better understand the experimentation that I am presenting, it could be useful to recall some geometric fundaments - presented here with simple proportional graphic

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diagrams - used in spherical photography. At a first level of description I would like to remember and clarify the process of mutual interaction between the equirectangular image (in the specific standard of the current digital spherical cameras available) and the digital optical space that is captured, and subsequently returned to the viewing or projection environments. Available; AR/VR and 360°Theatre (and/or planetary) (Fig. 1).

Fig. 1. a): equirectangular proiection: reticular grid of a sperical surface; flat planmap, with centred observer orientaton indices. b), c): equirectangular proiection views; reticular grid of a sperical surface: with centred observer orientaton indices; b) (left): concave interior, c) (right): below pole concentric inside. (2 and 3 dimensional graphic diagrams by the Author, made by equirectangular image processing software)

The spherical cameras – twin or multiple lens - capture all the image surrounding from the shooting point and implement all the specific different directions views into a single equirectangular digital matrix of 1:2 format (equivalent to the projection of two “front-rear” hemispheres into “panoramic” view composed of two squares placed side by side), with embedded metadata concerning picture orientation (Fig. 2 a), b), c)).

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Fig. 2. a), b), c): a) (left): 360° Sperical picture: extended flat planmap of the equirectangular proiection survey file, with centred observer orientaton indices; b) (center): concave interior view, c) (right): below pole concentric inside view (courtesy of: Liceo Artistico di Brera, Milano. Spazio Hajech: interior view of the conference and exhibition hall; spherical picture by the Author, digital elaboration with equirectangular image processing software).

With appropriate vector softwares it is possible to generate equirectangular raster image files from three-dimensional models of projects, as well as from 3D scans survey.

4 From Original Vintage Photographs to Wide-angle Architectural Photographic Views and Current Digital Spherical Experiments A second analysis here concerns some specific case study adopted, in the context of the exhibition “Boito architect Archivio Digitale”, examining the different types of architectural photographs taken in the evocative interior architectural environments. Two important interior spaces have been chosen as case studies for this investigation in reason of their peculiar typology as well as of the decorative apparatuses that characterise them designed by C. Boito: the complex and very articulated staircases of the staircase of Palazzo Cavalli Franchetti (1886) in Venice (Romanelli 1989) (also compared with some century pictures), and the former Museo Civici del Santo (1879) in Padua. The comparison below presented - some also with reference to rare 19th century photographic images from the historical archive of C. Boito - with those obtained with traditional wide-angle cameras, can be compared with equirectangular spherical shots. The progressive widening of the field of vision is immediately evident from the comparison in the previous images. This factor, originating from the geometric theories of perspective, (linear and later also curvilinear) has various causes and motivations. These qualities derive first of all from the intrinsic characteristics of the optical equipment used (evidently limited in the case of 19th century images, then more available in today’s wider conventional optics, and furtherly extended to the spherical through digital treatment in the equirectangular). It must be considered as well the technical relationship (photo-graphia) between the shooting optics and the storage medium (analogue digital), and finally from the sharing of visual canons (or iconographic codes) relating to the perception of images. These evolutions of the iconographic code can be identified in the convention in the span of these images (Fig. 3 a, b, c): 1) the limits of the average ocular field [30°–60°] (without any significant consequent aberrations of the sphericity of the retina, i.e. within the visual field of the perspective canons of scenography);

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2) the iconographic acceptance of the wide-angle extension [90°–120°] (with relative critical awareness of the percepted variation in of the communicated spatial amplitude); 3) Finally, the equirectangular [360° spherical] projection, developed by means of multiple optic camera device, managed by specific algorithm on a computational support (on a digital matrix in 1:2 format): currently usable through observation with interactive viewers using the metadata embedded in the file structure, or projected in a spherical theatre. The planar equirectangular presentation (as in the case of the pages of this publication) is also interesting because - if appropriately composed and after an adequate introduction and experience of geometric decipherment (see Sect. 1) it allows one to grasp the observer’s lateral and rear visual fields (Fig. 4 a, b, c)

Fig. 3. a: (Left quatrain): Venezia, Palazzo Cavalli Franchetti, 1886. Collezione fotografica of Camillo Boito, (Courtesy of¨Accademia di Brera – iconographic apparatuses for research doctorate Thesis of: PhD Federico Alberto Brunetti ©1997); view of exterior, staircase: industrial arts decoration sample details. b: (Right quatrain): Venezia, Palazzo Cavalli Franchetti: staircase interior wide views: zenith sight of muses decorations, frontal section between the floors, mid landing passage. Focal length: 12–24 mm. photography of architecture ©2021 by the Author. c: Venezia, Palazzo Cavalli Franchetti, staircase interior global views: decorations, mid landing passage. 360° Spherical picture, photography of architecture ©2021 by the Author

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The 19th century architects soon understood how the photography released a great opportunity of producing finely accurate images of their case studies, obtaining an important documentary supplement to the survey drawing of specific objects and artefacts; just as for the visual culture of the 20th century the wide-angle view represented a significant cognitive extension towards a broader and more complex understanding of the context and between the relationships present in the field of observation. Finally, the possibility - already theorised for a long time in the field of geometry and cartography - of capturing the global field of view from a point of view that is stored on a digital support and that can be interacted with through appropriate devices, represents for the current visual culture a mode of experience (see Sect. 5) consistent with contemporary awareness of the translation of reality into digital data.

Fig. 4. a: Padova. Camillo Boito; ex Musei Civici del Santo. Staircase interior global views: decorations, mid landing passage. 360° Sperical picture ©2021 photography by the Author. b, c: Padova. Camillo Boito; ex Musei Civici del Santo. Staircase interior wide views: b) (left quatrain): zenith sight: ceiling decorations, frontal section between the floors, mid landing passage. Focal lengt: 12–24 mm.; c) (right quatrain): 360° Sperical pictures ©2021 by the Author

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It is worth mentioning here the pioneering and original work of C.M. Escher (1898– 1972) who, in analogue era, explored through his brilliant graphic work the perceptual limits of the visual field, towards extended representation of space opened to spherical perspective, and some other the paradoxes of vision, inaugurating a new iconographic codes of a “surrounding vision”; his research was appreciated and also carried out in collaboration with various mathematicians, including Douglas Hofstadter. In fact, these widened images are not so much the responsibility of the frontal visual memory, but of the environmental perception in the mnemonic maps reconstructed in order to coordinate the spatial orientation - static and dynamic - and the ergonomic and postural movements of the individual. Considerable recent researches from the area of neuroscience, relating to vision and tridimensional and dynamic mnemonic modelling of space, are particularly concerned with the multisensory interactions that pertain to this cognitive domain. These mental activities, while based on direct vision, are integrated experiences through memory, by articulated movements of the eyes, head and whole body, and finally through hypothetical and deductive cognition. New behavioural measurements, integrated with the mapping of the specific activated brain areas, are making now possible to better understand the complex dynamics of vision, and not just of sight, in the global experience of space. For all these integrated reasons, which here are only summarised, I can consider that 360° vision (a broad definition that goes beyond the view) and spheric representation can be assumed - similarly to perspective for Renaissance culture - as a ‘symbolic form’ of the current digital era.

5 Spherical Projections Experience at Politecnico di Milano 360°Theatre Starting from these assumptions, and having available the spherical images of the Boito’s internal amplifiers, the third element of verification concerns the specific characteristics of the functioning of the new Teatro360° of LaborA in the Politecnico of Milano, both for the compositional and algorithmic aspects of the adopted geometry, and for the perceptive and environmental experience that this device has demonstrated to be able to offer to the visitors. The opportunity to project the spherical images created in Boito’s interiors in the theatre360° LaborA of the Politecnico di Milano, allowed me to test, and better understand, the compositional formats of the specific digital scenic device, and also to observe directly, and indirectly, the perceptive and experiential interactions of such a representation apparatus (Fig. 5). The partitions of the original spherical equirectangular image, in this surrounding circular digital room, are subdivided and managed by different algorithms on two contiguous, accurately coinciding surfaces as follows (Fig. 6a, b, c, d): A) a surrounding cylindrical panoramic projection, B) a circular planar projection at the base: A) i.e.: the overview towards the surrounding horizon (indicated in the diagram in the band between the double lines) is composed of a continuous sequence proportioned

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1:8 (in the diagram: oriented with the front view “C” into the centre, the two lateral views respectively to the left “SX” and to the right “DX”, the two symmetric positions behind the observer respectively “−R” and “+R”, with the respective index of angular directions (−180°, −135°, −90°, −45°, −0+, +45°, +90°, +135°, + 180°) (Fig. 6 a) B) i.e.: the horizontal surface report within the remaining circular base perimeter (therefore indicated in the diagram between the lower double line band and the horizontal line focusing in the “S” pole), which reports the surface converging to the shooting point and below the observer. A system of 4 panoramic projectors, positioned and obscured in the ceiling of the hall - and 2 towards the floor - provide the splitting and stitching of the original equirectangular picture. A digital post-production control unit allows to manage the files of still or dynamic video images - from real shots or virtual simulations - in the space of the immersive 360° theatre. Due to the typology and functionality of the hall, unlike what could take place in a semi-dome of a planetarium, the vertical space towards the “N” pole of the image above the horizontal panoramic band is not visible, as it is dedicated to locate the projectors system in the ceiling ((Fig. 6 a), b), c), d)). In any case, the observer’s perceptive experience of the surrounding and underlying visual field of an architectural context is decidedly realistic, interesting and involving. I realized a short series of verification maquettes, using the same geometric diagrams and the respective images shown here, finalized to understand analogically and illustrate this tridimensional procedure of subdivision, treatment and digital projection (Fig. 7 a, b), c) d)).

Fig. 5. Spherical immersive photographic projections at: Politecnico di Milano 360°Theatre at: LaborA - physical and virtual modelling. Spherical image composition calibration test, projection display for Boito Architetto Archivio Digitale exhibit presentation, visitors experiences 2022.0102.

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Fig. 6. a), Politecnico di Milano 360°Theatre at: LaborA Spherical image composition calibration test. b), c) d): up/down – image composition calibration test (compare with: Fig. 1 a) b), c):

Fig. 7. a): image composition calibration test, verification maquettes display for: Boito Architetto Archivio Digitale exhibit presentation, visitors experiences; .b), .c) .d): image composition calibration test, verification maquettes display; first layout, cilindrical and spherical planar projection

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6 Some Deductions from 360°Theatre Test It is definitely interesting to verify the representation by the shooting of spherical images, than re-presented in the environment of immersive 360° projection, compared to the spatiality described and prefigured by the original Boitian projects (digitalized, gathered and shared for the exhibition). The geometric description presented in the original orthogonal projections design templates, evidently prefigure a spatial complexity that would be fully revealed in the three-dimensional construction of the work. In methodological terms, and potentially in a more general sense, it is possible to identify some innovative elements that these visual technologies present for a greater understanding of the perception of the global space around the observer. These “spherical perspective” images allow to describe in detail, and therefore to be able to analyse, and prefigure, some further field of design intervention, both in optical-visual and mnemonic experience of designed spaces.

7 From Equirectangular Geometry to “Experience Design” Some methodological considerations therefore arise regarding some of the epistemological issues involved in these experiments. The new technologies offered by these scenarios and communication tools, in particular for vision, seem progressively oriented towards bridging the distance between the observer and the image of the represented data. The wearability of immersive visors has now become an emblematic individual point of contact between a peripheral device with a system of technological platforms remote and dominant – to a virtual world whose suggestion seems to be able to propose itself perceptively almost as a seductive alternative to the real world. The available power to those who manage these ‘augmented’ potentials of the ‘meta’real is even made evident by the socio-economic phenomena that this development has triggered, generating the enrichment and power over data, i.e. as an effective resource of the present and future economy. On other hand is evident the opportunity for a responsible educational collaboration by these new media is strategic in relation to global, ecological, social sharing, and newfound social and ethical values emergencies. The understanding and knowledge of these new (digital, and powerfully iconographic) languages is an argument of great responsibility for those who seek to orient present and future culture and education. Also these scenarios the opportunity offered by virtual theatres are particularly interesting for advanced geometric disciplines and neuro-visual sciences, which could find here a particularly relevant field of interaction (Tatler and Land 2011). For the specific representation of the urban environment and architecture, the possibility of presenting to the observer with a global field of view - from real life footage or virtual design models - analogous to the usual panoramic view of the surroundings (360° at the horizon of the field of view) and of the ground space (in a wide range of horizontal interaction), allows a perception that is decidedly verisimilar to the conditions of environmental visibility and practicability of the space around the individual. The reactions of the visitors into the “wunderkammer” of the 360° virtual theatre revealed spontaneously them led to “move” in the available horizontal space, thus experiencing not only the aspect of the complete enveloping overview, but also of the immediate

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viability of the represented environment. These experiences are developed with remarkable proxemic spontaneity by some intuitive actions (see: Fig. 5): gesturing to details that catch one’s eye, starting in trying to climb the projected steps, conversing willingly and commenting on the qualities of the place displayed with other person in the room. Even though this is not are not stereoscopic projections, the perception of the shown environment is felt as an experiential proxemics depth, probably because the observer’s entire body is free to generate an immediately and spontaneously effective interaction in the modification of the obtainable “points of view”. The increase in “degrees of freedom” that a virtual theatre offers, and the immediacy of the possible immediate and simultaneous sharing between several co-present users, requires a commitment to investment and technological management on a completely different range compared to the selfreferencing of AR/PC platforms, however much they may be networked remotely with other users. A first conclusion that I would like to underline is that the contemporary image is the increasingly dematerializing from a material support (a painting, a panel, a wall, an “object”) and presents itself in visual terms as an autonomous and individual interacting spatiotemporal experience. These dynamics has been well described in the recent literature defined “Experience Design”: such definition underlines the user or observer’s active ability to inwardly grasp the relations and relationships of a work, is tentatively replaced by a generation of artefact proposed as an active experience in itself Pine and Gilmore (1999). The 360°theatre, far away from a possible alienation and although requiring a decidedly significant technological apparatus, seems to offer a space of representation in which the observer can rediscover the (prio)perception of his own initiative and authority towards to the image he virtually inhabits. Equirectangular pictures and surrounding visual experience proves to be a system of representation which has the communicative possibility of sharing and disseminating knowledge of reality, leaving the irreducible experience of the direct encounter between the individual and concrete reality, establishing situations in which the interaction between perception, intelligence, astonishment and surprise can still be part of the true cognitive discovery of the world. “The horizon of sky and earth ends in the same line”. This essential phrase by Leonardo da Vinci seems to summarise precisely the dynamic between perception and concept (Trattato della pittura. Dell’orizzonte. §.928). This quotation remembers that the experience of reality is however synthesised by the intelligence of vision, in forms that are the basis of descriptive and re-creative knowledge of the world, that is, as a presupposition of the modes of representation and its design regeneration. As analogously in history, an innovation in the possibility of representation can reveal an innovative conception of the composition of the designed space. A new mode of representation, extended to the entire surrounding geometric space and visualised by the individual, can therefore advance the knowledge of the architecture experience, of its mental formalisations and therefore of the compositional processes in which geometry, vision and matter can interact by way of the project. Aknowledgements. Proff. Luca Monica, Stefano Cusatelli, Annalisa B. Pesando, Sandro Scarrocchia, curators of exibition: Boito Architetto Archivio Digitale, Politecnico di Milano (2021.122022.01); Politecnico di Milano: LaborA - physical and virtual modelling; Comunicazione &

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Relazioni Esterne, exibition staff prof. Alessandro Deserti, Director od Desig Department and: Fabio Peri, Planerario Civico di Milano; Mario Cigada Milano, Oftalmologist; Alessia Dorigoni (UNITN) Eye Tracking neuroscience resarcher.

References Boito, C.: I principi del Disegno e gli stili dell’ornamento. Hoepli, Milano (1882) Yang, Q., Snyder, J.P., Tobler, W.R.: Map Projection Transformation: Principles and Applications. Taylor & Francis Group, London (2000) Brunetti, F.A.: Spherical images: capture and visualization devices. Icons of a computational paradigm. In: Cheng, L.Y. (ed.) ICGG 2020. Advances in Intelligent Systems and Computing, vol. 1296, pp. 947–951. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-63403-2_92 Brunetti, F.A.: The Camillo Boito Historical Heritage Photo Collection as an Iconographical Fund for the “National Stile.” In: Amoruso, G. (ed.) INTBAU 2017. Lecture Notes in Civil Engineering, vol. 3, pp. 211–216. Springer, Cham. (2018). https://doi.org/10.1007/978-3-31957937-5_22 Pine, J., Gilmore, J.: The Experience Economy. Harvard Business School Press, Boston (1999) Tatler, B.W., Land, M.F.: Vision and the representation of the surroundings in spatial memory. Philos. Trans. R. Soc. B 366, 596–610 (2011) Romanelli, G.: Tra gotico e neogotico. Palazzo Cavalli Franchetti a San Vidal. Albrizzi, Venezia (1989) Zucconi, G.: L’invenzione del passato, Camillo Boito e l’architettura neomedievale 1855–1890. Marsilio, Venezia (1997)

Textile Drawing. A Geometric Matter Stefano Chiarenza(B) San Raffaele Roma Open University, 00166 Rome, Italy [email protected]

Abstract. This study investigates the relationships between textile design and geometry, focusing to this end on the concepts of structure and symmetry. These aspects, which affect both the graphic systems of the decorations and the textures of the fabrics, explain with great effectiveness the compositional logic of textile patterns whose complexity is often attributable to simple flat isometric transformations. The goal is to reveal geometric principles and construction rules underlying the design and the textile structure. These aspects are generally difficult to interpret for those who deal with textile design and who frequently follow a practical and intuitive approach and are not always supported by a real geometric awareness of graphic constructs. The paper thus intends to offer a tool with which to convey a scientific approach to the graphic design of fabrics, stimulating reflection on teaching and research in the discipline. Keywords: Weaving · Texture · Isometric transformations

1 Introduction The study of fabric design, apparently only a question of graphics, actually represents a wide-ranging research field. The graphic design of a textile pattern implies concepts such as structure, series, proportion, symmetry, which inevitably refer to other disciplinary fields, first of all, mathematics and geometry, and which are also reflected in the balance of natural forms, whose complexity is always attributable to a precise constitutive rigour. In reality, the possibility of resorting to basic regular schemes, on which to intervene with infinite and unpredictable variations to design complex final configurations, clearly expresses the close link between art and science which is so significant in textile design. There is a strong relationship between the basic formal structure and the decorative image derived from it, generally obtained by repetition or transformation of the elementary design that constitutes the composition module [1, p. 1]. The study, starting from a graphic-configurative approach, investigates the concept of structure and regularity in the design of fabrics. For this purpose, attention is paid both to the compositional structure of the printed design and its weave. These processes while highlighting different perceptual aspects of a fabric –only visual in the first case, also tactile in the second– are attributable to a similar geometric logic on which we intend to investigate.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 554–561, 2023. https://doi.org/10.1007/978-3-031-13588-0_48

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2 Patterns Geometry in Printed Decoration Graphic patterns are fundamental in the decorative design of the printed textile surface. Their characteristic is, as clarified by Hann, “the repetition (or translation) of a motif at a given distance on the plane. If the repetition is continuous in one direction only, between two lines (real or imaginary), the pattern obtained can be defined as a border pattern or linearly developed motif, frieze or one-dimensional design. When the repetition occurs in two independent directions, covering the plane, the resulting pattern can be referred to as all-over; it also includes wallpaper patterns, periodic patterns or crystallographic patterns” [2, p. 55]. The basic assumptions of a pattern are the structure [3] and the symmetry of the ornamentation. The structure corresponds to a basic grid underlying the design: it is an implicit network of points organized according to precise geometries, from which the so-called graphic units, or elementary modules, of the decoration, derive. In particular, within a plan, it is possible to identify five types of elementary units [4, p. 262]: parallelograms, rectangles, rhombuses, squares and hexagons (Fig. 1, left). These units are also called Bravais lattices, by the French physicist Auguste Bravais (1811–1863) who first classified them in his 1848 study of crystals. [5, pp. 27–28].

Fig. 1. On the left: the 5 Bravais lattices. On the right: some examples of basic motif’s geometric transformations.

From the lattice and the reticular units associated with it, it is, therefore, possible to subdivide the plane according to a matrix which, by determining the arrangement and movement of the basic motifs, dictates the rule for the repetition of the figurative composition [6–8]. Among the various geometric decorations, the all-over printed motifs make the key role of the pattern structure more effectively. In them, the modular figurative combinations are independent of the shape of the decorative surface frame and the decorations are conceived according to a logic of repetition, potentially endlessly. By anchoring itself

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to the base structure, the decorative module can proceed in all the main directions of the plane (vertical, horizontal, diagonal). The other distinctive aspect of the graphic layout of a pattern is, as observed, the symmetry, which comes into play with great clarity in the figurative composition. It is a question of considering this principle in a broad sense, including in it all the isomorphic geometric transformations –translation, rotation, reflection and glide reflection– which can be traced in a plane, both concerning the basic module and to the overall image (Fig. 1, right). First of all, focusing on elementary decorative motifs –also known as “finite figures”, “bounded figures” or “point groups” [2, p. 55]– we can say that these are distinguished in symmetrical and asymmetrical, as a function of their internal geometric logic. In the first case, at least two graphic elements with the same shape and content are recognized in the modular unit, except possibly for a different orientation (Fig. 2, left). Depending on the constituent symmetry characteristics, motifs may be classified using the notation cn (c for cyclic) or dn (d for dihedral). The cyclic ones have n-fold rotational symmetry; the dihedral ones have n distinct reflexion axes as well as n-fold rotational symmetry [9, p. 10]. In the second case, the basic module cannot be further broken down, representing in itself an irreducible figuration (Fig. 2, right).

Fig. 2. William Morris, patterns for wallpapers. On the left: Garden Tulip, 1885, symmetrical pattern (Printed: Jeffrey & Co. Image: V&A). On the right: Autumn Flowers, 1888, asymmetrical pattern (J.H. Dearle. Image: Brooklyn Museum).

Among the possible isomorphisms which lead to the final design of the decorative composition by repetition of the module, the most used is undoubtedly the translation. The combination of translation with one or more isometries results in 17 decorative patterns. Of these, four are obtained by resorting to translation, reflection, glide reflection, symmetry or combinations of them; five exploit, in addition to the isometries already mentioned, also the rotation (180°); three, on an exclusively hexagonal lattice, also

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resort to rotation (120°); three, based on a square lattice, use 90° rotations; finally, two on a hexagonal lattice, together with reflections and glide reflections, see the use of 60° rotations. It is evident that, depending on the choice of the reticular structure, the elementary decorative motif, with its intrinsic symmetry, and the compositional rules that respond to different combinatorial possibilities, we arrive at variegated decorative patterns. We can also say that motif and structure can influence the perceptual organization of a pattern. In other words, the motif and its structuring in the pattern can highlight some isometries rather than others. As it was observed in a specific study “each type of wallpaper pattern is defined by its lattice type and a list of isometric transformations (i.e., transformations that preserve distance: translation, rotation, reflection, and glide reflection) that map the pattern onto itself. When different perceptual organizations are induced in wallpaper patterns by different motifs, different isometries become salient to the observer. As a result, formally isomorphic wallpaper patterns can appear to be of different types, and formally different wallpaper patterns may appear to be similar” [10, p. 190]. These considerations reinforce and make more evident the perfect synthesis, in textile design, between art and science [11], between drawing and mathematics. As observed by Davis and Hersh “through intuition, the artist is often an unconscious mathematician, discovering, rediscovering, and exploring ideas of spatial arrangement, symmetry, periodicities, combinatorics and transformations and discovering, in a visual sense, theorems of geometry” [12, p. 43].

3 The Geometry of the Textures Geometry and mathematics are fundamental for all textile production. It is not just a matter of surface. A significant link is represented by the actual structure of the fabrics. Regardless of the printed motifs, which are also governed by principles of modularity and repetition, the weave of the fabric constitutes a strong link between textile art and geometric-mathematical principles. In the textile structure, the order in which the threads are intertwined defines a weft that repeats itself periodically. The unit of repetition is the smallest, single recurring part. In the unit, each wire must have at least two weaves. Unlike what happens in the printing of the surface, in weaving it is important to plan the sequence of the weaves that will define the texture. Through the alternating overlapping of the threads, it is possible to create a large number of motifs that can be both of a geometric modular type and based on the definition of figurative motifs (Fig. 3). The possibilities of obtaining a huge variety of designs are historically linked to the invention, in 1804, of the so-called Jacquard machine, named after its inventor Joseph Marie Jacquard (1752– 1854), which represents perhaps the most revolutionary innovation in textile art [13, p. 30]. The Jacquard machine offered the possibility to speed up the weaving work, to weave with a single movement, hundreds of threads with a single evolution (today more than 10000), allowing the realization of complex geometric drawings and to increase the dimensions of the fabric. The sequence of operations was controlled by a series of replaceable punched cards to be inserted in the head of the loom [14]. Precisely for this reason the Jacquard machine has been considered an innovation closely linked to the history of computing hardware [15].

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Fig. 3. Examples of weaving schemes. Taken from Donat, F. Methodik der Bindungslehre, Dekomposition und Kalkulation für Schaftweberei, Hartleben’s Verlag, Wien und Leipzig (1908).

The different possible sequences of wefts or woven structures are represented, in the design phase, through conventional graphic notations. The most commonly used method consists in representing the intertwining of the warp and weft threads on a squared grid. A black square indicates that one thread passes over the other [16, p. 2]. An empty square instead represents the passage of a weft thread over a warp thread. If two or more adjacent black squares follow one another in a vertical column, it means that a warp passes over two or more weft threads (Fig. 4, left). Such graphic notation already makes evident the structure of the motifs and the methods of repetition. The definition of a graphic structure with a unique meaning “combines two possible ways of reading: on the one hand, it renders the pattern intuitively comprehensible and allows the weaver to count its features as though on an arithmetic board. On the other hand, the unam - biguous determinacy of the notations renders them usable as a storage medium” [17, p. 150]. Similarly to what happens in the printed drawings, also in the weaving, the basic units are reiterated according to precise mathematical-geometric sequences, determining the composition of the pattern as a whole. Also in the warp, therefore, “at the base of the compositional system, there is the relationship, that is the geometric-chromatic-formal structure which, suitably combined in a compositional set of adjacent and interpenetrating modules, is repeated for the entire extension of the fabric” [18, p. 240] or the surface to be covered.

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It is therefore clear that the beauty of a pattern is not so much due to the nature of its elements as to the right use of them as a unit in a rhythmic scheme [1]. However, the relationships between textile art and geometry are not limited to the definition of flat patterns but also extend to the third dimension. Three-dimensional textures were already possible in the past thanks to weft or chain weaving or through the use of the jacquard loom. Today the methods of parametric design and 3D printing allow the creation of 3D textures starting from two-dimensional textures or through innovative additive processes that can simulate the morphologies of biological growth. Recent studies on the geometry of 3D textiles have also made it possible to define a taxonomy [19]. The textiles have been grouped into four categories (solid, hollow, shelllike and nodal) on which completely new surface joints are based. The results obtained, which are impossible through traditional weaving processes, consist for example of tessellations and patterns of the alveolar, polygonal, shed type. These elements, created with the aid of parametric infographic software, define a new aesthetic that evokes the microscopic morphology of biological structures found in nature. In this way, the link between geometry and form and between textile art and mathematics is further strengthened (Fig. 4, right). As Thompson wrote “the harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty. […] Moreover, the perfection of mathematical beauty is such that whatsoever is most beautiful and regular is also found to be most useful and excellent” [20, pp. 1095–1096].

4 Conclusions In artistic production, from the visual arts to architecture and design, it is often possible to find the use of geometric-mathematical principles for the definition of structures and form. If compared to the major arts there is a very rich panorama of studies and research, the role of these principles in the field of decorative textile art is less investigated. The present article, therefore, wanted to highlight a series of problems of a geometricmathematical type that characterize the fabrication of fabrics, enunciating and discussing them as a possible stimulus in the didactic and research fields. In fact, in recent years university courses aimed at training in fashion design have flourished and it, therefore, appears of great importance to recover those often transversal theoretical foundations that allow overcoming professionalizing approaches, orienting towards new horizons of research and innovation in design. In particular, in the field of textile design, where the structure and shape of the decoration play a key role, a theoretical void has often been identified with the consequent need to refer to mathematical literature that is in many cases difficult to understand. The importance of bringing back to the discipline of textile design some basic concepts of geometry certainly allows to simplify the symbolic-terminological abstraction and to transfer this knowledge effectively to those who approach this area within the new curricula, allowing a conscious graphic-design approach.

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Fig. 4. On the left: examples of textures and weaving schemes. On the right: 3D textures.

References 1. Horne, C.E.: Geometric Symmetry in Patterns and Tilings. Woodhead Publishing Ltd., Cambridge (2000) 2. Hann, M.A.: The fundamentals of pattern structure. Part I: woods revisited. J. Text. Inst. 94(1–2), 53–65 (2003) 3. Washburn, D.K., Crowe, D.W.: Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. University of Washington Press, Seattle (1991) 4. Grünbaum, B., Shephard, G.C.: Tilings and Patterns. Freeman, New York (1987) 5. Bravais, A.: On the Systems Formed by Points Regularly Distributed on a Plane or in Space. Crys-tallographic Society of America, Memoir, no. 1, pp. 1–113 (1949). Translated by Amos J. 1st edn. Sur les systèmes formés pardes points distribués régulièrement sur un plan au dans l’espace. Journal de l’École Polytecnique, 33(19), 1–128 (1850) 6. Stephenson, C., Suddards, F.: A Textbook Dealing with Ornamental Design for Woven Fabrics. Methuen, London (1897) 7. Christie, A.H.: Traditional Methods of Pattern Designing. Claredon Press, Oxford (1910) 8. Day, L.F.: Pattern Design. B.T. Batsford Ltd., London (1903) 9. Thomas, B.G., Hann, M.A.: Patterns in the Plane and Beyond: Symmetry in Two and Three Dimensions. University of Leeds International Textiles Archive, Leeds (2007) 10. Kubovy, M.: The perceptual organization of dot lattices. Psychon. Bull. Rev. 1, 182–190 (1994) 11. Kappraff, J.: Connections: The Geometric Bridge Between Art and Science. McGraw-Hill, New York (1990) 12. Davis, P.J., Hersh, R.: Descartes’ Dream. Harvester Press, Brighton (1986) 13. Hobsbawn, E.J.: The Age of Revolution, 1789–1898. Vintage Books Editions, New York (1996)

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14. Posselt, E.A.: The Jaquarde Machine. Posselt, Philadelphia (1893) 15. Essinger, J.: How a Hand-Loom Led to the Birth of the Information Age. Oxford University Press, Oxford (2004) 16. Watson, W.: Textile Design and Colour, Elementary Weaves and Figured Fabrics, 2nd edn. Longmans, Green and Co., London (1921) 17. Schneider, B.: Programmed images: systems of notation in seventeenth- and eighteenthcentury weaving. In: Bredekamp, H., Dünkel, V., Schneider, B. (eds.) The Technical Image: A History of Styles in Scientific Imagery, pp. 142–151. University of Chicago Press, Chicago (2021) 18. De Paolis, R.: Il disegno della superficie: dal tessuto d’arredo al rivestimento di interni. In Rossi, M. (a cura di). Il Disegno come Ricerca. Strumenti grafici e modelli rappresentativi per il progetto, pp. 237–242. Maggioli, Santarcangelo di Romagna (2012) 19. Chen, X. (ed.): Advances in 3D Textiles. Woodhead Publishing Limited, Cambridge (2015) 20. Thompson, D.W.: On Growth and Form. Cambridge University Press, Cambridge (1945)

Engineering Computer Graphics

Soccer Player Pose Recognition in Games Rodrigo G. Reis1 , Diego P. Trachtinguerts1 , Andr´e K. Sato1 , abio S. G. Tsuzuki2 , Rog´erio Y. Takimoto1 , F´ and Marcos de Sales Guerra Tsuzuki1(B) 1

Department of Mechatronics and Mechanical Systems Engineering, Computational Geometry Laboratory, Escola Polit´ecnica da Universidade de S˜ ao Paulo, S˜ ao Paulo, Brazil [email protected] 2 Media Portal, S˜ ao Paulo, Brazil

Abstract. Computational models and statistics are becoming more important in professional sports, particularly in soccer. With the advance in the Computer Vision area, information such as pose recognition and player orientation are useful for several applications: team statistics, player tracking, etc. Analyzing the data achieved can improve team performance. This article presents a technique to determine the pose of every player on a soccer ﬁeld throughout the match. Speciﬁc hand-held devices for soccer players can determine and track the position of the player only for their own team. However, it fails in determining the posture of the player. Conventional pose determination of the player does not work well when there is a large number of people in the image. This work overcomes such gaps by combining the person detector YOLO and the pose recognizer OpenPose. The proposed solution utilizes regular transmission recordings of previous matches. The developed approach can successfully determine the player’s posture through these videos. However, this is only the beginning of Computer Vision applied to sports. For future work, player orientation and automatic event detection and classiﬁcation are expected to be developed.

Keywords: Soccer

1

· Pose recognition · Image processing

Introduction

Having the correct pose and orientation of the body is essential in all team sports. Proper positioning, relative to both the ball and the other players, increases the chances of a successful play. However, pure analytics is no longer enough to assess a team or speciﬁc players. Instead, computational tools to automatically evaluate metrics and performance have gained increasing importance. The current stateof-the-art in sports analytics includes predicting a player’s next move; estimating the probability of success for a certain play, and tracking movements through player-held devices [1,12,13,19,22]. Determine the movement of the next player requires a reconstruction of the soccer ﬁeld based on the video broadcast. This c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 565–574, 2023. https://doi.org/10.1007/978-3-031-13588-0_49

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can be done by determining the mapped points and calculating a homography matrix [7–9,18,23–26]. Even though the use of tracking devices is helpful in understanding a player’s movement on the ﬁeld, it does not give any information about their pose and orientation or about the opposing team. The main goal of this article is to introduce a better alternative to pose estimation and recognition in soccer games, by using video data from the original live transmissions. It is important to note that this method can be easily applied to other sports. Initially, the real-time object detection system YOLO [21] was used to determine a boundary box that surrounds each player on the ﬁeld. Subsequently, the boxes were submitted to a reﬁnement stage, and, ﬁnally, the OpenPose pose recognition tool [5] was used to estimate the pose of the players inside each bounding box. Since there is no data from the tracking devices, the results were visually validated and showed great promise. However, as previously discussed, pose recognition is only part of the problem, and a complete solution also needs to tackle the player’s orientation. Both problems are intrinsically connected, so a natural extension of this paper is future research on estimating the orientation of the ﬁeld. This paper is structured as follows. Section 2 has a brief bibliographic review of pose estimation. Section 3 has two basic concepts related to pose estimation and convolutional neural networks. Section 4 describes the methodology: the YOLO network and the OpenPose network. Then, Sect. 5 explains how YOLO and OpenPose are combined. Section 6 shows the results obtained, showing that posture is estimated for all players present in the image. Finally, Sect. 7 presents the conclusions and future work.

2

Bibliographic Review

Applications of deep learning to pose recognition started with DeepPose [27]. It takes the full image as input, and as output it produces the pose without any speciﬁc part detectors. In general, current methods are divided into two categories: top-down [11,16,17,28] and bottom-up [3–5,10,15] approaches. Initially, the ﬁrst ones try to ﬁnd people, then recognize their pose; while the latter ones detect all joints to associate them in a later stage. For either approach, several methods can be applied to diﬀerent network architectures to generate heat maps for speciﬁc features. The sequential architecture, for example, uses a sequence of basic layers (convolution, pooling, fully connected) to detect joints in the input image. Fan et al. [10] proposed a dual source network to improve precision compared to previous models. However, with every diﬀerent method, new problems arise. To solve some of them, such as kinematically impossible predictions, spatial dependencies, and context information were added to the architecture in [19,20,29]. With increasing contributions to the topic, the CMU Perpetual Computing Lab developed OpenPose, which is now one of the fastest and most accurate pose estimators available [5]. The solution presented in this paper is closely related to the work developed by Arbu´es-Sanguesa et al. [3] with the Bar¸ca Innovation Hub. In the work

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proposed by them, a super-resolution network is used to preprocess the image, allowing the OpenPose to better detect the body parts of the players. The proposed work uses a real-time object detection network to determine a bounding box that surrounds every player on the ﬁeld and that will be used to detect the pose of the player.

3

Basic Concepts

This section presents the two basic building blocks of the approach discussed here. The ﬁrst is the Pose Estimation itself, which can be determined in two major ways. Then, the concept of Convolutional Neural Networks is introduced, an important algorithm in image processing with Deep Learning. 3.1

Pose Estimation

Pose estimation is a common problem in Machine Learning, due to its versatility and usefulness in various applications. In summary, a person’s pose is estimated from an image or a video by identifying and classifying the determined joints, called keypoints, from the human body, such as the neck, shoulders, elbows, and ankles. The neighboring key points are then connected to a line, forming a pair. If applied correctly, the result should be a skeleton-like image (see Fig. 1).

Fig. 1. Pose estimation result: a skeleton-like drawing.

For the estimation of single persons poses, there are various methods with diﬀerent architectures that have been shown to give great results [2,4,6,10,14, 15]. When multi-person pose estimation is considered, however, new diﬃculties need to be solved. These challenges include the lack of information regarding the exact number of people, and possible contact and overlap with diﬀerent people. To address these problems, the most intuitive and common approach is to detect each person and then use the single-person pose estimation method separately. The main issue with this top-down approach is that execution time is proportional to the number of people, so it can be very ineﬃcient. Because of that, other teams, such as CMU with OpenPose, came up with a bottom-up approach, which tries to process the entire image at once. Clearly,

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there are still some problems that include a long processing time per image and the inability to retain gains in eﬃciency. 3.2

Convolutional Neural Networks

A convolutional neural network (CNN) is an algorithm that takes an image as input and is capable of diﬀerentiating several aspects of it. It has three types of layers: convolutional layer, pooling layer, and fully connected layer. The third one is always the last, but there can be several convolutional and pooling layers. The previous layers identiﬁed simple attributes (low-level features), such as colors, edges, and gradient orientation. As more layers are used, the network starts to understand high-level features until it can recognize full-sized objects. The convolutional layer is where feature recognition happens. By applying a ﬁlter (a 2D array of weights, also known as feature detector or kernel) to a process called convolution, in which the ﬁlter will move across the pixels of the input image to check for the intended feature. The pooling layer performs a downsampling on the input, reducing the number of parameters. Similarly to the feature detector, this layer has a ﬁlter that moves across the entire input, but without weights. Instead, it either selects the maximum value from the input array or outputs the average value from the input. This process is used to reduce the complexity and improve the overall eﬃciency of the neural network. Finally, the fully connected layer does the feature classiﬁcation itself, based on extraction from the previous layers. It usually uses a softmax function to output a number between 0 and 1.

4

Methodology

The proposed method utilizes two diﬀerent well-known algorithms: YOLO, a people detector; and OpenPose, a pose estimator. Here is a brief explanation of both. 4.1

YOLO

YOLO, You Only Look Once is a one-stage real-time object detection algorithm. This means that it uses a convolutional neural network to make predictions of bounding boxes and class probabilities all at once. For it to work, the input must be resized to a square image. In this paper, the size used was 416 × 416. Given the image, the algorithm divides it into a grid N × N . In each cell, the CNN predicts boxes and classes for each target object. After that, a Nonmaximum Suppression (NMS) ﬁlter is applied, and an Intersection over Union (IOU) metric is used to properly adjust the bounding box to the object. Finally, the output is given by a list of attributes from each ﬁnal box: height, width, center coordinates, and detected class. Regarding the convolutional network architecture, the YOLO version applied in this work, YOLOv3, uses a DarkNet-53 as the model backbone, a 53-layer CNN, with consecutive 3 × 3 and 1 × 1 layers.

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Fig. 2. YOLO algorithm applied to a soccer image.

4.2

OpenPose

OpenPose is one of the ﬁrst to use a nonparametric representation (known as Part Aﬃnity Fields, or PAFs) to associate body parts with individuals. As already commented, this approach uses a bottom-up method, which means that it ﬁrst ﬁnds all possible joints in the image and then connects them according to the highest probability. It is important to note that it does not depend on a person detector to work. There are three main attributes that make this algorithm work: a convolutional neural network, the Part Conﬁdence Map, and the Part Aﬃnity Fields. Part Conﬁdence Maps. A conﬁdence map is a 2D representation of the machine’s belief that a certain body part is located in a speciﬁc pixel. Each part is represented on a diﬀerent map. Therefore, the number of maps is necessarily the same as the total number of body parts. Two of them are shown in Fig. 3. Part Aﬃnity Fields. PAFs are one of the building blocks of this bottom-up approach. They are a set of 2D vectors that represent the degree of association (aﬃnity) between parts. As a consequence, this information coincides with the location and orientation of the main limbs throughout the image (see Fig. 4). Convolutional Neural Network. The architecture used in OpenPose is a multi-stage CNN. The ﬁrst stage is based on the VGG-19 CNN. After that, the image features are fed into two new network branches simultaneously: one of them predicts the conﬁdence maps, and the other the PAFs.

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(a) Right wrist

(b) Right elbow

Fig. 3. Two diﬀerent conﬁdence maps

Fig. 4. Part aﬃnity ﬁeld between right elbow and right wrist.

5

Proposed Method

The method presented in this paper combines the two algorithms discussed in the Methodology section. However, it is applicable not only to single images but also to videos as well. For the image part, the frame is used as an input to the YOLO detector. It is important to note that the network was calibrated to detect only people, instead of all possible objects it can identify. This increased precision and eliminated any unnecessary information from the output. After the detection of the respective bounding boxes over each player, a dominant color identiﬁcation algorithm is run inside each box to classify the people into three types: one of two teams or a referee. This is important for future work to understand the player orientation in the ﬁeld. The boxes were then subjected to a treatment phase to eliminate superposition. If two were found to be in this condition, they would be merged into a larger box to help processing afterward. The next step is to cut each box out of the original image and run OpenPose on each one. At ﬁrst, this seems to defeat OpenPose’s bottom-up approach, since it ﬁrst needs a person detector. However, this is not the case. Because of the image’s resolution (up to 1080p only), individual players are very small and are often not detected when OpenPose is used on its own. However, when a smaller

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image is given as an input, the success rates are higher. YOLO is used to know where to cut the original frame to achieve the best results. After running OpenPose and drawing the recognized skeleton, all cuts are then re-selected in the input image, showing all players and their poses at once (see Fig. 5). Finally, to apply this technique to videos, a regular top-down approach is used: The image processing is applied to every frame sequentially, which is then saved in a new video.

Fig. 5. The steps taken are summarized here: after extracting each player with YOLO, OpenPose is run, and then the original image is overwritten.

6

Results

Because of the lack of position data from handheld devices, the results were visually validated. Therefore, so far, for both image and video, most of the players’ poses were recognized to a satisfying degree. Examples of the applied method are shown in Fig. 6. They were each taken from diﬀerent matches, all in 1080p 30FPS. The ﬁrst is between Manchester United and Chelsea in the Emirates FA Cup Semi-Final 2019–2020. In the second round, Manchester United played with West Ham United in the Emirates FA Cup Fifth Round 2020–21. Finally, the third was between Leicester City and Brighton in the Emirates FA Cup SemiFinal 2019–2020. For all three, 70 s of video (2100 frames) were processed, with an average of 88.25% of those wielding similar results to the presented. More detailed statistics are shown in Table 1. In this analysis, most of the frames contained between 10–18 players, and the success was arbitrarily considered the correct detection of at least 12 of these individuals. In unsuccessful situations, there are two possible causes: YOLO could not detect the player; or the bounding boxes were created, but the OpenPose network failed. These two scenarios were not taken into account individually.

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Table 1. Results achieved for the processing of 2,100 frames from three diﬀerent matches. TF = Total Number of Frames. SF = Number of Successful Frames. UF = Number of Unsuccessful Frames TF

SF

UF

Success %

Match 1

2,100

1,876

224 89.33

Match 2

2,100

1,691

409 80.52

Match 3

2,100

1,993

107 94.91

Fig. 6. Results given by the proposed algorithm on the three matches.

7

Conclusions and Future Works

This paper has reviewed important concepts and algorithms used in the world of Person Detection and Pose Estimation. After a discussion about the use of pose information in soccer, and in sports in general, a new method to extract such data from images and videos was presented. Using well-known tools, such as YOLO and OpenPose, this optimized algorithm combines both top-down and bottom-up approaches for maximum accuracy. As discussed previously, pose recognition alone is not enough information to analyze a soccer match. Therefore, it should be used as a building block for the next step: player orientation with respect to the ﬁeld and the ball. To achieve this, future work may start with classifying the players into two teams, then studying their position. Finally, after this development, a full 3D reconstruction of the game can be created.

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Acknowledgement. R. G. Reis and D. P. Trachtinguerts are supported by CNPq (procs. #138988/2021-8 and #143606/2021-2). Marcos S. G. Tsuzuki is partially supported by CNPq (proc. #311.195/2019-9). Andr´e K. Sato is supported by Shell/FUSP.

References 1. Fifa. https://www.ﬁfa.com/technical/football-technology/standards/epts 2. Andriluka, M., Pishchulin, L., Gehler, P., Schiele, B.: 2D human pose estimation: new benchmark and state of the art analysis. In: 2014 IEEE Conference on Computer Vision and Pattern Recognition, pp. 3686–3693 (2014) 3. Arbu´es-Sang¨ uesa, A., Haro, G., Ballester, C., Mart´ın, A.: Head, shoulders, hip and ball... hip and ball! using pose data to leverage football player orientation. In: 2nd Bar¸ca Sports Analytics Summit (2019) 4. Belagiannis, V., Zisserman, A.: Recurrent human pose estimation. In: 12th IEEE International Conference on Automatic Face Gesture Recognition, pp. 468–475 (2017) 5. Cao, Z., Hidalgo, G., Simon, T., Wei, S.E., Sheikh, Y.: Openpose: realtime multiperson 2D pose estimation using part aﬃnity ﬁelds. IEEE Trans. Pattern Anal. Mach. Intell. 43, 172–186 (2021) 6. Chen, Y., Shen, C., Wei, X.S., Liu, L., Yang, J.: Adversarial PoseNet: a structureaware convolutional network for human pose estimation. In: IEEE ICCV, pp. 1221– 1230 (2017) 7. Dang, B., Tran, A., Dinh, T., Dinh, T.: A real time player tracking system for broadcast tennis video. Lecture Notes in Computer Science 5991 LNAI(PART 2), 105–113 (2010) 8. Doria, F.F., et al.: Soccer ﬁeld lines determination and 3D reconstruction. In: Cheng, L.Y. (ed.) ICGG 2020 - Proceedings of the 19th International Conference Geometry and Graphics, pp. 568–579 (2021) 9. Doria, F.F., et al.: Determination of camera position in soccer games. In: 14th IEEE International Conference on Industry Applications, pp. 167–171 (2021) 10. Fan, X., Zheng, K., Lin, Y., Wang, S.: Combining local appearance and holistic view: dual-source deep neural networks for human pose estimation. In: IEEE CVPR, pp. 1347–1355 (2015) 11. Fang, H.S., Xie, S., Tai, Y.W., Lu, C.: RMPE: regional multi-person pose estimation. In: IEEE CVPR, pp. 2353–2362 (2017) 12. Garc´ıa-Aliaga, A., Marquina, M., Coter´ on, J., Rodr´ıguez-Gonz´ alez, A., LuengoS´ anchez, S.: In-game behaviour analysis of football players using machine learning techniques based on player statistics. Int. J. Sports Sci. Coach. 16, 148–157 (2020) 13. Goes, F.R., et al.: Unlocking the potential of big data to support tactical performance analysis in professional soccer: a systematic review. Eur. J. Sport Sci. 21, 481–496 (2021) 14. Gong, X., et al.: AutoPose: searching multi-scale branch aggregation for pose estimation (2020). https://arxiv.org/abs/2008.07018v1 15. Jain, A., Tompson, J., Andriluka, M., Taylor, G.W., Bregler, C.: Learning human pose estimation features with convolutional networks. In: 2nd International Conference on Learning Representations (2013) 16. Jin, S., et al.: Towards multi-person pose tracking: Bottom-up and top-down methods. In: IEEE ICCV (2017)

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17. Li, J., Wang, C., Zhu, H., Mao, Y., Fang, H.S., Lu, C.: CrowdPose: eﬃcient crowded scenes pose estimation and a new benchmark. In: IEEE CVPR, pp. 10855–10864 (2019) 18. Lima, F.B.C.L., et al.: Artiﬁcial neural network applied to soccer ﬁeld reconstruction. In: 2021 14th IEEE International Conference on Industry Applications, pp. 667–671 (2021) 19. Maksai, A., Wang, X., Fua, P.: What players do with the ball: a physically constrained interaction modeling. In: 2016 IEEE CVPR, pp. 972–981 (2016) 20. Pﬁster, T., Charles, J., Zisserman, A.: Flowing ConvNets for human pose estimation in videos. In: IEEE ICCV, pp. 1913–1921 (2015) 21. Redmon, J., Farhadi, A.: YOLOv3: an incremental improvement. https://pjreddie. com/yolo/ 22. Santos, A.B., Theron, R., Losada, A., Sampaio, J.E., Lago-Pe˜ nas, C.: Data-driven visual performance analysis in soccer: an exploratory prototype. Front. Psychol. 9, 2416 (2018) 23. Takimoto, R.Y., Martins, T.C., Takase, F.K., Tsuzuki, M.S.G.: Epipolar geometry estimation, metric reconstruction and error analysis from two images. In: IFAC Proceedings Volumes, vol. 45(6), pp. 1739–1744 (2012). 14th IFAC INCOM 24. Takimoto, R.Y., Vogelaar, R., Ueda, E.K., Tsuzuki, M.S.G., Gotoh, T., Kagei, S.: 3D reconstruction of large point clouds with a new point correspondence algorithm. In: 16th IASTED International Conference on Software Engineering and Applications, pp. 247–254, Las Vegas, USA (2012) 25. Takimoto, R.Y., Neves, A.C., Martins, T.C., Takase, F.K., Tsuzuki, M.S.G.: Automatic epipolar geometry recovery using two images. In: IFAC Proceedings Volumes, vol. 44(1), pp. 3980–3985 (2011) 26. Takimoto, R.Y., et al.: 3D reconstruction using low precision scanner. IFAC Proceedings Volumes, vol. 46(7), pp. 239–244 (2013) 27. Toshev, A., Szegedy, C.: DeepPose: human pose estimation via deep neural networks. In: IEEE CVPR, pp. 1653–1660 (2013) 28. Xiu, Y., Li, J., Wang, H., Fang, Y., Lu, C.: Pose Flow: eﬃcient online pose tracking. In: 29th BMVC (2018) 29. Zhang, Het al.: Human pose estimation with spatial contextual information (2019). https://arxiv.org/abs/1901.01760v1

Parametric Design and Development of Wood Roof Based on Revit Ziru Wang(B)

and Chao Yu

Dalian University of Technology, No. 2, Linggong Road, Ganjingzi District, Dalian, China [email protected]

Abstract. In BIM technology design, compared with other kinds of buildings, wood structure buildings possess a large number of holes, slots, embedded connections and other structures. In order to realize the parametric modeling of wood structure architectural design and solve a series of problems such as a large number of hole and groove modeling, material information setting and complex operation process, this paper interacts with external C# programming technology based on the API program commands provided by Revit, and designs the program code to receive or use data. It avoids inefficient methods such as manual drawing model and manual drawing data that rely on third-party plug-ins for model processing, provides automatic modeling of complex structures such as grooves and holes, reception and application of parameter variables, and adopts automation ideas such as automatic material matching, distributed modeling and visual process simulation for wood structure design, Gradually realizes the parametric modeling of wood structure roof structure. At the same time, this paper designs the corresponding algorithm according to the specification, designs the values of some program parameters, and further provides an example to prove the executable ability of the program. Keywords: Revit · API · Parametric design · Automatic modeling

1 Introduction With the development of economy, the construction engineering industry has also begun to transit from the traditional construction era to the era of assembly, industrialization and automation. The construction industry will continue to optimize the whole project life cycle and save equipment and labor costs with the acceleration of social festivals. Improving work efficiency has become the direction of industry personnel [1]. At the same the time, a concept of combining parameter information with architecture was born: Building Information Modeling (BIM). BIM is characterized by the using of digital technology to integrate the information of design, construction, operation, later maintenance and demolition of construction projects throughout the project and is widely recognized in the industry [2–5]. Based on Revit software, the researchers carried out parametric research in different ways for structural types such as steel structure or reinforced concrete building, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 575–586, 2023. https://doi.org/10.1007/978-3-031-13588-0_50

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bridge body and polyhedron. Among them, reference [5] initially provides a parametric modeling route of point line surface; Literature [6] optimizes the data structure of the expression model and designs an algorithm for checking duplicate elements, which reduces the number of transaction changes and improves the efficiency; Literature [7] provides a new idea of creating family library. As a traditional architectural structure type in China, wood structure occupies a leading position in structural types in scenic resort villages and national key scenic spots and historic sites [8]. In terms of wooden house structure, literature [9] preliminarily designed the wooden house model analysis system by using API interface, which optimizes the accuracy of relevant data of wood structure hole and groove processing from the perspective of machining, so as to facilitate users to extract wooden house model data. Literature [10, 11] provides a method to complete modeling by combining RevitAPI with dynamo plug-in, and effectively solves the problems of wall cutting. In conclusion, it can be seen that many achievements have been made in the details of wooden house surface and model data processing, but the parametric automatic modeling of wooden house structure is rarely mentioned. In the modeling process of wooden house structure, complex openings, groove structure and how to correspond to material information are its unique characteristics, which are also the researchers’ key points. Manually create hollow families on holes and slots in Revit, and then create family instances to generate corresponding grooves, which increases the user’s manual operation time. This process does not realize automatic modeling. In this paper, RevitAPI command and C# programming technology will be used to interact [5, 6]. Taking the triangular wooden roof as an example, it is accurate to small parts such as bolts and round nails. According to the parameters required by the operator, it will complete the automatic generation of hole and groove structure and automatically match the material information of the model. By designing and writing background code to generate UI window program plug-in, the generated model is demonstrated step by step, and finally the triangular square wooden roof model is presented in detail in the software, so as to truly realize the automatic modeling function of parametric wooden roof.

2 Revit Parametric Design Development Process There are three parametric design methods for Revit: researchers can use the built-in development environment of Revit through module_ Startup, Module_ The shutdown method compiles various software modeling operations, runs macro scripts, and executes their commands. The second is that users can visually compile operation commands by linking the third-party grasshopper tool with dynamo Revit functions. The last method is based on Visual Studio platform, using programming language for more detailed parametric program development.

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2.1 Development Platform and Framework This paper takes visual studio community 2019 as the development platform, selects. Net framework 4.6 as the target framework, automatically generates binding redirection instead of assembly unification, and compiles the model parametric design program. 2.2 RevitAPI and Development Language Revit modeling software provides users with rich API commands. Developers can directly create, modify or delete model elements through the control commands integrated in the API to achieve the expected functions [12]. This paper is developed based on revitapi2018 version and C# language. The process is mainly carried out in units of elements. Figure 1 lists the basic inheritance relationships between families, family categories and family instances in RevitAPI.

Fig. 1. Basic inheritance relationship of element type in RevitAPI

3 Revit Parametric Modeling of Wooden Roof The wooden roof is composed of roof, wooden roof truss, rafters and necessary horizontal transverse supports. The structure of rafters and horizontal transverse supports is relatively simple, while the structure of wooden roof truss is complex and diverse, and there are various hole digging, anchoring, connection and reinforcement measures. The following will focus on the parametric modeling of triangular wooden roof truss, and describe in detail the process of realizing automatic modeling. 3.1 Dimensional Data Control In order to realize parametric automatic modeling of wooden roof, we first need to determine the key data. According to the requirements of wood structure design manual and wood structure design standard: gb50005-2017, this paper designs relevant program codes to effectively calculate and control the size and structure of wood structure roof, including important information such as roof length, number of roof trusses, height of roof trusses [13, 14]. In terms of macro structure size design, the span of triangular wooden roof frame shall not exceed 18m, the height span ratio shall be limited to 1 / 4, and the truss spacing

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shall not be greater than 6m. In the design of component connection parts, the minimum spacing between two bolts shall be set according to the grain direction of wood, and shall not be less than the specified amount, so as to prevent stress concentration. By designing the WinForm form application in visual studio, creating the form object in the program and rendering the form, the user can directly input these parameters in the subsequent plug-in. The initial input parameters belong to string type. Use the Convert.ToDouble method to make the program receive user input and close the window, as shown in Fig. 2.

Fig. 2. Design parameters UI window interface of wooden house

In order to make the proportion of the parameters and the size of the model possesses an integer multiple, each value is converted to be consistent in the program. Considering that the proportion after the model is generated is too small to directly observe the details of the model, this paper has been tested, Increase the overall horizontal and vertical scale of the model generating the wooden roof to 2 times. It should be noted that for the TextBox control with filling function in the form application design, its modifiers property needs to be set to public, otherwise the program will not accept the parameters entered by the user. 3.2 Basic Unit Modeling The structural structure of wooden roof is relatively complex, which is subdivided into bolts, dowel pins, round nails, ribs, attached wood, sleepers, wood splints, base plates, cushion blocks, upper chord, lower chord and roof. In order to optimize the effect presented after the model is generated, the above basic unit parts are controlled in detail. For the relatively regular objects such as upper and lower chords, attached timber and sleepers, the coordinates of each key point are established first, and then the connection between two points is carried out by using the CreateBound method in line, and the line segment is wrapped in the sequence of

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Curvearrarray, using the NewExtrusion method. The selected plane is extruded from point to line and from line to body. As shown in Fig. 3, the variable cross-section model generation process of cushion block needs to use the NewBlend method in FamilyCreate to loft and fuse it along a specific path by creating profiles at the upper and lower ends. The specific design idea of this method is shown in Fig. 4.

Fig. 3. Cushion block model generated automatically by program

Fig. 4. Program design idea of variable section component model

For the model of dowel pin and long round nail, it is impossible to stretch and fuse them during modeling due to their irregular shape. Here, the NewRevolution method is used for these components to edit the segment contour and generate them by rotating them at a certain angle with the axis of the contour rotation axis. It should be noted that the contour and extrusion fusion required for rotation must be closed, and the rotation axis and contour segment must be on the same plane.

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In order to facilitate operation, new classes of dowel pin and long round nail are created in the project file, and the method with the return value of Revolution type is written in the class. Finally, variables are declared in the roof truss class for reception, and the two-point connecting line in the contour key points is regarded as the rotation axis, so that the contour of the line segment rotates 360° around the axis to generate the model. 3.3 Automatic Treatment of Holes and Grooves Compared with the roofs of other materials, the prominent difference of wooden roof is that there are a large number of grooves and holes in order to ensure the structural stability and structural requirements. Taking the grooves at both ends of the bolt cap and web member as an example, first generate the original stretching body according to the parameters, then conduct hollow modeling at the contact positions between the bolt and the bolt cap, the web member and the upper and lower chords through calculation, fill the IsSolid attribute in the NewExtrusion method as false, indicating that the created model belongs to the cutting body, and finally create a combined element set, collect the hollow part, bolt cap and upper and lower chords into the collection to connect the entity with the hollow model, that is, generate grooves and holes. Figure 5 shows the specific program design process, and the effect diagram of the generated groove model is shown in Figs. 6, 7 and 8.

Fig. 5. Program design idea of automatic generation of groove hole

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Fig. 6. Display of groove model of automatic block

Fig. 7. Program generated bolt cap hole model

Web member

Fig. 8. The drawing of the groove model of the top chord generated by the program

3.4 Element Editing Due to the symmetry of the wooden roof and the wide variety of repeated components, many editing element commands in the ElementTransformUtils class under the Revit.DB namespace are involved. Each command needs to provide the ID of the element to achieve the expected operation. Among them, the coordinate variables entered by the move, copy and mirror commands are the coordinates relative to the original instance id, and these commands have their own corresponding batch operation methods. The element IDs to be edited are packaged into a collection, and the ICollection collector is used for unified operation. 3.5 Material Texture In the traditional manual setting of model materials in projects, each family category of components needs to be set separately, and it needs to be adjusted individually in case of special instances, which greatly increases the manual operation time. The research target materials are fixed wood materials and other metal components such as bolts, round nails and dowel pins. In order to distinguish the wood model from other components in visualization, this paper designs a series of methods to automatically

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set the model material. Figure 9 shows the design flow and idea of the implementation method process.

Fig. 9. Flow chart of automatic matching material information program design

Through the above process, the user can fill in the material type to be set after “wood material” through the WinForm pop-up window, and then the program will automatically

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search the entered material name through the material browser to complete the model material setting. 3.6 False Positive Information Processing Due to the complex detail processing and fine coordinate ratio in the process of automatically generating the model through the code, when debugging the code operation results, Revit will recognize by default that the two line segments close to each other or the edge of the model do not coincide, and falsely report the warning of “the element deviates slightly from the axis”, which seriously affects the executability of the program code and the appearance of the user interface. In order to solve this problem, this paper designs and writes the FailureSolution class that inherits the IFailuresPreprocessor preprocessor interface, traverses and receives the error information and establishes the if statement. When the GetDescriptionText description text receives the warning text of “the entity deviates slightly from the axis…”, the message box can be prohibited. At the same time, continue to write the corresponding FailedHandler method for this program. This method is realized by defining the corresponding processor and inputting the structure name parameter in the code, and finally is eliminated the false alarm pop-up window. 3.7 Making Plug-ins After editing the background API code file successfully, use the external application interface to realize the plug-in function in Revit. This paper reconstructs a project, in which the class inherits the interface of IExternalApplication under the Autodesk.Revit.UI namespace, and uses OnStartup function to generate plug-in modules. Next, need to put the edited addin file into the file directory of the corresponding version, guide Revit to identify the GUID ID through the assembly path, find the external application interface DLL project file, and then reopen the Revit application. If signature is not set for the plug-in, click “Always Load” to complete the production.

4 Parametric Design Example In this paper, a Korean pine square wood roof in a certain place is selected as the modeling example. The roof is 24 m long and eight roof trusses, each 12 m wide and 3 m high. In terms of program interface code, the left half of each roof truss is generated during modeling, and then it is mirrored through the MirrorElements instance method in the program code to generate a complete single roof truss model. It should be noted that after mirroring, it is found that the original hollow shear in the right half model no longer exists, so it is necessary to mirror and connect the original hollow model in the program to consolidate the model groove effect in the right half. Figure 10 is the rendering diagram of the single wooden roof truss model generated by the program.

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Fig. 10. Simulation rendering of wooden roof truss

Create a metric profile mullion family, load the family into a project, use the extrude roof command to initially build the roof, and change its properties to sloped glazing. Set the interior type of “network 1” and “boundary type 1” and “boundary type 2” in the sloped window properties as the newly loaded ceramic tile mullion family, and simulate the effect of wooden roof brick, as shown in Fig. 11.

Fig. 11. Effect picture of wood roof tile

In terms of Revit plug-in operation, create a new family file, open the “wooden roof” module, click the plug-in, as shown in Fig. 12, and establish a for cycle in the program according to the roof length distance and the number of pieces entered by the user, so as to realize the corresponding number of roof truss arrangement, edges and rafters. Add transverse support and roof on the corresponding number of frames, and the square wood roof modeling is completed, as shown in Fig. 13.

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Fig. 12. Parametric overlapping model of partial components of wood roof truss

Fig. 13. Generating a wooden roof model

5 Conclusion Based on the structural characteristics of wooden roof, this paper collects the design parameters input in the UI process through WinForm window, designs the program by using RevitAPI and C# programming technology, simplifies the design idea of wooden roof modeling, and realizes the parametric automatic modeling process of wooden roof. At the same time, according to the manually input parameters, the key coordinate points and line segments of the model are constructed and different element sets are formed. Through code operations such as merging, processing, setting out, connection and condition filtering, the structural structure of grooves and holes in the wooden roof, accurate model coordinate position, simulation of step-by-step modeling visualization process, automatic matching of model materials and so on are realized. This helps designers get rid of a lot of repetitive work of manual drawing, and improves the work efficiency of wood structure house design.

References 1. Deng, L., Zhou, Z., Ye, X., et al.: Detailed design process of fabricated components based on building information model [J/OL]. J. Guilin Univ. Technol., 1–9, 16 Dec 2020. http://kns. cnki.net/kcms/detail/45.1375. N.20201029.1103.002.html

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2. Li, H., Guo, H., Huang, T., et al.: Research on the application mode of Bim in construction projects. J. Eng. Manage. 24(5), 525–529 (2010) 3. Chen, X.: Research on the idea and application of construction project life cycle information management (BLM). Tongji University, Shanghai (2006) 4. Naamane, A., Boukara, A.: A Brief Introduction to Building Information Modeling (BIM) and its interoperability with TRNSYS. Renewable Energy Sustainable Dev. 1(1), 126–130 (2015) 5. Ma, B.: Research on parametric design and construction process simulation of cable-stayed bridge based on Revit. Dalian University of technology, Dalian (2017) 6. Li, C., Prince, R.: Polyhedron parametric modeling based on Revit secondary development [J]. Civil Architectural Eng. Inf. Technol. 12(01), 110–116 (2020) 7. Hu, X.: Design of Single Storey Industrial Plant Based on BIM. Dalian University of Technology, Dalian (2018) 8. Hu, J., Zhu, J.: Analysis of characteristics of wood structure healthy buildings. Constr. Technol. 21(418), 130–133 (2020) 9. Xue, L., Yang, L., Zhang, L.: Design and development of CAD/CAM system for wooden house model based on Revit. Mech. Eng. Automation 6(6), 91–93 (2019) 10. Guo, Y., Zhang, Y., Zhu, Y.: Rapid modeling of light wood structure frame system based on Revit software and dynamo visual programming. Forestry Machinery Woodworking Equipment 46(08), 18–24 (2018) 11. Qiu, Y.: Parametric Modeling and Data Extraction of Wooden House Structure Based on Revit. Shenyang University of Technology, Shenyang (2020) 12. Autodesk Asia Pte Ltd. Autodesk ® Revit ® Basic course of secondary development. Tongji University Press, Shanghai (2015) 13. Editorial Committee of wood structure design manual. Wood structure design manual (Third Edition). Beijing: China Construction Industry Press (2005) 14. Design standard for timber structures: gb50005-2017. Beijing: China Construction Industry Press (2018)

Computation and Amendment Method of Surface Deformation Based on Welding Theory Pengfei Zheng1,2,3(B) , Jingjing Lou1 , Yunhan Li1 , Xiyuan Wan1 Qingdong Luo1 , and Linsheng Xie2

,

1 Yiwu Industrial and Commercial College, Yiwu 322000, China

[email protected]

2 East China University of Science and Technology, Shanghai 200237, China 3 HC Semitek Corporation, Yiwu 322000, China

Abstract. There are numerous forming processes for complex surfaces, and different surface forming processes all may cause material changes, such as stretching or shrinking, etc., whether they are additive or subtractive manufacturing techniques. If the material wear caused by the forming process is not considered in the sheet metal stamping process, it will certainly produce a reduction in the forming size of the surface or an increase in the assembly error, and even lead to forming failure. This paper takes welding forming process as an example, based on welding deformation theory to analyze the influencing factors that cause deformation, build a welding deformation calculation model for linear and curved welds to propose the corresponding deformation correction calculation method to compensate for the amount of surface deformation caused by welding. Finally, on the basis of theoretical analysis, a method of calculating and correcting surface deformation based on welding theory is proposed, and the proposed method is verified by experiments. Through experimental data analysis, the surface deformation calculated by the method proposed in this paper is close to the experimentally measured deformation, and the error between the two is about 8%. As a result, the proposed method has good engineering application value. Keywords: Welding theory · Surface deformation · Computational model · Amendment · Surface flattening

1 Introduction There are many surface forming technologies, each with its owncharacteristics and different applications, for which there are also many surface flattening methods. Most of the surface flattening methods are based on theoretical curved surfaces, and less consideration is given to the influence of the surface forming process. However, some scholars have already began to research the surface processing technology. Ren et al. [1] developed a generic method, suitable for arbitrary part geometries and various incremental © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 587–597, 2023. https://doi.org/10.1007/978-3-031-13588-0_51

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sheet forming (ISF) processes, for addressing one of the main causes of geometric inaccuracy. Peter et al. [2] proposed a new testing method which overcome these deficits: neglecting forming-specific tribological boundary conditions such as continuous infeed of new sheet metal material and lubricant as well as high local contact stresses, and allowed for a high resolution of wear formation in time. Wang et al. [3] adopted an isogeometric membrane element to predict the flattened contour of the initial blank from the energy-based initial solution estimation approach, and presented the development of one-step inverse isogeometric analysis (IGA) based on the total deformation theory of plasticity. A novel analytical model in predicting the twisting angle was developed based on membrane analysis by Chang et al. [4]. Similarly, researchers have extensively studied sheet metal surface forming techniques and proposed corresponding optimization methods [5–11]. Welding is an indispensable surface forming technology in modern industrial high quality and high efficiency manufacturing technology, which is widely used due to its advantages of easy forming, low production cost and material saving. Especially in auto body processing, automated welding of bodies using industrial robots is the most typical application. However, the welding process will cause bending deformation of the material, and how to calculate the amount of welding deformation is a problem worth studying. For this reason many scholars have carried out related research work. Chen et al. [12] investigated the nonlinear deformation behavior of welded stainless steel I-section flexural members. Heidari et al. [13] explored pore pressure, stresses, and deformation near a vertical weld by a forward finite-element model (2D, forward, hydro-mechanical model). Meng et al. [14] and Wang et al. [15] observed and measured welding deformation using different methods. In addition, some researchers analyzed the welding deformation mechanism, and proposed the corresponding welding deformation computation methods for related fatigue crack formation and extended life prediction in welded structures [16–20]. Consequently, in order to make the surface flattening method more adaptable to actual production and fully consider the impact of surface deformation caused by welding processing, this paper proposes a surface deformation computation method based on welding theory, according to the mechanism of welding deformation, build a deformation computation model, and according to this in the surface flattening computation for timely amendment to improve the accuracy of surface flattening.

2 Welding Process and Deformation The welding production process refers to the process of welding the finished welded structure from metal materials (including plates, profiles and other parts, etc.) through a series of processing procedures and assembly. The plate before and after welding will produce different forms and degrees of deformation phenomenon, as shown in Fig. 1 [21]. Accordingly, from the factors affecting the welding deformation, we analyze the sources of various factors and study the laws and characteristics of welding deformation, thus concluding that: the amount of welding deformation is impacted by the arc voltage, welding current, welding speed, groove form, plate thickness and many other factors [21]. According to the Chinese national standard GB/T986–1988, the groove forms are

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generally I-shaped groove, V-shaped groove, double V-shaped groove, X-shaped groove, U-shaped groove, J-shaped groove and other forms, and some of the groove forms and sizes are shown in Table 1.

(a)

(b)

(c)

(d)

Fig. 1. Common forms of welding deformation, (a) vertical and transverse shrinkage, (b) angular deformation, (c) bend deformation, and (d) distortion deformation.

Table 1. Common groove types

No.

Thickness T/mm

Groove dimension Groove

Types

Interval

Blunt

Angle

c/mm

edge

α(β)/(o)

p/mm 1-3 1

0-1.5 I-shaped

3-6

0-2.5

3-9 2

0-2

0-2

65-75

0-3

0-3

55-65

0-3

0-3

55-65

0-3

1-3

65-75

V-shaped 9-26

3

12-60

X-shaped

4

20-60

5

30-60

U-shaped

2-3

1-4

4-10

6

16-60

J-shaped

2-3

2-4

4-10

Bouble V-shaped

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In summary, the amount of deformation is subject to a number of objective and random factors of the comprehensive constraints. As a result, the amount of welding deformation can be computed by approximation, and the surface flattening process to compensate for the plate cutting correction to improve assembly accuracy.

3 Surface Deformation Computation Method Based on The Welding Theory 3.1 Construction of Welding Deformation Computation Model The idea of finite element analysis is used to analyze the stresses and strains that occur during the welding process due to changes in the temperature field as an example of a welding unit in a single-layer weld. As shown in Fig. 2, the weld seam is segmented and simplified by using the unit diagram method to take a unit length of the weld seam and analyze it for weld stress and deformation. As shown in Fig. 3, according to the theory of welding deformation, the amount of plastic deformation of welding can be computed by formula as: εp = fp (x) ≈ −εp = −(|εe − α · f (x)|) − εs

(1)

where, εp is the amount of depression deformation, εe is the deformation rate, α is the coefficient of thermal expansion of the metal, f (x) is the temperature field function, εs is the strain limit. The corresponding residual stresses are computed by formula as: σ = E(εe − εp ) = E(εe − fp (x))

(2)

Fig. 2. Analysis model of welding element

where, σ is the residual stress, E is the modulus of elasticity, εe is the amount of residual appearance deformation, εp is the central depression deformation. Welding stress and deformation are a pair of mechanical concepts closely related to each other. Uneven heating of the welded parts leads to uneven thermal deformation, and the welding process is characterized by high temperature, large variation, nonstationarity, non-linearity and variable physical properties, which can only be simplified and combined with similar principles for approximate calculation. As consequence, the

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(a)

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(b)

Fig. 3. Plastic deformation analysis of welding element, (a) strain analysis without plastic deformation (εmax < εs ), (b) strain analysis with plastic deformation (εmax > εs ).

analysis of welding residual stresses will be further simplified and the approximate calculation of welding deformation is derived as follows. w w 2 2 F = σ · δdx = E · δ (εe − εp )dx − w2

Since, = E · δ

− 2c

− w2

εe dx + E · δ

− w2

c 2

− 2c

= E · δ · εe (w − c) + E · δ

(εe − fp (x))dx + E · δ

c 2

− 2c

− 2c

=

c 2

− 2c

w 2 c 2

εe dx

(εe − fp (x))dx = 0

where, F is the total load, σ is the residual stress. c 2 Then, εe (w − c) = − (εe − fp (x))dx =

c 2

− 2c

fp (x)dx −

c 2

− 2c

εe dx

fp (x)dx − c · εe

(3)

Thus, the residual appearance deformation expression can be obtained as follows. c c c 2 2 2 εe (w − c) = − (εe − fp (x))dx = fp (x)dx − εe dx = εe =

as:

1 w

c 2

− 2c

− 2c

c 2

− 2c

− 2c

− 2c

fp (x)dx − c · εe

fp (x)dx ≈ −

1 w

c 2

− 2c

εp dx = −

1 w

c 2

− 2c

(|εe − α · f (x)|) − εs dx (4)

Since in the interval (− 2c , 2c ), α · f (x) > εe , the Eq. (4) can be further simplified εe = −

1 w

c 2

− 2c

1 (|εe − α · f (x)|) − εs dx = − w

c 2

− 2c

α · f (x) − εe − εs dx

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=

c α (εe + εs ) − w w

c 2

− 2c

f (x)dx ≈

c α·c (εe + εs ) − · T w w

(5)

Accordingly, it can be seen from Eq. (5), the welding deformation is determined by the aforementioned arc voltage (T ), welding current (T ), welding rod diameter (c), welding speed (T ), groove form (w), joint form (w), material properties (α, εe , εs ) and other factors.

Fig.4. Welding deformation calculation model.

For calculation convenience, the corresponding calculation model is constructed according to the characteristics of the weld seam, as shown in Fig. 4, which is a buttjoint single-sided welding model of two plates, the model contains linear and curved welds, which are connected by arc welding process. The relevant parameters involved in the model are: arc voltage (U ), welding current (I ), welding speed (v), thermal efficiency coefficient (η), weld material (α, C, E, σS ), analysis unit size (l, w), welding rod (c), bevel form (c1 ), and welding shrinkage direction (D). 3.2 Calculation of Deformation Amendment Combined with Eq. (5), the strain in the transverse direction of the weld can be calculated as. εe ≈

c c α·c α·c·U ·I ·η (εe + εs ) − · T = (εe + εs ) − w w w w·C ·M ·v

(6)

where, α is the thermal expansion coefficient of metal, U is the arc voltage; I is the welding current; η is the thermal efficiency factor, C is the specific heat capacity, M is quality,v is the welding speed. For open groove butt welds, the left side of the flat plate weld is blocked from expanding and is not completely free to deform, shows a deformation of exactly half of the groove gap c1 . Therefore, the same for the right-hand side of the solder plate, εe = cl1 . To simplify the calculation, the parameter c in Eq. (6) can be replaced by an approximation of the Gaussian heat source radius for the Gaussian heat source. The

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parameter wc in Eq. (6) is now defined as the heat source radiation deformation coefficient, where its value is taken as 0.85. In addition to linear welds, some curved welds are often encountered in production. Based on similar principles, it is known that the curve weld is finite-differentiated to approximate the curve segment by a short linear segment, so that the curve weld deformation calculation problem can be considered as a collection of a series of linear welds to carry out the amendment calculation, which can significantly reduce the complexity of the curve weld deformation amendment problem. For example, a certain curve weld C can be represented by the following formula. C=

n

Li,j , i = 1, · · · , n − 1

(7)

i=1,j=i+1

where, Li,j is the line between point i and point j on the weld curve C. According to Eq. (6), the transverse shrinkage deformation of the i-th segment of the linear weld on the curved weld C can be expressed as εe (Li,j ), whose direction is perpendicular to Li,j . εe (Li,j ) ≈

c c α·c α·c·U ·I ·η (εe + εs ) − · T = (εe + εs ) − w w w w·C ·M ·v

(8)

As a result, the transverse shrinkage deformation caused by curved welds can be expressed by the following equation.

L =

n i=1,j=i+1

L · εe (Li,j ) ≈

i,j

n

L·

i=1,j=i+1

i,j

c α·c·U ·I ·η (εe + εs ) − w w·C·M ·v

(9)

Due to the factor of groove gap, the dimensional deviation caused by the groove gap also needs to be considered when the transverse dimensional amendment of surface flattening, so the cumulative contraction of transverse deformation shrinkage and groove influence Lt can be expressed as. Lt = l +

c1 c1 =εe · l+ 2 2

(10)

The direction of shrinkage of a curved weld along its principal normal direction can be calculated by the spatially arbitrary curved Frenet frame calculation method. Each small section of linear weld deformation shrinkage can be computed by Eq. (9), and its overall shrinkage can be expressed as. c1 c1 Lt = l + =εe · l+ = 2 2

n i=1,j=i+1

L

i,j

c1 ·εe (Li,j )+ 2

(11)

In contrast, because the amount of longitudinal shrinkage in the welding process generally along the direction of the weld shrinkage, and only affect the weld start and end position, the overall size of the weld brings relatively small impact, so its deformation in the welding process is often ignored.

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3.3 Construction of Surface Flattening Amendment Method Based on the above theoretical analysis, it is known that the welding process generates deformation. In order to introduce this deformation factor into the calculation method of surface flattening, we construct a surface flattening amendment method that considers the welding deformation. In order to facilitate the research and simplify the calculation of welding deformation amendment, we make the following assumptions about the corresponding performance and process requirements of the welding workpiece: (1) The material of the workpiece to be assembled is uniform and of the same thickness. (2) Dimensional errors caused by the punching and shearing process are not considered. (3) The weld is uniform and the deformation of the weld is the same at all points during the welding operation. (4) The deformation of the workpiece to be welded along the weld seam in the transverse and longitudinal directions remains consistent. We use the surface flattening calculation method [22] to obtain the unfolded graph of the welded parts, according to the requirements of the use of welded parts, select the appropriate groove form, discrete segmentation of the groove curve, use the welding deformation calculation method proposed in this paper for amendment, and finally reverse compensation of the flatten graph to obtain the revised unfolded graph after considering the welding deformation, the corresponding calculation workflow is shown as Fig. 5.

Fig. 5. Amendment calculation method flowchart.

As shown in Fig. 6, the theoretical unfolded outline graph produces a shrinkage deformation l after the welding process. In addition, the design of the welding groove causes a welding gap 0.5c1 . Therefore, the combined amendment in surface flattening is Lt = l + 0.5c1 , which is the basis for dimensional compensation in sheet blanking.

4 Example and Analysis The deformation amendment method proposed in this paper was tested in a case study at a pressure vessel manufacturing company, as shown in Fig. 7, it is a ball crown head part

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Fig. 6. Schematic diagram of welding deformation flattening and amendment.

of a pressure vessel with a radius of 3000 mm and a wall thickness of 15 mm. We use Eq. (10) to find its amendment amount of 61.22 mm, and according to this compensation amount to correct the theoretical unfolded graph, to obtain the two corrected welded parts punching graphics, as shown in Fig. 8. Lt = l +

c1 c1 =εe · l+ = 2 2

n i=1,j=i+1

L ·εe (Li,j )+

i,j

c1 ≈ 61.22 mm 2

Fig. 7. The size of head, (a) is the shape of head, (b) is the drawings of head.

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Fig. 8. Theoretical flattened drawing and amendment flattened drawing of the head.

The material of head is mild steel, and CO2 shielded welding is used. The relevant welding parameters are as follows: welding current is 150 A; voltage is 22 V; welding speed is 5 mm/s; welding wire model is E501-1, and welding wire diameter is 1.2 mm. Through example experiments, it can be clarified that the surface deformation calculation and amendment method proposed in this paper can effectively correct the impact of deformation brought about by the welding process and improve the accuracy of surface forming, which has certain guiding significance for predicting the quality of welded forming and the service life of pressure vessels.

5 Conclusion (1) Through a comprehensive analysis of the impact of welding deformation factors, the main factors affecting welding deformation are summarized, and how these factors affect welding deformation are theoretically analyzed, which lays a theoretical foundation for the establishment of a calculation method for welding deformation shrinkage. (2) According to the mechanism of welding stress and welding deformation, the approximate calculation method of welding shrinkage is analyzed, and the calculation model of welding deformation is constructed based on this theory. Based on the original surface flattening algorithm, the dimensional shrinkage caused by the deformation of the welded part is compensated and amended before the punching of the sheet, and the corresponding amendment flattening calculation method is proposed. (3) The experimental verification is carried out by taking the welding of the elliptical head parts in the pressure vessel. The test results show that: the welding deformation computation method proposed in this paper is close to the experimentally measured deformation, and the error between the two is about 8%. As a result, the proposed method has good engineering application value.

Acknowledgements. This study was supported by the Domestic Visiting Engineers Project of Zhejiang Education Department in 2020, China (Grant No.FG2020196, FG2020197), the Second Batch of Teaching Reform Research Projects in the 13th Five-Year Plan of Zhejiang Higher Education, China (Grant No.jg20190878), and the Public Welfare Science and Technology Research Project of Jinhua, Zhejiang Province, China. (Grant No.2021–4-386).

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References 1. Ren, H., Xie, J., Liao, S., et al.: In-situ springback compensation in incremental sheet forming. CIRP Ann-Manuf. Tech. 68, 317–320 (2019) 2. Groche, P., Christiany, M., Yutian, W.: Load-dependent wear in sheet metal forming. Wear 422–423, 252–260 (2019) 3. Wang, C., Zhang, X., Shen, G., et al.: One-step inverse isogeometric analysis for the simulation of sheet metal forming. Comput. Methods Appl. Mech. Engrg. 349, 458–476 (2019) 4. Chang, Z., Chena, J.: Mechanism of the twisting in incremental sheet forming process. J. Mater. Proc. Tech. 276, 116396 (2020) 5. Li, J., Ning, T., Xi, P., et al.: Smoothing parametric design of addendum surfaces for sheet metal forming. Chin. J. Mech. Eng. 33, 4 (2020) 6. Pham, Q.T., Kim, J., Luyen, T.T., et al.: Application of a graphical method on estimating forming limit curve of automotive sheet metals. Int. J. Auto. Tech. 20(S), 3–8 (2019) 7. Wang, M., Guo-long, L., Cai, Z., et al.: Linear distribution principle for sheet forming using continuous roll forming process. Int. J. Prec. Eng. Manuf. 21, 557–564 (2020) 8. Guo, X., Tao, J., Wang, H.: Research progress on an advanced forming technology for aviation tube. J. Nanjing Univ. Aero. Astro. 52, 12–23 (2020) 9. Jiang, T., Gong, H., Shi, W., et al.: Process parameters optimization of u-shaped bending based on response surface methodology. J. Shanghai Univ. Eng. Sci. 33, 278–282 (2019) 10. Zhang, H., Wang, X., Chen, X., et al.: Study on process parameters in transverse bending of tailor rolled blanks. J. Northeast Univ. 40(5), 728–733 (2019) 11. Zhang, M., Tian, X.T., Li, B.: Shape control for press bend forming of integral panel. Acta Aeronaut. et Astronaut. Sinica. 41 (2020) 12. Chen, X., Yuan, H., Du, X., et al.: Nonlinear deformation behaviour and design of welded stainless steel I-section flexural members. Eng. Struct. 252, 113683 (2022) 13. Heidari, M., Nikolinakou, M., Hudec, M., et al.: Impacts of vertical salt welding on pore pressure, stresses, and deformation near the weld. Mar. Petrol. Geol. 133, 105259 (2021) 14. Meng, D., Zhang, L., Li, Y., et al.: Welding deformation of unequal thickness tailor welded blanks under multiple factors. Earth Environ. Sci. 508, 012223 (2020) 15. Wang, G., Yang, F., Zhang, L., et al.: Optical dynamic measurement of welding deformation. J. Phys. 2012, 012018 (2021) 16. Saternus, Z.: Computer methods for determination of deformations in welded closed profiles. Procedia Eng. 177, 188–195 (2017) 17. Lu, Y., Lu, C., Zhang, D., et al.: Numerical computation methods of welding deformation and their application in bogie frame for high-speed trains. J. Manuf. Process. 38, 204–213 (2019) 18. Luca, R., Paolo, F., Filippo, B.: Estimation of multi-pass welds deformations with Virtual Weld Bead method. Procedia. Struct. Integ. 25, 149–158 (2020) 19. Luca, R., Paolo, F., Filippo, B.: A novel method for welding residual deformations prediction. Procedia. Struct. Integ. 28, 171–179 (2020) 20. Yi, J., Lin, J., Chen, Z., et al.: Prediction and controlling for welding deformation of propeller base structure. J. Ocean. Eng. Sci. 6, 410–416 (2021) 21. Zhou, W., Zhang, N.: Practical Manual of Welding Process, 1st edn. Chemical Industry Press, Beijing, China (2020) 22. Zheng, P.-F., Lou, J.-J., Lin, D.-J., An, Q.: A curved surface flattening computing method combined with machining process. In: Cheng, L.-Y. (ed.) ICGG 2021. AISC, vol. 1296, pp. 186–198. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-63403-2_17

Construction of Diﬀeomorphisms with Prescribed Jacobian Determinant and Curl — That Forms a New Deﬁnition of Averaging Diﬀeomorphisms Zicong Zhou1(B) 1

and Guojun Liao2

Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China [email protected] 2 Math Department, University of Texas at Arlington, Arlington, TX 76019, USA [email protected]

Abstract. The Variational Principle (VP) is designed to generate nonfolding grids (diﬀeomorphisms) with prescribed Jacobian determinant (JD) and curl. The solution pool of the original VP is based on an additive formulation and, consequently, is not invariant in the diﬀeomorphic Lie algebra. The original VP works well when the prescribed pair of JD and curl is calculated from a diﬀeomorphism, but not necessarily when the prescribed JD and curl are unknown to come from a diﬀeomorphism. In spite of that, the original VP works eﬀectively in 2D grid generations. To resolve this issue, in this paper, we describe a new version of VP (revised VP), which is based on the composition of transformations and, therefore, is invariant in the Lie algebra. The revised VP seems to have overcome the inaccuracy of the original VP in 3D grid generations. In the following sections, the mathematical derivations are presented. It is shown that the revised VP can calculate the inverse transformation of a known diﬀeomorphism. Its inverse consistency and transitivity of transformations are also demonstrated numerically. Finally, a new deﬁnition of averaging diﬀeomorphisms based on the revised VP is proposed. Keywords: Adaptive grid generation · Computational diﬀeomorphism · Jacobian determinant · Curl

1

Introduction

Computational construction of diﬀeomorphisms is an active research ﬁeld in computational geometry. For instance, conformal diﬀerential geometry achieved remarkable success in surface diﬀerential geometry [7]. Whereas, this study is about how to characterize and control a meaningful distribution of grid points c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 598–611, 2023. https://doi.org/10.1007/978-3-031-13588-0_52

∇φ and φ Diﬀeomorphic φ Formed by det∇φ

599

over a volumetric domain, such as in [8,12]. This is the problem of adaptive generation of non-folding grids. One approach to the task is to ﬁnd a diﬀerentiable and invertible transformation T , i.e., a diﬀeomorphism, by controlling its JD which models local cell-size, such as in [6,9]. Its idea has been applied in constructing deformable image registration methods, such as in [3,10]. A grid generation method in [2], the deformation constructs T with prescribed JD, by a scalar function 0 < f ∈ C 1 , whose key component is the solution to a divergence − curl system (similar to the constraint equations in (4)). Its divergence is approximated by f − 1 and curl is assigned to 0 due to the challenge of realizing curl beforehand, where curl models the local cell-rotation. Consequently, grids generated by the deformation method are not unique when lacking of curl information, in [11]. To overcome this problem, the original VP was proposed in [4] and studied further in [14]. In 3D grid generations, the original VP only provides inaccurate approximations around the true solution. The authors attempted to analyze the uniqueness [13] of such transformations, but it turned out cannot be completed for the reason that JD of a transformation is merely an approximation to the divergence of the transformation. Fortunately, this does not undermine VP to produce well approximated grids and in turn allows mismatched values of prescribed JD and curl under certain range. A novel image atlas construction method proposed in [14] utilizes the original VP but requires (i) accurate solution in 3D grid generations and (ii) optimize JD and curl in a separable manner so one may investigate how each of them aﬀect the corresponding diﬀeomorphism. Another limitation of the original VP is its formulation of small deformations T = id + u ∈ H02 (Ω) where u is the displacement ﬁeld and id is the identity map (uniform grid), such as in [10], whose function composition “◦” is approximated by T 2 ◦ T 1 ≈ T 1 + u 2 = id + u 1 + u 2 = id + u 2 + u 1 = T 2 + u 1 ≈ T 1 ◦ T 2 .

(1)

From a computational perspective, (1) risks having function composition maps T } ⊂ H02 (Ω) that forms a diﬀeomorphism group [1]. outside of the collection {T Therefore, in this paper, to resolve (i), (ii) and to broaden the transformations that VP can characterize, we fundamentally revise VP to consider transformations taking composition as the left-translation. Surprisingly, this revision also overcome the inaccuracy of the original VP in 3D grid generation. The structure of the paper is organized as follow. In Sect. 2, we reformulated VP to cope with composition as left-translation. In Sect. 3, examples are provided to demonstrate the eﬀectiveness of the revised VP. In Sect. 4, a new deﬁnition of averaging diﬀeomorphims is proposed.

2

New Version of Variational Principle

Let a simply-connected, bounded Ω ⊂ R3 (similar in R2 ) be the domain and ω =< x, y, z >∈ Ω. Let a scalar function fo > 0 and a vector-valued function g o on Ω satisfy ω )dω ω = |Ω| and ∇ · g o = 0, respectively, fo (ω Ω

(2)

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where the ﬁrst indicates fo like JD and the second requires g o like curl. Given φo = φm (φ φo ) ∈ φ o ∈ H02 (Ω), we look for a diﬀeomorphic transformation φ = φm ◦φ u is an intermediate transformation that left-translates H02 (Ω), where φm = id +u φm = δu u, such that, the cost functional—sum of squared φ o to φ and implies δφ diﬀerences (SSD) is minimized: 1 φ) = φ − fo )2 + |∇ × φ − g o |2 ]dω ω SSD(φ [(det∇φ (3) 2 Ω ∇ · φm = f − 1 φm = ∇f − ∇ × g = F (f, g ) in Ω, (4) ⇒ Δφ subjects to ∇ × φm = g with u = 0 on ∂Ω where f , g and F are control functions. Its variational gradient with respect to the control function F can be derive as follows. Denote P = φ − fo and Q = ∇ × φ − g o , then, for all δF F vanishing on ∂Ω, det∇φ φm ◦ φ o ) δSSD(φ φ − fo )δdet∇(φ φm ◦ φo ) + (∇ × φ − g o ) · δ∇ × (φ φm ◦ φo )]dω ω = [(det∇φ Ω

⎛

⎞ φo )y − ∇φm 2 · (φ φ o )z ∇φm 3 · (φ ⎝ φm det∇φ φo + Q · δ −∇φm 3 · (φ φo )x + ∇φm 1 · (φ φo )z ⎠]dω ω = [P δdet∇φ Ω φo )x − ∇φm 1 · (φ φ o )y ∇φm 2 · (φ φo (δu1x φm 2y φm 3z + φm 1x δu2y φm 3z + φm 1x φm 2y δu3z = [P det∇φ

Ω

− δu1x φm 2z φm 3y − φm 1x δu2z φm 3y − φm 1x φm 2z δu3y − δu1y φm 3z φm 2x − φm 1y δu3z φm 2x − φm 1y φm 3z δu2x + δu1y φm 3x φm 2z + φm 1y δu3x φm 2z + φm 1y φm 3x δu2z + δu1z φm 2x φm 3y + φm 1z δu2x φm 3y + φm 1z φm 2x δu3y − δu1z φm 2y φm 3x − φm 1z δu2y φm 3x − φm 1z φm 2y δu3x ) ⎛ ⎞ δu3x φo1y + δu3y φo2y + δu3z φo3y − δu2x φo1z − δu2y φo2z − δu2z φo3z Q· ⎝ −δu3x φo1x − δu3y φo2x − δu3z φo3x + δu1x φo1z + δu1y φo2z + δu1z φo3z ⎠]dω ω +Q δu2x φo1x + δu2y φo2x + δu2z φo3x − δu1x φo1y − δu1y φo2y − δu1z φo3y ⎛ ⎞ ⎛ ⎞ φm 2y φm 3z − φm 3y φm 2z Q2 φo1z − Q3 φo1y ⎝ ⎠ ⎝ φ = [(P det∇φ o φm 3x φm 2z − φm 2x φm 3z + Q2 φo2z − Q3 φo2y ⎠) · ∇δu1 Ω φm 2x φm 3y − φm 2y φm 3x Q2 φo3z − Q3 φo3y ⎛ ⎞ ⎛ ⎞ φm 3y φm 1z − φm 1y φm 3z −Q1 φo1z + Q3 φo1x φo ⎝φm 1x φm 3z − φm 1z φm 3x ⎠ + ⎝−Q1 φo2z + Q3 φo2x ⎠) · ∇δu2 +(P det∇φ φm 3x φm 1y − φm 1x φm 3y −Q1 φo3z + Q3 φo3x ⎛ ⎞ ⎛ ⎞ φm 1y φm 2z − φm 2y φm 1z Q1 φo1y − Q2 φo1x φo ⎝φm 2x φm 1z − φm 1x φm 2z ⎠ + ⎝Q1 φo2y − Q2 φo2x ⎠) · ∇δu3 ]dω ω +(P det∇φ φm 1x φm 2y − φm 2x φm 1y Q1 φo3y − Q2 φo3x A1 · ∇δφm 1 + A 2 · ∇δφm 2 + A 3 · ∇δφm 3 ]dω ω , where δφ φm = δu u. = [A Ω

Here, the “big vector” s are now denoted as A i , where i = 1, 2, 3. By Green’s identities with ﬁxed boundary condition and for some Bi such that ΔBi =

∇φ and φ Diﬀeomorphic φ Formed by det∇φ

601

−∇ · A i and i = 1, 2, 3, then it can be carried to, ω δSSD = [−∇ · A 1 δφm 1 − ∇ · A 2 δφm 2 − ∇ · A 3 δφm 3 ]dω Ω ω = [ΔB1 δφm 1 + ΔB2 δφm 2 + ΔB3 δφm 3 ]dω Ω ω = [B B · δΔφ φm ]dω ω = [B1 δΔφm 1 + B2 δΔφm 2 + B3 δΔφm 3 ]dω Ω

Ω

B · δF F ]dω ω [B

=

⇒

Ω

∂SSD = B. F ∂F

(5)

In the case of needing to minimize SSD with respect to JD and curl in a separable manner, the followings are the derivation of partial gradients with respect to f F = δΔφ φ = ∇δf − ∇ × δgg , then, and g . For arbitrary δf and δgg , one may get δF from (5), it leads to, B · (∇δf − ∇ × δgg )]dω ω = [B B · ∇δf ]dω ω + [−B B · ∇ × δgg ]dω ω δSSD = [B Ω Ω Ω ω + [−∇ × B · δgg ]dω ω = [−∇ · B δf ]dω Ω

⇒

Ω

∂SSD = −∇ · B ∂f

and

∂SSD = −∇ × B . ∂gg

(6)

A gradient descent pseudo-code is summarized below. t is the step-size of gradient. Major computational costs occur in solving P oisson equations by a Fast Fourier Transform P oisson solver (FFT). Minor incremental computations happen in the function composition by interpolation of step 7. Deﬁne ratio = SSDf inal /SSDinitial and let F be the control function to be determined.

φ, φm ] = revisedVP(fo , g o , φ o ) Algorithm 1. [φ Ω |/ Ω fo ); • 1: initialize F = 0 , φ = φ o , t and normalize fo = fo ∗ (|Ω • 2: while stopping criteria is unmet • 3: if better • 4: solve for B from ΔBi = −∇ · A i by FFT where i = 1, 2, 3; • 5: update F new = F − t ∗ B ; u = F by FFT ; • 6: solve for u from Δu φo ) by interpolation, where φm = id + u ; • 7: update φ = φm (φ • 8: if SSD decrease, • 9: better; • 10: t = t ∗ tup (For example, set tup = 1.05 and tdown = 0.95); • 11: F = F new ; else • 12: better false; • 13: t = t ∗ tdown .

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The experiments are implemented with MatLab codes on a laptop with Intel Core i5-7300HQ Processor, 16 GB RAM and NVIDA GeForce GTX 1050Ti GPU.

3

Numerical Examples of Revised VP

Example 3.1 is an example of a 3D grid reconstruction by the revised VP which demonstrates this work can be applied in 3D scenario. For more intuitive and clear visualizations, the rest of the examples are provided in 2D. E.g. 3.2 shows the revised VP is capable of constructing inverse transformation; E.g. 3.3 and E.g. 3.4 conﬁrm that the computational solutions of the revised VP satisfy inverse consistency and transitivity, which are expected in a diﬀeomorphism group. In all ﬁgures, “a” vs “b” means that “a” in Red is superposed on “b” in Black. 3.1

Example: 3D Grid Reconstruction Comparison Between the Original and Revised VPs

This example demonstrates revised VP achieves better solutions over the original VP proposed in [4]. Given a 3D grid Φ in Black, in Fig. 1, Φ is manually built by multiple times of applying cutoﬀ rotation and displacement over Ω such that Φ > 0.1734 and Φ is non-folding grid. For clean visualizations, the 3D grids det∇Φ are only plotted on the 25-th frame on z-axis and the 37-th frame on x-axis. Φ and g o = ∇ × Φ as the prescribed JD and curl. Deﬁne fo =det∇Φ

(a) Φ

(b) (a) on xz-plain

Fig. 1. Given ground truth (GT)

Figure 2(a–d) is the solution of original VP, Φ ovp , and is superposed on the GT, Φ . It can be visually seen in Fig. 2(e) that Red grid lines of Φ ovp do not lineup well with Φ in Black grid, i.e., a more accurate solution would have covered more Black grid. Comparing to the solution of revised VP, Φ rvp , in Fig. 2(a– d), that also is superposed on Φ , in Fig. 2(e–h), there is only a small portion of the Black grid stay uncovered by Φ rvp which indicates revised VP provides

∇φ and φ Diﬀeomorphic φ Formed by det∇φ

603

much better solutions over the original VP. This observation is conﬁrmed by measurements in the following Table.1. It also recorded the revised VP had reached the ratio-tolerance with less iterations and computational time. Table 1. Performance of Fig. 2 Solution

Ω

ratio

Sec

Iteration

Max diﬀerences of Fig. 2(a,c) φ − fo | |det∇φ

||∇ × φ − g o ||2

φ − Φ ||2 ||φ

φ = Φovp

[1, 51]3

5.2 ∗ 10−5

1765.89

3016

0.7279

0.0866

1.9454

φ = Φ rvp

[1, 51]3

9.0 ∗ 10−6

788.21

576

0.0104

0.0136

0.0403

(a) Φ ovp vs Φ

(b) (a) on xz-plain

(c) Φ rvp vs Φ

(d) (c) on xz-plain

Fig. 2. Solutions by original VP Φ ovp and revised VP Φ rvp vs GT Φ

This example demonstrates a three-fold improvement of the revised VP: (i) the revised VP is eﬀective and capable in generating 3D grids with prescribed JD and curl; (ii) the revised VP ﬁnds more accurate solutions in terms of the prescription of JD and curl (it is more obvious in 3D examples) compared to the original VP; (ii) the revised VP reaches the desired tolerance faster than the original VP (in fact, the computational cost of the revised VP for an eﬀective iteration is very close to the original VP, where main extra computations occur on the interpolation of step 7 in the provided Algorithm 1).

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Example: Construction of Inverse

id = 1 and ∇×id id = 0 for the identity map id It is known that det∇id id. To construct the inverse of a given grid, in this example a tiger grid T in Fig. 3(a), 1 = fo , 0 = g o and T = φ o are fed to Algorithm 1. It is expected to ﬁnd a transformation that left-translates T and outputs id id. As Fig. 3(b) shows, D = φm is found. And, φ = D ◦ T is compared with id in Fig. 3(c), where the composition in Red grid line covers well over id in Black grid line. The tiger grid T is “inverted” by D . In Fig. 3(d), the left inverse of D is guessed by T and, as T ◦ D looks, it is very close to id as well. This indicates that D approximates quite accurately to T −1 . Table 2. Performance of Fig. 3 Orientation Ω

ratio

Second Iteration Max diﬀerences of Fig. 3(c) |JD| |curl| L2 -norm

D : T → id [1, 97]2 1.41 ∗ 10−5 858.2

(a) T

(b) D : T → id

50000

0.0072 0.0074 0.0166

(c) D ◦ T vs id

(d) T ◦ D vs id

Fig. 3. Inverse by left-translation

3.3

Example: Inverse Consistency

Given a grid, Ψ , of an ox and a grid, Φ , of a Mona Lisa portrait in Fig. 4. Feed Φ = fo , ∇ × Φ = g o and Ψ = φ o to Algorithm 1, then φ = D 1 ◦ Ψ and det∇Φ φm = D 1 are outputted. By reversing the order, D 2 and D 2 ◦ Ψ are found. It is expected that D 1 and D 2 are the inverses of each other. It is shown in Fig. 4(g,h) id. where the compositions of D 1 and D 2 are close to id

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Table 3. Performance of Fig. 4 Orientation Ω

ratio

Second Iteration Max diﬀerences of Fig. 4(g, h) |JD|

|curl|

L2 -norm

[1, 101]2 4.1916 ∗ 10−5 974.6

44302

0.0044 0.0338 0.0327

D 2 : Ψ → Φ [1, 101]2 3.0864 ∗ 10−5 874.1

41624

0.0035 0.0032 0.0115

D1 : Φ → Ψ

Fig. 4. Inverse consistency

3.4

Example: Transitivity

A problem of transitivity is presented in this example. C , P and R are grids of circle, pentagon and rectangle shapes, respectively, shown in Fig. 5(a, b, c). First, D 1 from C to P , D 2 from P to R and D 3 from C to R are found and shown in Fig. 5(d, f, j), respectively; second, D is formed in Fig. 5(h) by left-translating D 1 with D 2 , then it is compared with D 3 as shown in Fig. 5(l). They are very close to each other. Figure 5 has shown what is expected in the sense of transitivity. Table 4. Performance of Fig. 5 ratio

Orientation

Ω

Second Iteration Max diﬀerences of Fig. 5(d, f, j)

D1 : C → P

[1, 97]2 1.28 ∗ 10−3 421.5

17367

0.0332 0.0222 0.2507

D 2 : P → R [1, 97]2 1.99 ∗ 10−3 470.3

21126

0.0385 0.0326 0.3414

D 3 : C → R [1, 97]2 9.64 ∗ 10−4 462.6

20559

0.0283 0.0177 0.1444

|JD|

|curl|

L2 -norm

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Fig. 5. Transitivity

3.5

Example: The Inverse of an Image Registration Deformation

This example connects the revised VP to the proposed image registration method in [14]. It suggests that the solutions by the image registration method are potentially of the same diﬀeomorphism group that the revised VP focuses. An extensive and detailed study of this insight will be included in a diﬀerent work. Image registration is the task of aligning a moving image, Im , to a f ixed image, φ) is close to If . In general, it is done If , by a transformation, φ , so that Im (φ with a minimization problem: φ) − If ]2 dω. min [Im (φ φ

Ω

Ideally, a diﬀeomorphic (non-folding) φ is expected because the registered image φ) should not be distorted by φ . Im (φ

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Fig. 6. Inverse deformation φ −1 guessed by revised VP

In Fig. 6, Im is the moving image to be moved to the ﬁxed image If ; φ is φ) is the registered image which the diﬀeomorphic solution found by [14]; Im (φ is close to the ﬁxed image If . Next, φ −1 is the guessed inverse transformation of φ found by the revised VP in E.g. 3.2. In (f), φ is left-translated by φ −1 , in Red grid, and superposed on Black grid id but the Black grid barely shows. This id, which is also conﬁrmed by shows the composition φ −1 ◦ φ is very close to id Table 5. Therefore, φ −1 is a very good guessed inverse of φ and is of the same diﬀeomorphism group where φ belongs. An interesting question is whether φ −1 is also an valid inverse registration deformation that moves the ﬁxed image If back φ−1 ) to the moving image Im . The answer is YES, at least in this example. If (φ −1 can be treated as a valid registration is indeed close to Im . This means φ deformation from If to Im . Table 5. Performance of Fig. 6(e) Orientation

Ω

ratio

Second Iteration Max diﬀerences of Fig. 3(c) |JD| |curl| L2 -norm

φ −1 : φ → id [1, 129]2 1.33 ∗ 10−5 1680.7 80000

4

0.0583 0.0280 0.0749

Averaging Diﬀeomoephisms Based on Revised VP

The original averaging diﬀeomorphisms method [5] is based on taking arithmetic means for both JDs and curls of the given diﬀeomorphisms. It is noticed that

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the geometric mean ﬁts better to the property of JD. Here, a modiﬁed deﬁnition of averaged diﬀeomorphism is presented by using geometric mean for JDs and arithmetic mean for the curls. The eﬀectiveness of this deﬁnition of averaging diﬀeomorphisms is demonstrated in the following example. T i=1,...,n }, compute the geometric mean of JDs Given n diﬀeomorphisms {T T i=1,...,n } and the arithmetic mean for curls {∇ × T i=1,...,n }, namely, {det∇T fo = (

n

1

n

T i) n det∇T

and

go =

i=1

1 ∇ × T i. n i=1

(7)

As mentioned in Sect. 1, this work is motivated to enhance the image atlas method [14]. With the partial gradients (6) found, a faster algorithm of the revised VP, Algorithm 2, is summarized by the pseudo-code below. φ, φm ] = revisedVPfaster(fo , g o , φ o ) Algorithm 2. [φ • 1: do Alg.1 with the following changes: • 2: replace F = 0 by f = 1 and g = 0 in step 1 of Alg.1; B = −∇·B • 3: in step 4 of Alg.1, once B is found, compute partial gradients ∂SSD ∂f and ∂SSD = −∇ × B ; g ∂g • 4: replace step 5 of Alg.1 by fnew = f − t ∗ ∂SSD and g new = g − t ∗ ∂SSD g , ∂f ∂g then deﬁne F = ∇fnew − ∇ × g new ; • 5: replace step 11 of Alg.1 by f = fnew and g = g new ;

Its higher convergence rate is totally by observations on numerical tests. It is still of our curiosity why Algorithm 2 tends reduce SSD faster than Algorithm 1. However, Algorithm 1 may continuously drop SSD without easily stagnated, as the comparison shows in the next example. 4.1

Example: Averaging Two Diﬀeomoephisms

The averaged diﬀeomorphism T avg can be generated by feeding f0 and g o to VP. In Fig. 7, T is given as the ground truth, T 1 and T 2 are manually generated 1 T = (det∇T T 1 det∇T T 2 ) 2 and g o = ∇ × T = 12 (∇ × T 1 + ∇ × such that fo = det∇T T 2 ) are satisﬁed. We want to see if the averaged diﬀeomorphism T avg is close to the ground truth T . Aﬃrmatively, as it shows in Fig. 7(d,e), the averaged alg2 diﬀeomorphisms T alg1 avg and T avg , generated by Algorithms 1 and 2, respectively, in Red almost overlap the ground truth T in Black. Table 6. Performance of Fig. 7 Solution via Ω

ratio

Second Iteration Max diﬀerences of Fig. 7(d,e) |JD|

|curl|

L2 -norm

Algorithm 1

[1, 97]2 2.20 ∗ 10−4 912.7

50000

0.0053 0.0294 0.0705

Algorithm 2

[1, 97]2 2.32 ∗ 10−4 400.4

21184

0.0085 0.0297 0.0707

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609

Fig. 7. Averaging T 1 and T 2

As recorded in the Table 6, and the ratios plotted in Fig. 8 both Algorithms 1 and 2 worked in constructing the averaged diﬀoemorphism and their results are of a comparable quality. However, Algorithm 1 stopped when max iteration is reached and Algorithm 2 ended while SSD can not be reduced any more. Hence, it is up to user to determine which one ﬁts better to their needs. Furthermore, it is feasible to construct another algorithm for VP which takes advantages of both Algorithms 1 and 2 by optimizing along a combination of gradients (5) and (6).

(a) Plot of ratios

(b) Plot of log(ratio)s

Fig. 8. Performance of Algorithm 1 v.s. Algorithm 2

In Fig. 8, (a) is the original plot of ratios and (b) is plot in the natural logarithm values of ratios for a better visualization. Please note that, Fig. 8 recorded both eﬀective and ineﬀective iterations and they are plotted all the way to the 41000-th performance, which is suﬃcient to see the Algorithm 2 converges faster than Algorithm 2.

5

Discussion

The original VP is unsuccessful in characterizing of its solutions being the diﬀeomorphism group H02 (Ω) and has poor performance in 3D grid generations. These limitations leave the original VP acceptable in the realm of grid generation for

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certain engineering purposes, but once it comes to the areas that demand higher accuracy such as medical image processing, undesired results can be accumulated. Actually, this revision of VP also aims at providing a better theoretical insight and computation tool to enhance the novel image atlas construction method proposed in [14], as is shown in E.g. 3.5. The studies in this regard will be included in future works.

6

Conclusion

In sum, the revised VP is enabled to construct transformations that cope with composition of transformations as the left-translation of the diﬀeomorphism group and welcomes more candidates to be characterized by using only JD and curl. It handles 3D grid generations much better than the original VP. Analytical study of how the provided algorithm converges is of our interests. In a future study, building eﬃcient algorithms for 3D medical image registration is planned. Acknowledgment. The second author is partially supported by the NIH grant R03MH120627 as the grant PI. The ﬁrst author is an approved collaborator of this grant. The content is solely the responsibility of the authors and does not necessarily represents the oﬃcial view of the National Institute of Health. Appreciation goes to Prof. Xiaoqun Zhang and her Medical Imaging team in the Institute of Natural Sciences, Shanghai Jiao Tong University, where brilliant ideas were shared and explored.

References 1. Bauer, M., Joshi, S., Modin, K.: Diﬀeomorphic density matching by optimal information transport. SIAM J. Imag. Sci. 83, 1718–1751 (2015) 2. Cai, X., Fleitas, D., Jiang, B., Liao, G.: Adaptive grid generation based on the least-squares ﬁnite elements method. Comput. Math. Appl. 48, 1007–1085 (2004) 3. Chen, Y., Ye, X.: Inverse consistent deformable image registration. In: Alladi, K., Klauder, J., Rao, C. (eds.) The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, pp. 419–440. Springer, New York (2010). https://doi.org/10.1007/ 978-1-4419-6263-8 26 4. Chen, X., Liao, G.: New variational method of grid generation with prescribed Jacobian determinant and prescribed curl (2015). arxiv.org/pdf/1507.03715 5. Chen, X., Liao, G.: New method of averaging diﬀeomorphisms based on Jacobian determinant and curl vector (2016). arxiv.org/abs/1611.03946 6. Dacoragna, B., Moser, J.: On a partial diﬀerential equation involving the Jacobian determinant. Ann. Inst H Poincar´e 7, 1–26 (1990) 7. Gu, X., Yau, S.T., Computational Conformal Geometry. International Press of Boston, Inc., Boston (2008) 8. Grajewski, M., Koster, M., Turek, S.: Mathematical and numerical analysis of a robust and eﬃcient grid deformation method in the ﬁnite element context. SIAM J. Sci. Comput. 312, 1539–1557 (2009) 9. Huang, W., Sun, W.: Variational mesh adaptation II: error estimates and monitor functions. J. Comput. Phys. 184, 619–648 (2003)

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10. Joshi, S., Davis, B., Jomier, M., Gerig, G.: Unbiased diﬀeomorphic atlas construction for computational anatomy. Neuroimage 23, 151–160 (2004) 11. Liao, G., Cai, X., Liu, J., Luo, X., Wang, J., Xue, J.: Construction of diﬀerentiable transformations. Appl. Math. Lett. 22, 543–1548 (2009) 12. Liseikin, V.: Grid Generation Method, Springer, Dordrecht (1999). https://doi.org/ 10.1007/978-90-481-2912-6 13. Zhou, Z., Chen, X., Cai, X., Liao, G.: Uniqueness of transformation based on Jacobian determinant and curl-vector (2017). arxiv.org/abs/1712.03443 14. Zhou, Z.: Image analysis based on diﬀerential operators with applications to brain MRIs. Ph.D. dissertation, University of Texas at Arlington (2019)

The Cuneiform Brick Parametric Techniques for Brick Wall Configurations Francesco Di Paola(B)

, Calogero Vinci , and Fabrizio Tantillo

Department of Architecture (DARCH), University of Palermo, Viale delle Scienze, 90128 Palermo, Italy {francesco.dipaola,calogero.vinci}@unipa.it, [email protected]

Abstract. The study presents some results of an experimental research on brick wall cladding, proposing methodological approaches for the creation of new geometric-compositional configurations. Citing innovative case studies of historical and contemporary architecture, the research has allowed to determine a different geometric shape of the brick, which takes its cue from the traditional wedge-shaped brick in majolica brick, widespread in Sicily since the sixteenth century for the realization of cusps and domes. Through the definition of procedural parametric algorithms for the analysis and geometric-spatial control, it is described the path that led to the digital prototype of a modular component of new design, able to adapt in shape and size to design patterns of cladding surfaces with simple and variable curvature free-form. The design of complex geometric and organic shapes, through the visual programming of digital algorithms (generative modelling, algorithmic modelling, computational modelling), as well as bringing a methodological and applicative renewal, has initiated interdisciplinary insights. Starting from a support grid for installation, we report several structural solutions of wall tessellations that highlight and validate its potential applications. The re-proposal of the building element with materials and techniques of the latest generation is in line with the goals of sustainable development OSS (Sustainable Development Goals SDGs) of the ONU 2030 Agenda. Keywords: Parametric design · Geometric patterns · Free-form surface

1 Introduction The brick has always been the most commonly known and used in building construction, considered one of the oldest construction elements dating back to 7000 BC. In the art of building, the brick masonry wall is obtained from the assembly of individual prefabricated and standardized modular elements mounted in situ following different assembly techniques in relation to the evolution of construction and production. This multi-millennial building system, which unifies historical and contemporary architecture, distinguished in the various eras by morphological type and construction practice used, is an interesting area of investigation surprisingly current that offers new and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 612–623, 2023. https://doi.org/10.1007/978-3-031-13588-0_53

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stimulating insights with the use of modern tools for graphic representation, design and geometric-spatial control. Bricks have been used since the times of the ancient Mesopotamians, Egyptians, Greeks and Romans, becoming very common all over the world, up to the present day. Over the centuries, in relation to different needs and traditions, the brick wall face, the load-bearing element of the structure in elevation, has assumed a static function and, at the same time, of aesthetic-formal cladding. In masonry, the color of the clay, the surface treatment of the face, the arrangement of individual elements and their geometric shape have over time determined the formal expression of architecture, linking construction technique and aesthetic interpretation. Even if the use of brick in the built environment as a construction material has a millennial history, the potential of new digital technologies for the design of masonry elements applied in the creativity of contemporary design and architecture is still underdeveloped. Contemporary architectural practices employ a wide range of digital tools for the exploration of complex forms. The design of complex, organic geometric forms, through the visual programming of digital algorithms (generative modelling, algorithmic modelling, computational modelling), enables the generation of complex formal solutions that bring methodological and applicative renewal. So, at the present time, it is of interest to implement tools that provide interactive techniques for designing brick models and that can handle constant changes in a parametric and automated manner to inform and influence the design process of the entire assembly, providing design feedback responsive to the constraints and construction and structural requirements of the overall model and the proposed individual bricks. In addition, the algorithmic definitions and plug-ins made available do not yet have a general validity in relation to a generic surface geometry, and the automatic processes of modification of the standard model of parallelepiped form generate topological errors that invalidate the design choices. This study is part of this field, proposing methodological approaches for the creation and control of new geometric-compositional configurations of patterns composed of bricks different from the canonical ones, through the visual programming of digital algorithms.

2 The Brick: From Tradition to Innovation In Eastern Architecture, especially in buildings of Islamic matrix, the building tradition provided for the use of bricks mainly as decorative cladding for surfaces with variable curvature. The individual elements characterized the facing according to geometric patterns and plastic effects to create light and shadow effects, due to indentations, projections, discontinuities and the treatment of the mortar joints [1]. Two examples among all are the Twin Towers of the 11th-century Kharraqan Mausoleum in Qazvin, and, later, in the late 14th century, the Mausoleum of Khoja Ahmed Yasawi in present-day Turkistan, which show in the treatment of the surface envelope an artistic variety of plastic configurations typical of the Eastern tradition, including calligraphic strips with specially shaped bricks (Fig. 1) [2].

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Fig. 1. Use of bricks as decorative cladding in Islamic Architecture, details: left, Mausoleum of Khoja Ahmed Yasawi; right, Tower of Kharraqan Mausoleum.

In the complex socio-cultural landscape, in the long period of time from the fall of the Roman Empire to the Renaissance, each era introduced specific characteristics in the compositional layouts using brick as a building material. In architectural journals there are many significant examples of brick buildings made by distinguished architects of the modern era with technical repertoires that have defined new expressive and creative styles, employing the canonical form of parallelepiped. Among the masters who worked in the 20th century certainly include: Antoni Gaudi with the Crypt of Santa Coloma di Cervellò in Barcelona, 1898–1915; Frank Lloyde Wright with the Johnson Offices in Racine, 1936; Peter Beherens with the Höchster Farbwerke Offices in Frankfurt (1920–1924); Alvar Aalto with the Experimental House in Muutratsalo, Finland (1952–1953); Louis Kahn with the Indian Institute of Management in Ahmedabad, India (1962–1974); Mario Botta with the Cumbre de las Americas Monument, Santa Cruz della Serra (1996); Renzo Piano and Cristoph Kohlbecker with the Daimler-Benz in Postdamer Plastz, Berlin, Germany (1998). In the last decades, in the world of architectural design and more generally of design, there has been a real technological revolution, thanks to the development of increasingly advanced digital procedures (CAD and BIM) that have introduced automated and robotic

Fig. 2. Construction of wall faces with robotic arm systems (Gramazio and Kohler at ETH Zürich) or with the use of drones.

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assembly and construction solutions in the field of construction such as robotic arms and drones into the production processes (Fig. 2) [3]. The pioneers in the field of robotic construction applied to architecture are certainly the architects of the Swiss firm Gramazio&Köhler, who have been engaged in research in the field of Architecture and Digital Fabrication in collaboration with the Department of Architecture (D-ARCH) of ETH Zurich in Switzerland [4]. Starting from the first half of the 2000s, the studio’s research focused on the possibility of realizing nonstandardized facades, mainly made of brick, thanks to the aid of a robotic arm arranged for the construction of panels whose shapes, or surface textures, could vary automatically following specific programming algorithms [5]. One of the first experimental examples of architectural application of this technique is the facade of the Gantenbein Vineyard in Fläsch, Switzerland, 2006. In 2015, construction began on the Administrative Building of the Textile and Apparel Industry Association in Münster, Germany, a project by Behet Bondzio lin architekten. The treatment of the facades is of interest, with the parametrically controlled arrangement of 70,000 brick bricks that, as a whole, give the observer the perception of a light drape with folds. Through collaboration with brick manufacturer Deppe Backstein-Keramik, seven types of shapes designed with gradually angled faces were produced (Fig. 3).

Fig. 3. Administrative Building Textilverband/Behet Bondzio Lin Architekten (2015, Germany). An example of the use of a brick of a particular shape, different from the canonical parallelepiped one.

2.1 Algorithmic-Generative Approach to Geometric Pattern Configuration The wall face is defined by the reciprocal geometric-spatial relationships between the bricks that represent its modular units. In the authoring tools of Building Information Modelling (BIM), as in the recently adopted parametric modelling techniques, walls, slabs, windows are defined as “components” to represent a whole [6]. When using these types of parametric components, for example a wall, changes made to the object affect the entire geometric shape as a single entity. For example, in the case of the “wall” object, these tools are currently limited in that these do not allow the selection and modification of wall components independently, giving less freedom of action and interaction with the basic components (the bricks) and their semantic relationships within their specific domain. Therefore, pattern control must

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be based on parametric modelling that gives the designer the tools for interactive modification, especially in the presence of complex shapes with a large number of elements. This feature makes the parametric design of brick walls possible, giving designers the possibility to parametrically test formal concepts and manage continuous changes. The new feature of this approach is not so much the formal and aesthetic results, but rather the logical process that constitutes the substructure of the project. From an additive method, in which independent marks are superimposed on the drawing sheet, we move on to an associative process, in which each element is related to the others. Parametric applications define a different approach to the generative design process and to the representation of complex geometric shapes, optimising them for use in multiple fields of application (structural, energy, economic, etc.). The work of Gramazio and Köhler and Zwarts and Jansma has demonstrated the advantage and potential of automating this process, not simply to speed up the construction process, but with the aim of making new contributions to growth and knowledge in construction techniques as much as in design aesthetics [7]. Even though this is an interesting research topic, there are still few academic studies documenting these new techniques for creating brickwork [8, 9]. Current digital tools allow computational solutions to interactively create models on brick walls by rotating the bricks around the axes of the spatial triaxis. In addition, it is possible to map a digital image onto the masonry wall (Fig. 4) within an interactive and iterative solution that can manage design changes and designer feedback in real time (BrickDesign plug-in for Rhinoceros, ROB Technologies).

Fig. 4. BrickDesign plug-in, example of parametric control of the mapping of an image on a brickwork.

2.2 The Cuneiform Brick in Sicily: Structure and Form In Sicily, the production of wedge-shaped majolica bricks for the construction of complex structures such as cusps and domes with the most original configurations is attested as early as the 16th century. These particular architectural elements define the end of high bell towers or towers, marking the landscape and often becoming characteristic in the historical iconography of urban centres. The production of maiolica bricks for making cusps and domes is common to almost all the ceramic production centres on the island, from Enna to the Nebrodi hills, but

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particularly the Madonie area. The most interesting aspect is certainly the complexity of the construction of these structures. The unusual and original forms of the individual basic brick elements and the modes of the masonry apparatus have to deal with geometrically complex surfaces. Through the aggregation of identical pieces, or with a few morphological variations, it is possible to generate real walling apparatuses of variable geometric-constructive typology, with the possibility to realise different laying configurations; these assembled elements constitute both the structure and the polychromatic finish (Fig. 5).

Fig. 5. Sicily, examples of spiers with different geometrical type of cladding: conical, hemispherical and bulboid conformation.

The types found can be classified according to the horizontal section and the geometry of the elevation: there are conical conformations (more or less elongated), pyramidal (with a square, hexagonal or octagonal base, with equal or symmetrically paired sides, more or less depressed), bulbous (with simple curvature in plan or in vertical section, or with double curvature), hemispherical (with a continuous extrados or with a horizontal mistilinear section), with or without a vertical tiled structure. In fact, the majolica brick is at the same time a structural element, a surface finish and a piece of a mosaic that develops according to complex geometric patterns with clear references to the Middle Eastern tradition. These elements, in which the maiolica part is limited to the visible end, vary in thickness and length from three to six centimetres and from twelve to twenty-five centimetres respectively. A common feature of all the elements examined, hence the term “wedge-shaped”, is the presence of a tapering, more or less accentuated, of the long vertical sides, in some cases associated with a tapering also for the larger horizontal faces to ensure geometric congruence with the forms that were to be composed. In general, the same element could be used in different applications with different functions (structural, collaborating, cladding, simply decorative) (Fig. 6).

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Fig. 6. Sicily, examples of wedge-shaped bricks used in the same spire with different tapering angles.

3 Methodology 3.1 The Cuneiform Brick, New Design Approaches To design boundary condition-compatible solutions that satisfy specific requirements, it is necessary to permute the relationships of the elements to determine alternative configurations. Digital procedures that allow local intervention while keeping the modular unit connected to the entire assembly still need implementation to be truly effective. To represent and control the wall face in relation to the shape of the cladding surface and the geometric pattern, it is essential to know the relative dimensions of each modular unit and to understand their positioning. The canonical form of the parallelepiped brick is often unsuitable for free-form cladding surfaces, because it is common to encounter spatially incompatible solutions which lead to topological errors (overlaps, intersections and interconnections) or structural instability. In fact, the laying of traditional parallellepiped bricks on circular trajectories or mathematical NURBS curves creates limits and problems that invalidate the design choices (Fig. 7).

Fig. 7. Use of parallelepiped bricks on free-form surface facing, topological errors; schemes in plan and in space.

Thus, the challenge is twofold. The first step is to determine a generic methodology to translate the design flow into an appropriate set of parameters capable of generating the required parametric behaviours, interacting with the individual models (the bricks) and maintaining the geometric coherence of the overall structure (the wall). The second step requires the design of a parametric brick shape that identifies specific types that can expand the range of possible design configurations, respecting aesthetic and structural requirements.

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On the basis of the above considerations, the study describes the methodological path to design a digital prototype of a brick, able to adapt in shape and size to design patterns of cladding of geometric surfaces with simple and variable free-form curvature. We report several structural solutions of wall textures that highlight and validate the potential applications. The newly designed modular element takes its idea from the traditional special wedge-shaped brick made of majolica, which was widespread in Sicily from the 16th century to build cusps and domes [10]. The algorithm, as a system of relationships and connections, provides the coherence between the parts and the respect of an overall logic. The defined algorithm definitions required a finite number of input data, called geometric-mathematical inputs or parameters, and provided as a result an univocal output, which in the context of this study, is a particular arrangement of the wall face. The modification of parameters is propagated throughout the entire system, taking into account the relationships within it. Rather than acting directly on the design of the individual brick element, this approach gives the user the possibility of defining a logical process, valid for the same set pattern configuration. As a direct consequence of the associative logic, it has been possible to create conceptual and effective links between the different levels of design detail. The modification of a parameter on a larger scale (for instance, the choice of the cladding surface) is able to generate a propagation of modifications such as to reach the congruent redefinition of details on a smaller scale (element-brick). It was possible to hypothesise a direct link between the parameters relating to the general form of a complex surface and the geometric characteristics of a structural node, all guided by the relationship logic defined by the designer. The tool chosen for algorithm generation is Grasshopper (a well-known plug-in developed since 2007 by Finnish programmer David Rutten for Robert McNeel&Associates’ Rinhoceros modelling software). Inputs and instructions were provided to the computer through a specific visual scripting editor, while the output of the algorithm was displayed within the modelling software workspace. In this way, by virtually linking the various components (which may be geometric entities or primitives, mathematical operations and geometric-spatial transformations) in a sequence of commands (associative logic), the nodal diagram is generated which encodes the parametric geometry associated with it. The resulting nodal system tree within the editor is characterised by a flow of data in one direction: each operation is influenced by those that precede it and it is not possible to initiate loops, unless additional components are created ad hoc. The methodological process adopted has led to the definition of an algorithm divided into three fundamental parts: – the parametric definition of the shape of the modular brick element; – the design of the pattern logic of the wall texture; – the generation of the variable texture wall cladding made up of the modular elements (associative modelling). The first step involved the modelling of the brick shape 2.0, defined starting from the traditional solid parallelepiped. Thanks to the parametric components of Grasshopper

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and LunchBox, it was possible to extract the constituent topological elements (faces, edges and vertices) from the initial model (“deconstruct Brep” component). Once the main elements had been extracted, plane and solid geometric transformation components (tapering, rotations along the X, Y and Z axes) were used in order to make the final model parametrically modifiable and editable. Four different types of brick were identified to compose geometrically complex spatial configurations (Fig. 8). Geometric patterns are one of the most interesting topics to be developed with the help of generative algorithms in Grasshopper. Through the creation of modular grids one has the potential to design patterns that can be used as the basis for other design products. Within the design study it was decided to analyse and develop certain types of planar and three-dimensional geometric patterns [11]. With regard to the setting of distribution grids, a useful tool for the generation of different configurations was the already mentioned LunchBox plug-in. For textures on surfaces with variable curvature, the algorithm was tested for applications on primitive surfaces (cylinder, cone, sphere) and more complex surfaces such as the free-form conoid (Fig. 9).

Fig. 8. Example of parametric design of a free-form surface wall face. Types of wedge-shaped brick used.

Fig. 9. Parametric algorithmic definition of a free-form conical surface with variable curvature and wedge-shaped bricks.

In accordance with the geometric-formal properties of the surface to be created, each component will assume a particular position in the digital space, rotated and translated according to variable angles, directions and layouts dictated by specific rules imposed by the initial geometric constraints. Any modification of the surface, even complex and with variable curvature, will give the possibility to update in real time the arrangement of

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all the constituent elements respecting the topology of the overall geometric shape (main curvatures, normal directions at a given point, changes of position) and the reciprocal semantic relations within a specific domain. 3.2 The Cuneiform Brick, New Technological-Constructive Solutions With regard to the use of wedge-shaped brick 2.0 as a wall cladding, it was decided to use a prefabricated steel structure to accommodate the individual elements. The choice was oriented towards a punctual connection on the single steel element, characterised by an “omega” profile of variable dimensions bolted onto horizontal aluminium L-section profiles, fixed directly to the building wall (Fig. 10).

Fig. 10. Concept of the technological solution for the installation of wedge-shaped brick.

The “omega” profile is characterised by two tongues, which stress the brick in a barycentric area, so that if there are any deformations of the iron, the brick will be stressed in the middle area with less risk of breaking. It will be possible to choose an omega of a different height depending on the thermo-physical calculations made on the building wall. Among the various examples analysed, we have represented claddings on flat surfaces with elements having the joint in line and/or staggered; curved surfaces such as the cylinder and surfaces with double curvature such as the spherical casing (Fig. 11) [12].

Fig. 11. Brick anchoring solutions to the cladding envelope.

4 Conclusions and Further Developments The methodology employed and the parametric procedures adopted for the generation of the model constitute a valid approach of investigation, analysis and control. The process

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makes it possible to explore in a single algorithmic definition the complex spatial articulation of the system in the various geometric typologies tested. The generative algorithm, defined according to an associative logic, describes the geometric relationships of the elements that make-up the general structure (the wall face), allowing the exploration of multiple configurations. Moreover, it is possible to modify in real time the geometrical features of each single component (the brick). The implementation of digital computational tools, able to create variable geometric patterns through the modification of parameters, can innovate technical-constructive solutions for an effective use of the wedge-shaped brick in contemporary architecture. The potential also lies in the multi-scalarity of the design method, through the use of energy-efficient technological solutions (in line with the strategic objectives of sustainable development of the UN 2030 Agenda) applied to the unit/brick for the architectural design of underground stations, urban spaces, exhibition halls and islands for recharging electric mobile devices (Fig. 12).

Fig. 12. Use of energy-efficient technological solutions. Design example of a bicycle charging station with wedge brick cladding.

Appendix Author Contributions: All authors conceived and designed the methodology; all authors wrote the introduction, and the conclusions; F.T. performed and wrote “1.1_The brick: from tradition to innovation” and “1.2_Algorithmic-generative approach to geometric pattern configuration”; F.D.P. performed and wrote the “2_The Cuneiform Brick 2.0_Methodology” and “2.2_Algorithmic-generative approach to geometric pattern configuration”; F.D.P. and C.V. performed and wrote “2.1_The cuneiform brick in Sicily: structure and form”; C.V. performed and wrote “Technological-constructive solutions”.

References 1. Grube, E.J., et al.: Architecture of the Islamic World. Thames & Hudson, Londra (1995)

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2. Panahi, A.: Application of geometry in brick decoration of Islamic architecture of Iran. Seljuk period. J. Am. Sci., 814-821 (2012) 3. Gramazio, F., Kohler, M.: Made by Robots: Challenging Architecture at a Larger Scale, Architectural Design, Londra (2014) 4. MAS ETH, master’s in advanced studies in Architecture and Digital Fabrication presso il Politecnico di Zurigo. https://www.masdfab.com/. Accessed 23 Jan 2022 5. Dell’Endice, A., Odaglia, P., Gramazio, F.: Prefabbricazione robotizzata e innovazione. Struttura su due livelli con assemblaggio robotico. MD Journal (2017) 6. Goedert, J., Meadati, P.: Integrazione della documentazione del processo di costruzione, in Building Information Modeling. Journal of Construction Engineering and Management (2008) 7. Kereshmeh, A., Matthew, E., Swarts, S.T., Gentry, T.R.: Integrated generative technique for interactive design of brickworks. ITcon, Jeddah 19, 225–247 (2014) 8. Lee, G., Sacks, R., Eastman, C.M.: Specifying parametric building object behavior (BOB) for a building information modeling system. Automation in construction, 758–776 (2006) 9. Shea, K., Aish, R., Gourtovaia, M.: Towards integrated performance-driven generative design tools. Automation in Construction, pp. 253–264 (2005) 10. Di Paola, F., Fatta, G., Vinci, C: Il mattone cuneiforme maiolicato. Procedure algoritmicoparametriche digitali come strumento di indagine e progettazione: dall’architettura storica all’innovazione del design. In FrancoAngeli. 42th International conference of representation disciplines teachers CONNECTING. DRAWING FOR WEAVING RELATIONSHIPS, pp. 429–444, Milano (2020) 11. Schumacher, P.: Parametric Patterns. Architectural Design, pp. 28–41 (2009) 12. Sullivan, C., Horwitz-Bennett, B.: Building with Bricks. Building Design+Construction, pp. 48–57 (2008)

PerForm – Designing Adaptable Furniture Eva Hagen(B)

and Benedikt Blumenröder

Fatuk – Faculty of Architecture, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany {eva.hagen,benedikt.blumenroeder}@architektur.uni-kl.de

Abstract. How can we bring more flexibility into sitting? This was the initial question of the seminar “PerForm” of the department Descriptive Geometry and Perspective, Faculty of Architecture of the TU Kaiserslautern (TUK) in 2021. The course started with a workshop providing the students with crucial competencies in digital design, presentation and fabrication tools. After analyzing related studies and projects, the students then developed their own conceptual drafts based on their acquired knowledge and skills. As a central part of the seminar, the research involved designing kinetic and flexible furniture that encourages motion and agility using parametric design tools. In addition, the use of robotic fabrication techniques building 1:1 wooden prototypes was a part of the complex task. Finally, an exhibition was held on campus including the presentation of the final furniture, mockups, documentations and posters. This approach combining topics such as geometry of movement, construction techniques and graphic presentations shows possibilities for a future-proof teaching concept. Keywords: Parametric design · Adaptable furniture · Digital fabrication

1 Introduction The seminar “PerForm” took place between April and September 2021 as part of the courses on digital architectural geometry. The main focus was to develop concepts towards a more flexible and animated way of sitting, and how furniture can be used by means of geometric modelling, movement analysis and representation. This approach led to the design of various adaptable and kinetic furniture. In this course, we wanted to integrate research, design and fabrication stage. Thus, the participating students began by analyzing studies and projects on kinetic art and furniture, different human movements and how these motions can be captured and displayed. Following this, the students developed their own projects. These designs were optimized through guidance and discussions during various presentations. Selected designs were finally implemented as a team task, and the results were presented at an exhibition where the built furniture could be tested. The seminar was supported by Campus Plus, an organization whose main focus is to create an attractive campus life and enabling a healthy way of studying. The realized projects are now part of their outdoor learning spaces. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 624–635, 2023. https://doi.org/10.1007/978-3-031-13588-0_54

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2 Workshops and Research 2.1 Compact Course on 3D-Modeling and Graphic Workshops Before the start of the semester, we covered the basics of 3D modeling in a block course. Digital tools supported the creative work in the design process. Workshops were held on computer-aided design (CAD) software and graphics programs (including visualizations and animations), as well as on the interface to computer-aided manufacturing (CAM). The workshops imparted profound understanding of how to use digital tools in order to control complex geometries in a creative way. Part of the course comprised of a module where the participants had to acquire relevant work processes and explain them to their fellow students. As a final step, the students designed a pavilion or freeform structure using 3D modeling software, and displayed it with a chosen presentation method (Fig. 1). In detail, the following tools and software were covered: • Rhinoceros®/Grasshopper®: (parametric) modeling and vector based technical drawings; • V-Ray for Rhino® and Cinema 4D®: result visualizations, renderings and animations; • Adobe Photoshop®: post production and atmospheric presentations; • Adobe Illustrator® and InDesign®: advertising posters, documentation and layout.

Fig. 1. Results of the graphic workshop. Projects by the students Carolin Schreiner, Arutiun Papikian and Cihangir Kazan.

In addition to the technical input, information regarding the health and study situations of students at our university was provided by CampusPlus (a student wellbeing organization) according to their current Health Survey [1]. According to the results, only 22.3% of the architecture students are physically active at least 2.5 h per week and 16% report having health issues. The gathered information was considered for the further design process of the individual furniture. The block course was held online due to COVID-19 restrictions, and the students were able to share their screens in order to show their progress and illustrate the given tasks. The workshops and information courses were held by tutors from our department, CampusPlus and StudioDragusha (a studio for architectural renderings).

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2.2 Analyzing Kinetic Principles and Motion-Picture Projections In the next step, the students studied various topics around human proportions and movements, the basic requirements for furniture [2], kinetic art, motion-picture projection and its history. The photographic pioneer and inventor of chronophotography Eadweard Muybridge documented movements and processes photographically for the first time [3]. He experimented with time-lapse photography and 360-degree panoramas [4]. Muybridge began developing camera shutters that captured entire sequences of motion on a single plate, producing photo sequences of animals in locomotion [5]. By compiling an encyclopedia of postures and poses, showing animals as well as people [6] engaged in a variety of different activities, he aimed to create detailed models for painters and sculptors (Fig. 2).

Fig. 2. Galloping horse and rider; woman walking and flirting a fan; a man walking – photographs by Eadweard Muybridge [7].

The French scientist Etienne-Jules Marey developed single-frame photography to reconstruct sequences of movements [8]. He made three-dimensional reconstructions possible by using rotating photographic plates in a “chronophotographic gun”, lightsensitive paper strips or celluloid, and a 35 mm camera [9, 10]. His research concerned the movements of animals and humans [11] (Fig. 3).

Fig. 3. Flying pelican; Salto con l’asta; The study of Movement – photographs by Etienne-Jules Marey [12].

Inspired by the works of the photographic pioneers Etienne Marey and Eadweard Muybridge, the students worked on the photographic reconstruction of motion sequences

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with continuous shooting (or “burst mode”). The students recorded their own movements sequentially, and combined the partial movements into a complete motion capture picture (Fig. 4). They collected and combined sports equipment and objects (including parts of an outdoor gym) to animate movements for further analysis and possible integration into their drafts. Kinetic, multifunctional and adaptable furniture were used as sources of inspiration and various constructions were analyzed (for example, works by Sebastian ErraZuriz, Ian Stell and Theo Jansen). The Cabinets of Sebastian ErraZuriz challenge human perceptions of furniture and sculptures with surprising transformations, by rotating, expanding or sliding into different shapes and forms [13]. The expandable seating and tables of Ian Stell are inspired by the pantograph, a mechanical instrument for transferring drawings on the same, larger or smaller scale [14]. Theo Jansen developed kinetic art objects, known as “strandbeests” (beach animals) who “walk” across the beach moved by wind energy [15].

Fig. 4. Motion capture of different movements by the students Nina Gusenburger and Dominik Diehl.

3 The Design Stage The first step of the design process was to gain an empiric understanding of a given problem. In the above-mentioned approach, the students had to analyze their own movements to generate ideas for more dynamic seating. They transferred their sketches to a threedimensional draft, to illustrate their ideas applying the knowledge acquired during the workshops. Digital modeling and parametric design offered opportunities to investigate complex geometries and allowed rule-based optimization processes. The student projects were enhanced and new ideas incorporated during weekly meetings. Input was given by both the tutors and external guests. The drafts had to be detailed enough to allow tests for feasible fabrication methods. Material performance had to be considered in connection with the technical requirements for production. In addition, the students created advertising posters, built models and prepared renderings to engage potential clients. CampusPlus showed a great interest in some of the designs, and supported the projects by funding building materials and offering to include selected furniture in their new outdoor learning spaces. After the final presentation, the best projects were built by the robot and manually post-processed (cf. Sect. 5).

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4 Selected Projects 4.1 “The Cube” Concept. “The Cube” was designed to combine movement, fun and flexibility, inspired by “The Infinite Cube” [16] and “Rubik’s Cube” [17]. These special cubes are toys which can be twisted and moved to create a new form. This concept was implemented in a larger scale with the student project. By combining several cubes as a basic form, a variety of complex structures can be built. Equipping these elements with hinges and connectors on certain adjacent edges enables different arrangements by rotating, folding and overlapping the individual cubes. These structures feature seating and tables for diverse activities (Fig. 5). Some surfaces are decorated with game boards for playing chess, backgammon or ludo. Each wooden cube consists of four square plates and eight triangles, which serve both as bracing and handles. The cube is hollow and lightweight; thus, it can be easily transformed. The interior also provides a storage area (for example, for bags). The cube consists of nine modules, each composed of three basic elements which are connected by hinges. There are three different modules, which depict the colors of the TUK: • Blue module: the hinges are opposite each other, generating a long bench or lounger • White module: the hinges are on the same side, generating two chairs and a table which can be put on top of another cube • Red module: two cubes are connected and can only be moved together as one (they can also be put on top of other cubes)

Fig. 5. Project “The Cube” by the students Svenja Brehm and Christine Chen: possible uses, unfolding the different modules and assembled furniture.

Construction and Dismantling. Each element is built by four plates and eight triangular frame corners (Fig. 6). The modules are put together using plug connections on the lowest positioned elements, offering possibilities to build up and dismantle the cube. Recesses on the bottom parts support the lifting of the modules.

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Construction and dismantling follow a certain sequence. The module in the middle is built first, then the neighboring modules are put in place, followed by the corner modules (the dismantling is done in reverse order). Marking the interiors of the lowest cubes facilitates the positioning during assembly.

Fig. 6. Project “The Cube”: Painting the individual parts, the built structure, using one of the game boards.

4.2 “Turning Bench” Concept. The idea of the “Turning Bench” was to create opportunities for flexible usage of benches with different seating positions. Starting with an equilibrated position, it is up to the user to choose a side by tilting the elements. The first position allows an upright sitting, whilst the second position allows more relaxed sitting (comparable to a recliner). The sitting positions differ not only in direction and height, but also in the possibilities of how to use them. The center of gravity of each seat element was calculated during the design phase with Grasshopper®, and used as a guideline for the rotation centers obtaining the equilibrium (Fig. 7). Rotations by 75° around the middle axis generate the different positions.

Fig. 7. Project “Turning Bench” by the students Nina Gusenburger and Dominik Diehl: equilibrium position, the different seating heights and assembled furniture.

Construction. The bench consists of 16 individually turning seat elements. The gaps between them are filled with corresponding spacers. The racks on both ends take up the forces, which are working in the rotation axes and on the supporting pipes (Fig. 8).

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Each rack has three elements, built up by a plug-in system. Both seating positions are predefined by supports made of stainless-steel tubes, which are held in place between the racks. The outer racks are offset inwards by two elements, in order to achieve an unobtrusive appearance and emphasize only the S-shaped design of the seating elements. The four elements at the edges are fixed by threaded rods. This grouping of four elements generates an ideal seat width of 42 cm. The optimal diameter for the rotation axes is crucial for the movement of the wooden parts along the stainless-steel tubes, which was tested during the milling process. All the seating elements and the necessary spacers were milled by the KUKA® robot. Subsequent grinding prevented injuries and enabled a more comfortable sitting. The racks are designed as a plug-in system. A 20 mm deep blind hole was milled into the floor plate to fit in the racks and to fix them through screws from the bottom. After preparing the rack, the parts were sorted in order. The rotation axes were pushed back and a seat element and a spacer were threaded alternately.

Fig. 8. Project “Turning Bench”: threading the first element, the assembled furniture in use and balancing different weights.

4.3 “Object O” Concept. Inspired by a “tilting doll” [18], “Object O” is a hemispherical rocking chair that can wobble to all sides and always tilts back to an upright position. Its center of mass is below the center of hemisphere, which causes the object to right itself when pushed in any direction. The user can influence the tilting by leaning to any side. The low positioned center of gravity prevents turning over and helps keeping the balance. Its location is achieved by integrating two hollow spaces into the hemisphere. A smooth, dynamic form had been chosen, which guaranteed optimal freedom of movement. Furthermore, this organic surface provides a comfortable seating. A cavity in the back part minimizes its weight, so it does not tilt backwards. A second cavity in the lower part of the front (filled with a denser mass) supports the upright position (Fig. 9). The object is sliced vertically for fabrication out of wooden plates, which can be milled in an optimal way. These plates were glued together and filled with concrete in the hollow spaces, resulting in the correct weight distribution.

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Fig. 9. Project “Object O” by Celine Lahaye and Arutiun Papikian: technical drawing showing the plates and concrete weight, keeping the balance and detailed rendering of the surface.

Construction. 14 cm plywood panels were used to mill the parts. After milling a prototype form consisting of 10 layers, five layers each were glued together, generating two half-parts of the furniture. A second milling process smoothed out the surface (Fig. 10). The lower cavity was filled with concrete before assembling both parts of the Object O.

Fig. 10. Project “Object O”: smoothing the surface of one of the half-parts, the assembled furniture and being in use.

4.4 “Swipe & Bleib” Concept. The idea of “Swipe & Bleib” (swipe and stay) was based on the artwork “Roll Bottom” by Ian Stell [19]. This furniture combines a chair and table, where moving metal slats along a defined path builds either the seat or the desk cover. The student project transferred the concept of the sliding elements to a larger scale (Fig. 11), a different material and usage.

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Fig. 11. Project “Swipe & Bleib” by Dorota Rudowicz and Jannis Rickertsen: different seating positions, assembling the furniture and detailed drawing of a bendable plate.

Construction. The main focus was on the multifunctionality of the furniture, which has been achieved by “swiping” bendable plates along curved grooves in order to rearrange them for flexible use. The upper silhouette of the external contour resembles a cloud-like shape. The grooves are milled into solid side parts and follow their outline to provide various seating, backrest and table positions. The plates are made flexible by continuous parallel slots. Ball bearings at the corners of the plates support a smooth sliding (Fig. 12). Putting weight onto the plates prevents them from moving while in use as a table or seat.

Fig. 12. Project “Swipe & Bleib”: cutting the slots, assembled mock-up and possible usage situation on campus.

5 The Fabrication Stage 5.1 Designing in the Context of Digital Fabrication Digital fabrication offers wide-ranging opportunities to implement projects that would often be impossible or very difficult to achieve manually. Nevertheless, like all fabrication processes, digital fabrication has its strengths and weaknesses that must be considered. Selected designs of the “PerForm” seminar were to be built with help of the 7-axis milling robot. For this purpose, the manufacturing possibilities had to be considered and the design had to be checked for its compatibility with the robot. For example, it is

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almost impossible to generate sharp inner edges parallel to the cutter axis with a milling cutter. This can be particularly difficult at joining points. Such restrictions have to be considered at an early design stage. 5.2 Preparation for Production and Milling with the Robot Before manufacturing, the students had to prepare the CAD models in detail for production. Tolerances had to be determined and incorporated into the model, before modelling the raw workpiece and placing the individual components upon it. All movements of the robot had to be simulated and optimized. The CAM software Powermill® was used, allowing to plan all milling paths based on the CAD model, checking whether the component and the milling paths could be reached by the robot, and ensuring that the robot would not collide with the component, the table, itself or the cell walls (Fig. 13). In most cases, the mounting of the components on the milling table could be solved by using screws; these had to be planned and placed correspondingly, so that the milling cutter would not collide with any screw. For the production of the “Object O”, a clamping device for the second milling pass had to be developed in order to make the component accessible to the robot from all sides, and to be able to position it precisely. When the program had been simulated and written, it was transferred to the robot controller. After the part had been clamped according to the CAD model and the correct program was selected, the actual milling process began.

Fig. 13. Robot simulation in Powermill; Robot and component screwed to the milling table; milling process.

6 Documentation and Exhibition Following the fabrication of the mock-ups and prototypes, all the participants had to document their projects (including the design and building process). A handout recording these documentations was printed for distribution at an exhibition of the PerForm seminar taking place on campus. Drawings, construction guidelines, visualizations, animations, posters, mock-ups and the 1:1 fabricated furniture were all used to present the result of the projects. During the event, the guests had the chance to experience the particular qualities (Fig. 14) of each furniture, such as their adaptable, multifunctional and/or kinetic features. These final prototypes are on display and part of the outdoor learning areas, where they provide space for outdoor lectures and social gatherings.

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Fig. 14. Details of the features of “The Cube”, “Turning Bench” and “Swipe & Bleib” (photos taken at the exhibition).

7 Conclusions The seminar “PerForm” covered a wide range of complex tasks: researching geometric, kinetic and design principles; gaining computational competence in applying parametric and CAM software; developing innovative seating structures; and fabricating and postprocessing on a 1:1 scale. Following these steps supported the conceptual creativity of the architecture design process, leading to outstanding results. The teaching experiences gained during this seminar are preconditions for further application-oriented design projects, in order to achieve successful, sustainable and quickly built architecture through cooperative manufacturing (both digital and manual). Challenging the students with tasks (for example, applying complex parametric CAD and CAM software, detailing implementation planning, operating production machines, and experiencing manual post-production) lead to a rapid increase in the students’ knowledge. The students developed such well-engineered and sophisticated projects, we decided to build more prototypes than initially planned. This unconventionally conducted project aroused great interest and commitment in the students, and showed the importance of originality in architectural teaching methods. The content covered in this course equipped the students with important knowledge for future tasks as architects. A major difference between this seminar and other design courses offered at architectural faculties is that many steps from the first preliminary draft up to the manufacturing process were covered in a single course. However, project management regarding schedule and coordination could be improved in the next seminar; for example, by increasing participative decisions and adding an interactive platform for information exchange, especially during the fabrication stage. New aspects regarding the design-fabrication process included cooperation with external companies and institutions, facilitating the creation of long-term partnerships. The success of this seminar proves the potential of the multidisciplinary approach for sustainable holistic methodology, applicable to educational concepts, applied research and architectural purposes in the future. Acknowledgments. The valuable cooperation with CampusPlus supported the successful progress of the seminar. Special thanks to Sebastian Kirn and Robert Bachmann for the instructive workshops, as well as Julia Müller and Max Sprenger for the management and timber supply. CLTECH GmbH & Co. KG donated part of the building material for the mock-ups. We appreciate

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the contribution of Hajdin Dragusha from StudioDragusha, who held a visualization workshop, and our colleague Viyaleta Zhurava assisting in the block seminar. We also like to thank Prof. Dirk Bayer from the Department of MEE (Methodik des Entwerfens und Entwerfen) for providing the robot and Florian Lapport, who spent many hours programming and operating the robot. We are grateful for the commitment of the students and their elaborated designs.

References 1. Töpritz, K., Lohmann, K., Gusy, B., Farnir, E., Gräfe, C., Sprenger, M.: Wie gesund sind Studierende der Technischen Universität Kaiserslautern? Ergebnisse der Befragung 06/15 (Schriftenreihe des AB Public Health: Prävention und psychosoziale Gesundheitsforschung: Nr. 01/P16). Freie Universität Berlin, Berlin (2016) 2. Jocher, T., Loch, S.: Raumpilot Grundlagen, 1st edn. Kraemer Verlag Wüstenrot Stiftung, Ludwigsburg (2010) 3. Eadweard Muybridge. https://de.wikipedia.org/wiki/Eadweard_Muybridge. Accessed 17 Feb 2022 4. Eadweard Muybridge. https://www.walthercollection.com/de/collection/artists/eadweardmuybridge. Accessed 17 Feb 2022 5. Muybridge, E.: Animals in Motion. Dover Publications Inc., New York (1957) 6. Muybridge, E.: The Human Figure in Motion, 1st edn. Dover Publications Inc., New York (2000) 7. https://www.thoughtco.com/thmb/dvt2Mq2xQrLNhJ8spyxI6IE5aA=/4314x3318/filters:fil l(auto,1)/high-speed-sequence-of-a-galloping-horse-and-rider-680806289-59c0259c68e1a 20014827f5f.jpg, https://nucius.org/en/gallery/the-human-figure-in-motion-women, and https://upload.wikimedia.org/wikipedia/commons/c/c8/A_man_walking._Photogravure_ after_Eadweard_Muybridge%2C_1887._Wellcome_V0048618.jpg. Accessed 17 Feb 2022 8. Marey, É.: https://de.wikipedia.org/wiki/%C3%89tienne-Jules_Marey. Accessed 17 Feb 2022 9. Marey, É.: La Chronophotographie. Hachette Livre BNF Location, Paris (2013) 10. Marey, É., Ankele, D., Ankele, D.: 100 Photographic Reproductions. Ankele Publishing, LLC, Austin (2011) 11. Marey, É.: Movement. University of Michigan Library, Ann Arbor (1895) 12. Wikimedia Commons. https://upload.wikimedia.org/wikipedia/commons/e/e0/Marey_-_ birds.jpg, http://www.fotographiaonline.com/etienne-jules-marey-1830-1904/#prettyPho to/0/ and http://photographyhistory.blogspot.com/2009/02/etienne-jules-marey-1830-1904study-of.html. Accessed 17 Feb 2022 13. Sebastian ErraZuriz Artist. https://sebastian.studio/. Accessed 02 Dec 2022 14. Ian Stell Artist. https://ianstell.com/. Accessed 02 Jan 2022 15. Theo Jansen Artist. https://www.strandbeest.com/. Accessed 21 Feb 2022 16. Infinity Cube. https://infinitycubefidget.com/. Accessed 14 Feb 2022 17. Rubik’s Cube. https://www.rubiks.com/en-us/about. Accessed 15 Feb 2022 18. Stehaufmännchen. https://de.wikipedia.org/wiki/Stehaufm%C3%A4nnchen. Accessed 17 Feb 2022 19. Roll Bottom. https://ianstell.com/RollBottom. Accessed 12 Feb 2022

Automatic Lung Segmentation with Seed Generation and ROIFT Algorithm for the Creation of Anatomical Atlas Jungeui Choi1 , Edson Kenji Ueda1 , Guilherme Cortez Duran1 , Paulo A. V. Miranda2 , and Marcos de Sales Guerra Tsuzuki1(B) 1

2

Computational Geometry Laboratory, Escola Polit´ecnica da USP, S˜ ao Paulo, Brazil [email protected] Instituto de Matem´ atica e Estat´ıstica da USP, S˜ ao Paulo, Brazil

Abstract. This paper describes the development of an algorithm to automatically generate seeds for lung CT (Computed Tomography) segmentation. The segmentation algorithm used is ROIFT (Relaxed Oriented Image Foresting Transform), a seed-based method for segmenting 3D images. Internal and external seeds are required for ROIFT. The internal and external seeds are automatically generated using the 2D watershed segmentation algorithm. Segmented images are transformed into a polyhedral model using the marching cubes algorithm. The segmented lungs will be used to create an anatomical atlas of the thoracic region. In this initial phase, 100 DICOM images were segmented. The anatomical atlas will be used as a regularization to solve the electrical impedance tomography of the human chest. Future work considers the segmentation of the ribs, skin, airways, and heart. Keywords: Three-dimensional mesh · Anatomical atlas · Seed generation · Thoracic CT images · Watershed · Relaxed oriented image foresting transform

1

Introduction

The image segmentation is very useful for medical and biological image analysis. However, to ensure reliable and accurate results, human supervision is necessary in several tasks, such as the extraction of poorly deﬁned structures in medical imaging [4,5,17]. These problems have motivated the development of many semiautomatic segmentation methods with the aim of minimizing user involvement and time required without compromising accuracy. Some authors used temporal coherence to reduce user interaction [18,21–24]. In this research, segmentation will be performed automatically using an algorithm for automatic seed generation.

c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 636–647, 2023. https://doi.org/10.1007/978-3-031-13588-0_55

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An important class of interactive image segmentation comprises seed-based methods, which were developed based on diﬀerent, supposedly unrelated theories, such as Watershed [26] and Image Foresting Transform [3]. These methods can be used directly for automatic targeting when seeds are found automatically. Segmented images will be used to create an anatomical atlas to regularize the electrical impedance tomography (EIT) problem [12,13,15,20]. This work is an initial step for the creation of the anatomical atlas. It addresses lung segmentation. However, similar algorithms are expected to be created for the ribs, airways, heart, and skin in the near future. EIT consists of a non-invasive technique to obtain images of the interior of a contour through the knowledge of information in the contour itself [9,10,13,14]. The anatomical atlas will be used to regularize the EIT problem [11,15]. This paper will focus on the development of a fully automatic lung segmentation algorithm. Internal and external seeds will be automatically generated and will be used as input to the 3D Relaxed Oriented Image Foresting Transform (ROIFT) [2] segmentation algorithm. Internal and external seeds are generated using a 2D watershed segmentation algorithm and considering some geometrical assumptions. This work is structured as follows. Section 2 presents a brief introduction of some basic concepts, such as DICOM images, watershed, Marching Cubes (MC) algorithm, Image Foresting Transform algorithm, Oriented Image Foresting Transform algorithm, and Relaxed Oriented Image Foresting Transform algorithm. Section 3 presents the proposed methodology to automatically generate seeds to be inputted into the ROIFT algorithm. In Sect. 4 the results will be presented, as well as the discussions, and the conclusions and future works are drawn in Sect. 5.

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Basic Concepts

In this section, some basic concepts are presented. Initially, the medical images used in this work are described, which are 3D CT DICOM images of the human chest freely available on the Internet. After that, the Watershed and MC algorithms will be explained in detail, which are necessary processes for the generation of seeds used for segmentation via ROIFT, explained at the end of this section. 2.1

Medical Images

DICOM (Digital Imaging and Communications in Medicine) is a set of standards for the management and transmission of medical information (medical imaging) in a digital format. The DICOM standard is widely used in CT, MRI, radiography, ultrasound, etc. In this work, images of the thoracic region of the human body obtained by CT will be used for segmentation. Figure 1 shows an example of a CT scan, a slice of the human chest. It is a 2D image, but it is possible to obtain a sequence of 2D images, where each image is a slice that makes up a 3D image.

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Fig. 1. Example of a CT DICOM image. It can be seen that the thoracic region is represented in three planes, a sagittal plane, a coronal plane and a transverse plane.

2.2

Watershed and Marching Cubes Algorithm

Watershed is an Image Segmentation Algorithm to detect basins in images [26]. The DICOM image is represented in shades of gray, and every grayscale image can be considered a topographic surface according to the watershed segmentation algorithm. The image is divided into two diﬀerent sets: the catchment basin and the basin line, by ﬁlling the water surface with its minimum value and preventing the water from joining other sources. The Marching Cubes Algorithm [1,25] computes a triangular mesh from discrete sample volume data and uses a divide-and-conquer approach to determine the isosurface inside a cubic cell. The vertices of the cell are classiﬁed as internal or external, with 28 = 256 possible conﬁgurations for each cell. Thus, the Watershed algorithm is used to segment each slice of the DICOM image, and all segmented images are converted into a 3D mesh STL (STereoLithography) ﬁle with the Marching Cubes algorithm. It was possible to segment the organs separately once each assumed a diﬀerent grayscale value. Figures 2(a), (b) and (c) represent, respectively, a rasterized representation of the lung, ribs, and airways that have been segmented from a CT image sequence. It is important to note that, using the watershed algorithm, it was not possible to segment the heart and skin. In addition, there were several problems related to noise in bone segmentation, in addition to airway failures. Another problem of this approach is that continuity of the structures cannot be ensured

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Fig. 2. (a) Lung. (b) Rib. (c) Airways.

between the slices, as noticed in Fig. 2(b), in which parts of the ribs are not correctly segmented. 2.3

IFT and OIFT

The Image Foresting Transform (IFT) [3] is an algorithm used in the development of image processing operators, with ﬁltering, segmentation, and form representation, in which an image is transformed into a graph and the graph into a forest of optimal paths (minimum cost path forest) The concept of a derivated image graph is used directly or indirectly in a lot of diﬀerent techniques of image analysis, which, although they have similar concepts, are presented as unrelated methods, leading to diﬀerent frameworks, such as watershed, random walk, fuzzy connectedness, and graph cuts. Depending on the framework used, the arc weights between a pixel of the graph can be interpreted as a similarity measure, velocity function, aﬃnity, cost, or distance. The IFT uniﬁes and extends many of these techniques by making use of more general search and connectivity optimization. A multidimensional and multispectral image Iˆ is a pair I, I where I ⊂ Zn is the image domain and I(t) assigns a set of m scalars Ii (t), i = 1, 2, ..., m to each pixel t ∈ I. For a given image Iˆ = I, I, an adjacency relation in ˆ Thus, in I deﬁnes a directed graph G = V, A, ω derivated in the image I. IFT a 2D/3D image can be interpreted as a directed graph G, in which the vertices V are the pixels/voxels of the image in its domain I ⊂ Zn , the arcs are the ordered pairs of pixels s, t ∈ A ⊂ I × I and each arc s, t ∈ A has an associated weight ω(s, t) ≥ 0, which can be given by a dissimilarity measure between the pixels s and t, as well as each vertex V ∈ I may have an associated weight ω(t) = ωv (t) ≥ 0. The set of adjacent nodes to s is denoted by N (s) = {t ∈ V : s, t ∈ A}. The Oriented Image Foresting Transform (OIFT) [7,16] is built on a dual version of the IFT framework considering that the arc weights ωst represent a directed aﬃnity function, given by the power of complement of a combination

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of an undirected dissimilarity function κst between neighboring pixels s and t, multiplied by an orientation factor as ⎧ ⎨ [K − κst × (1 + α)]β , if I(s) > I(t) ωst = [K − κst × (1 − α)]β , if I(s) < I(t) (1) ⎩ [K − κst ]β otherwise . such that α ∈ [−1, 1] and K is the maximum weight value used for the complement calculation. Diﬀerent procedures can be adopted for κst , such as using the absolute value of the diﬀerence in the intensities of the image (that is, κst = |I(t) − I(s)|). With α > 0, the OIFT segmentation favors transitions from light to dark pixels and for α < 0 favors the opposite orientation. The parameter β > 0 does not aﬀect OIFT, but will be important for the ROIFT component. 2.4

Relaxed OIFT

The ROIFT algorithm [2] is a hybrid method between the OIFT and the Random Walks (RW) extension for directed graphs [19], producing segmentation results in more smooth and intuitive ways. As proposed by Demario and Miranda [2], extending the work of Malmberg et al. [6], for an initial computed segmentation L0 , a sequence of fuzzy segmentations L1 , L2 , ..., LN by iterative relaxation is obtained, as i i t∈N (s) W (s,t)·L (t) , if s ∈ So ∪ Sb i+1 i L (s) = (2) t∈N (s) W (s,t) i L (s) otherwise ⎧ ⎨ [K − κst ]β if Li (s) = Li (t) i W (s, t) = ωst if Li (s) > Li (t) ⎩ if Li (s) < Li (t) ωts

(3)

such that So ⊂ V and Sb ⊂ V are two seed sets, in object/background segmentation, respectively. The ﬁnal crisp L is obtained, by assigning L(s) = 1 to all s ∈ V with LN (s) ≥ 0.5 and L(s) = 0 otherwise. This solution converges to RW for suﬃciently high values of N , but the idea here is to consider a lower amount of relaxation to get a hybrid result, so that N controls the smoothness of the segmentation boundary, thus improving the results in the presence of noise and weak edges.

3

Proposed Automatic Seed Generation Algorithm

In this section, seed segmentation is presented in a more intuitive and visual way, and then the algorithm methodology for use in lung segmentation will be presented.

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Seed Generation

The IFT/OIFT algorithm is initialized from background and object seeds. We can see in Fig. 3, that through the seeds, the algorithm searches for regions of the image that belong to the same object as the speciﬁed seed. For example, if the algorithm receives a seed of the background in an iteration, it will travel around the surroundings of that seed, guiding itself through similar regions to classify them as also being regions belonging to the background of the image.

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Fig. 3. (a) An image with two seed sets (markers), where the internal seed set So = {r1 , r2 } (internal markers) represents the object and the external seed set Sb = {r3 , r4 } (external markers) represents the background. (b) The segmentation result showing two forests of optimal paths rooted in the inner and outer seeds identifying the object and background respectively. The ﬁgure was taken from [8].

3.2

Algorithms for Seed Generation

Python libraries Numpy and Numpy-stl were used to develop the algorithm, which is useful for manipulating arrays and meshes. The seeds are represented as a list of three numbers that indicate the coordinates on the x, y, and z axes. The outer seeds are located in the center and on the side of the lung, while the internal seeds were determined as a set of seeds that form a sphere to obtain better segmentation accuracy, as shown in Fig. 4. Initially, it is important to point out that the STL mesh resulting from the watershed algorithm returns inverted on the z-axis, which can bring seeds of wrong coordinates. In this way, the lung rotated 180o with respect to the z-axis of origin and shifted to its original position. The hypothesis that the center of mass of the lung is on its external side is considered to calculate the coordinates of the ﬁrst external seed. Therefore, the ﬁrst outer seed (seed 1) was determined by calculating the center mass of the mesh, as shown in the line 4 of the simpliﬁed algorithm below. with the addition of a constant C to move the seed downwards, as some unexpected events may occur, such as having the seed inside the lung. The second outer seed (seed 2) was then found by calculating the value on the x axis of the side of the lung, a constant so that it is not so far from the lung (line 8). For the values y and z, the center-of-mass values were reused.

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Fig. 4. The relative position of the seeds in the lung. The yellow dots represent the set of internal seeds (seeds 3 and 4), while the external ones (seeds 1 and 2) are represented by pink dot.

The center of mass calculation was also used for each lung to determine internal seeds (seeds 3 and 4). To split the lung, it was necessary to center it and then ﬁnd the voxel coordinate closest to the origin to split it perfectly (line 7). After that, the single mesh was separated into two, they were returned to their original positions, and the seeds were calculated as the center of mass of both meshes (lines 9–12). Finally, to calculate the adjacent seeds, constant values were added until forming a seed sphere (lines 14–15).

Algorithm 1. Proposed methodology to determine the four seeds for the lung 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16:

Input: STL ﬁle CenterOfGravity ← lung.getMassProperties() Centralizes the STL using the current Center Of Gravity Seed1[x, y, z - C] ← CenterOfGravity Find the vectors (Three-dimensional array) from the lung Divide the lung in two, using the vectors and the CenterOfGravity properties rightlung, leftlung ← lung.divide() Seed2[x, y, z] ← [rightlung.max(x)-C, rightlung.mean(y), rightlung.mean(z)] RightCenterOfGravity ← rightlung.getMassProperties() LeftCenterOfGravity ← leftlung.getMassProperties() Seed3[x, y, z] ← RightCenterOfGravity Seed4[x, y, z] ← LeftCenterOfGravity for i in range(seedsinsphere): AdjSeeds3[x, y, z] ← Seed3[x+1, y+1, z+1] or Seed3[x-1, y-1, z-1] AdjSeeds4[x, y, z] ← Seed4[x+1, y+1, z+1] or Seed4[x-1, y-1, z-1] return Seed1, Seed2, Seed3, Seed4, AdjSeeds3, AdjSeeds4

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Results

A dataset of DICOM images of a total of 100 CT images of the human chest was used for segmentation, taken from the Data Science Bowl 2017 competition. Unfortunately, the data are currently not available for access. In these images, 6 were not possible for segmentation via the watershed. Thus, the seed generation algorithm only created seeds with viable coordinates in 94 images. Among all of them, the ROIFT algorithm was able to perfectly segment and convert to STL ﬁles. However, of the 94 images, 4 meshes had problems segmenting only one side of the lung, as shown in Fig. 5. The problem was due to the seed generation program, which will be corrected in future research.

Fig. 5. Failed segmented meshes

Compared to the watershed, the ROIFT algorithm was superior in fundamental aspects: it was able to correct for noise that appeared in unexpected regions of the lung in the watershed, ﬁling incomplete regions in the lower region of the lung, as shown in Fig. 6. ROIFT performed better in terms of processing time. However, if the size of the ﬁle is something to consider, you can increase the number of iterations of fuzzy segmentation, increasing the running time, and decreasing the ﬁle size. The Table 1 shows the details of each procedure. Two ROIFT segmentations were performed, one with 5 iterations and another with 25, to show the diﬀerence in the ﬁle size and processing time of the segmentations. Visually, one might notice a diﬀerence in mesh smoothing, as shown in Fig. 7.

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Fig. 6. (a) Watershed lung segmentation. (b) ROIFT lung segmentation.

Table 1. Algorithm comparison in 100 DICOM images. SA = Segmentation Algorithm. ROIFT 5 it = ROIFT with 5 iterations. ROIFT 25 it = ROIFT with 25 iterations. RT = running time. The accuracy represents the value in percentage of correctly segmented lungs divided by the total segmented images. SA Watershed

RT 3,893.56 s

File size 7.1 GB

Accuracy 94%

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Fig. 7. (a) Segmentation with 5 iterations. (b) Segmentation with 25 iterations.

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Fig. 8. (a) Watershed rib segmentation. (b) ROIFT rib segmentation.

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Fig. 9. (a) Front view of skin segmentation. (b) Side and rear representation of skin.

5

Conclusions and Future Work

In this work, we developed algorithms for automatic seed generation and used the ROIFT algorithm to segment the lungs in the thorax region. The method used in this paper showed better results in terms of segmentation time and noise reduction compared to the watershed segmentation algorithm. Automatic seed generation methods are in the process of being developed for the rib and for the skin. When testing with seeds manually, ROIFT has already presented signiﬁcant results. Figure 8 shows the diﬀerences between the two rib segmentation methods. Skin segmentation was not possible using the watershed. However, it is feasible to use ROIFT for the procedure, as shown in Fig. 9. Other regions of the chest will still be automatically segmented via ROIFT, such as: ribs, airways, skin, and heart. In this way, alternative ways of generating seeds will be developed.

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References 1. Chernyaev, E.V.: Marching cubes 33: Construction of topologically correct isosurfaces. Technical report (1995) 2. Demario, C.L., Miranda, P.A.V.: Relaxed oriented image foresting transform for seeded image segmentation. In: IEEE ICIP, pp. 1520–1524 (2019) 3. Falcao, A., Stolﬁ, J., de Alencar Lotufo, R.: The image foresting transform: theory, algorithms, and applications. IEEE T Patt. Anal. 26(1), 19–29 (2004) 4. Iwao, Y., Gotoh, T., Kagei, S., Iwasawa, T., Tsuzuki, M.S.G.: Integrated lung ﬁeld segmentation of injured regions and anatomical structures from chest CT images. In: IFAC Proceedings Volumes (IFAC-PapersOnline), vol. 45, pp. 85–90 (2012) 5. Iwao, Y., Gotoh, T., Kagei, S., Iwasawa, T., Tsuzuki, M.S.G.: Integrated lung ﬁeld segmentation of injured region with anatomical structure analysis by failurerecovery algorithm from chest CT images. Biomed Signal Process. 12, 28–38 (2014) 6. Malmberg, F., Nystr¨ om, I., Mehnert, A., Engstrom, C., Bengtsson, E.: Relaxed image foresting transforms for interactive volume image segmentation. In: Proceedings of the SPIE: Progress in Biomedical Optics and Imaging, vol. 7623, pp. 1–11. SPIE, United States (2010) 7. Mansilla, L.A.C., Miranda, P.A.: Image segmentation by oriented image foresting transform: handling ties and colored images. In: 18th International Conference on DSP, pp. 1–6 (2013) 8. Mansilla, L.A.C.: Segmenta¸ca ˜o de Objetos via Transformada Imagem-Floresta Orientada com Restri¸co ˜es de Conexidade. Ph.D. thesis, Instituto de Matem´ atica e Estat´ıstica da USP, S˜ ao Paulo, Brasil (2018) 9. Martins, T.C., Tsuzuki, M.S.G.: Simulated annealing with partial evaluation of objective function applied to electrical impedance tomography. In: IFAC Proceedings Volumes, vol. 44(1), pp. 4989–4994 (2011). 18th IFAC WC 10. Martins, T.C., Tsuzuki, M.S.G.: Electrical impedance tomography reconstruction through simulated annealing with total least square error as objective function. In: 34th IEEE EMBC, pp. 1518–1521, San Diego, USA (2012) 11. Martins, T.C., Tsuzuki, M.S.G.: Electrical impedance tomography reconstruction through simulated annealing with multi-stage partially evaluated objective functions. In: 35th IEEE EMBC, pp. 6425–6428 (2013) 12. Martins, T.C., Tsuzuki, M.S.G.: Investigating anisotropic EIT with simulated annealing. IFAC-PapersOnLine 50(1), 9961–9966 (2017). 20th IFAC WC 13. Martins, T.C., Tsuzuki, M.S.G.: EIT image regularization by a new multi-objective simulated annealing algorithm. In: 37th IEEE EMBC, pp. 4069–4072, Milan, Italy (2015) 14. Martins, T.C., Fernandes, A.V., Tsuzuki, M.S.G.: Image reconstruction by electrical impedance tomography using multi-objective simulated annealing. In: 11th IEEE ISBI, pp. 185–188, Beijing, China (2014) 15. Martins, T.C., et al.: A review of electrical impedance tomography in lung applications: theory and algorithms for absolute images. Annu. Rev. Control 48, 442–471 (2019) 16. Miranda, P.A.V., Mansilla, L.A.C.: Oriented image foresting transform segmentation by seed competition. IEEE T Image Process 23(1), 389–398 (2014) 17. Olabarriaga, S.D., Smeulders, A.W.: Interaction in the segmentation of medical images: a survey. Med. Image Anal. 5(2), 127–142 (2001) 18. Sato, A.K., et al.: Registration of temporal sequences of coronal and sagittal MR images through respiratory patterns. Biomed Signal Process. 6, 34–47 (2011)

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19. Singaraju, D., Grady, L., Vidal, R.: Interactive image segmentation via minimization of quadratic energies on directed graphs. In: International Conference on CVPR, pp. 1–8 (2008) 20. Tavares, R.S., Martins, T.C., Tsuzuki, M.S.G.: Electrical impedance tomography reconstruction through simulated annealing using a new outside-in heuristic and GPU parallelization. J. Phys. Conf. Ser. 407, 012015 (2012) 21. Tavares, R.S., et al.: Temporal segmentation of lung region from MRI sequences using multiple active contours. In: 32nd IEEE EMBC, pp. 7985–7988, Buenos Aires, Argentina (2011) 22. Tavares, R.S., Sato, A.K., Tsuzuki, M.S.G., Gotoh, T., Kagei, S., Iwasawa, T.: Temporal segmentation of lung region MR image sequences using Hough transform. In: 32nd IEEE EMBC, pp. 4789–4792, Buenos Aires, Argentina (2010) 23. Tavares, R.S., Tsuzuki, M.S.G., Gotoh, T., Kagei, S., Iwasawa, T.: Lung movement determination in temporal sequences of MR images using hough transform and interval arithmetics. In: Proceedings of the 7th IFAC Symposium on Modelling and Control in Biomedical Systems, pp. 192–197, Alborg, Denmark (2009) 24. Tsuzuki, M.S.G., et al.: Animated solid model of the lung constructed from unsynchronized MR sequential images. CAD 41, 573–585 (8 2009) 25. Tsuzuki, M.S.G., et al.: Propagation-based marching cubes algorithm using open boundary loop. Vis. Comput. 34(10), 1339–1355 (2018) 26. Vincent, L., Soille, P.: Watersheds in digital spaces: an eﬃcient algorithm based on immersion simulations. IEEE T Patt. Anal. 13(6), 583–598 (1991)

A Pre-processing Tool for Particle-Based Fluid Dynamics Simulations Cezar Augusto Bellezi1,2

, Liang-Yee Cheng1,2

, and Lucas Soares Pereira1,2(B)

1 Department of Construction Engineering, University of São Paulo, São Paulo, Brazil

[email protected] 2 Numerical Offshore Tank Laboratory, University of São Paulo, São Paulo, Brazil

Abstract. A pre-processing tool for fully-Lagrangian particle-based computational fluid dynamics methods is presented in this work. It adopts relatively simple strategies to generate particle models of the elements of the computational domain from boundaries represented by triangular meshes. Essentially, it is based on identification of the relative positions between cubic lattice nodes and the meshes to create particles that represent the geometry of the elements and assign the material type of the particles that are required for the simulations. Focusing on the recognition of the lattice nodes inside closed surfaces, variants of ray-casting and rasterization algorithms were proposed and evaluated. As a result, the rasterization considering three different orthogonal reference planes is the most accurate one with relatively low computational cost. Examples of the successful applications of the pre-processing tool are also provided. Keywords: Pre-processing · Particle-based method · CFD

1 Introduction Fully-Lagrangian particle-based methods are promising Computational Fluid Dynamics (CFD) techniques for the investigation of highly nonlinear hydrodynamic phenomena, such as impulsive hydrodynamic impact [1–3], multiphase flows and fluid-structure interaction [4–6]. Among the existing approaches, smoothed particle hydrodynamics (SPH) method [7] and moving particle semi-implicit (MPS) method [8] are the most used ones. In such techniques, the computational domain is discretized by particles, which allows the modeling of very complex geometries without the toilsome mesh generation process of traditional mesh-based CFD techniques. However, as inputs for a particle-based simulation, all the elements present in the domain must be modeled by particles. Despite much easier than mesh-generation, the creation of the particle model could be a challenging process in large-scale high-resolution simulations dealing with tens or hundreds of millions of particles. Thus, a simple, effective, and efficient technique for the generation of the initial particle grid should be developed in order to take full advantage of the particle-based CFD techniques. Domínguez et al. [9] developed a pre-processing software for SPH method based on regular particle distribution and using either predefined geometrical shapes or CAD files © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 648–660, 2023. https://doi.org/10.1007/978-3-031-13588-0_56

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formats as inputs. On the other hand, non-uniform spacing particle distributions based on a volume element mesh, the Voronoi tessellation or the centroidal Voronoi tessellation and the body-fitted scheme have also been proposed [10–12]. Since computer-based surface and solid modeling tools are widely adopted in engineering design, a pre-processing tool for MPS method that converts the output of CAD systems to a particle grid is presented in the current work. The pre-processing tool contains a set of functions based on relatively simple algorithms to generate a particle grid from the surfaces and solids represented by triangular meshes. For sake of simplicity, the Stereolithography (STL) file format was chosen as input format. Since uniform particle distribution is required for MPS and most of the particle-based CFD methods, the simplest approach is to initially consider a square lattice, for 2D models, or a cubic lattice, for 3D models, and identify lattice nodes that best describes a given surface or solid. In the case of solids, the algorithm should distinguish between particles inside and outside the solid, as well as recognizing and setting particle types such as dummy particles required for the modeling of the solid boundaries. The pre-processing tool was developed focusing on MPS method, but the concept can be extended and applied to other particle-based methods. In what follows, a brief description of the MPS method is given. After that, the pre-processing tool is shown in detail. Focusing on identification of lattice nodes inside closed surfaces, variants of ray-casting and rasterization algorithms were proposed and evaluated. Finally, applications of the tool are presented.

2 Moving Particle Semi-implicit The moving particle semi-implicit (MPS) is a meshless method that discretizes the computational domain in particles, which are Lagragian observation points. Originally proposed to simulate incompressible free surface flows, it solves the governing equations of the continuum replacing the differential operators by discrete differential operators on irregular nodes using a weight function, which defines the contribution of neighboring particles within an effective radius (re ) [8]. The MPS method adopts a semi-implicit algorithm. In each time step, for all fluid particles, velocity, position, and particle number density, which is the summation of the weights of all the particles inside the neighborhood, are estimated explicitly considering the external forces and viscous terms. After that, a system of Poisson equation for pressure (PPE) is solved implicitly, and correction of velocity and position of fluid particles are carried out considering the pressure gradient term. As the particle number density is proportional to fluid density, generally it is used as a reference to ensure incompressibility in the implicit calculation. The basic particle types of the MPS are showed in Fig. 1. The fluid domain is represented by fluid particles, shown in light blue, and the blue particles represent the free surface, a subset of fluid particles that can be detected using techniques based on neighborhood incompleteness criterion, such as neighborhood particles centroid deviation [13] and others. As Dirichlet dynamic boundary condition, the pressure of the freesurface particles is set to zero. The solid walls are usually modeled using three layers of particles. The external one in contact with fluid is formed by wall particles (pink), and their pressures are calculated together with the fluid particles when solving the PPE. The other two layers are formed by dummies particles (grey) to ensure the neighborhood completeness and correct calculation of the particle number density of wall particles.

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Fig. 1. Basic particle types of the MPS method.

In this way, the essential requirements of the pre-processing task are the generation of uniformly distributed particles with correct assignment of material ID (particle types) of all the particles: fluid, free surface, wall, dummies, and other that might be introduced according to modeling demands. Since the free surface deforms continuously, free surface particles are detected at every time step throughout the simulation. Thus, it is not necessary to detect free surface particles during the pre-processing.

3 Pre-processing Routines The objective of the pre-processing tool developed herein is to generate, from surfaces or solids represented by triangular meshes, an initial particle grid to be used as input data for the fully-Lagrangian particle-based CFD simulations. For sake of simplicity, the STL format is adopted as input file format because it is a commonly used one and able to represent complex surfaces in a relatively simple way. In summary, it adopts the face-vertex mesh data structure, which contains a list of the coordinates of the vertex of the mesh in a Nv × 3 matrix, where Nv is the number of vertices in the mesh, and a list that consists of the indexes of the vertices of each triangular face in a Nt × 3 matrix, where Nt is the number of triangular faces of the mesh. The initial particle grid consists of set of particles that model the elements of the computational domain. Beside representing the initial geometry of the elements (initial position of the particles), it also contains the material ID of the particles. To assure uniform particle distribution, a simple way is to generate the particle grid based on a regular cubic lattice of which the data structure could only contain a nx × ny × nz 3D matrix of integer numbers that stores the material ID associated to each particle. The lattice spacing is then equal to initial distance between particles (l0 ). The pre-processing tasks are carried out in two stages. First, it identifies the position of the lattice nodes in relation to the meshes, i.e., inside, outside, above, below, or close, etc. After that, correct material IDs are assigned to the lattice nodes identified as particles. 3.1 Particle Grid from Solids Represented by a Closed Surface Mesh The pre-processing routine that generates particle grids from a closed surface is essentially based on the verification of whether a lattice node is inside or outside the region delimited by the surface. This routine can be used to create particle model of solids, such as floating or submerged bodies, or to model liquid containers.

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Two algorithms to verify if a lattice node is inside a given closed surface were considered and analyzed in the present work. The first one, herein called “ray-casting”, is based on the number of intersections of rays casted from each node [14]. Figure 2 depicts the concept of the “ray casting” algorithm in 2D. The second algorithm, herein called “rasterization”, is based on the coordinates of the intersection between the closed surface and rays, which are lattice lines, casted from lattices nodes outside the surface. Figure 3 depicts the concept of the “rasterization” algorithm in 2D.

Fig. 2. Conceptual sketch of the ray-casting algorithm for a two-dimensional geometry.

Ray-Casting Algorithm (RCT). In this algorithm, a ray is casted from the position of a given lattice node to an arbitrary direction. Then, the total amount of intersections between the ray and the mesh are counted. The calculation of the intersection between the ray and a triangle of the mesh consists of two steps: first, the intersection point between ray and the plane that contains the triangle is obtained and, second, the barycentric coordinates of the intersection point are calculated to verify if the intersection lies inside the perimeter of the triangle. A lattice node is considered inside a closed mesh if the ray has an odd number of intersections with all the triangles of the mesh; otherwise, it is located outside. For instance, in Fig. 2, the red lattice node is outside the mesh because the ray casted from it intersects the mesh twice, while the green node is considered inside because the ray casted from it intersects only once. In the algorithm, a total of nr = nx × ny × nz rays should be casted, for each ijk-th node of the lattice, in which nr is the number of rays casted. Such algorithm is suitable for convex and non-convex closed surfaces. However, when the ray intercepts the edges or vertices of the mesh, misdetection may occur. This is because in a closed surface every edge is shared by two triangles so that a single intersection between the ray and the edge of the mesh is counted twice. The case of the vertices is even more challenging because the number of triangles that share a single vertex is unknown. To deal with these situations, three different approaches were considered in the ray casting routine and evaluated in the present work. • Ray-casting with correction factor for edges and vertices (RCT-C). In this algorithm, an initial check of the input mesh is carried out to identify the number of triangles (nm ) associated to a given m-th vertex, for all the vertices of the mesh. From the barycentric coordinates of the intersection point between ray and mesh it is verified if the ray intersects an edge or a vertex. Then, after the calculation of all the intersections of the ray from a given lattice node is done, a correction factor is subtracted from the total amount of intersections in the case of edges (−1 for each edge) or vertices (−(nm − 1) for a given vertex).

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• Ray-casting with unique intersection points (RCT-U). In this algorithm, the positions of the intersections between ray and mesh are stored. After determining the intersections of a ray to all the triangles of the mesh, the distances between the intersection points are calculated. If a distance is shorter than a threshold value (l0 was adopted herein), one of the intersections is not counted in the total amount. • Ray-casting with three rays (RCT-3). In this algorithm, instead of a single ray, two rays are casted in different directions from each lattice node. In the case the outcomes based on these two rays disagree (i.e., one has an odd number while the other has an even number), a third ray is casted in a different direction to define the number of intersections between ray and mesh.

Fig. 3. Conceptual sketch of the rasterization algorithm for a two-dimensional geometry.

Rasterization Algorithm (RST). In this algorithm, the coordinate axes aligned bounding box of the closed mesh is considered. A face of the box (parallel to XY , YZ or ZX ) is used as reference plane. Rays are casted from the lattice nodes of the reference plane, which are known points outside the mesh, in direction normal to the plane to intersect the mesh and the opposite face of the box, which is also outside the mesh. The intersection between the rays and the mesh are obtained. In the case of a convex closed surface, among the intersection points, the maximum and minimum coordinates normal to the reference plane are obtained. The lattices nodes between such minimum and maximum coordinates are considered inside the closed surface. In case of a non-convex closed surface, the coordinates of all the intersection points between the rays normal to the reference plane and the mesh are stored in an array. Then, the intersection points are sorted in ascending order in the ray direction. By considering an array with an even number of N intersection points, the lattice nodes between the coordinates of the n-th and the (n + 1)-th element, for n = {1, 3, . . . , (N − 1)}, are considered inside the closed surface. Otherwise, the nodes between the coordinates of the n-th and the (n + 1)-th element, for n = {2, 4, . . . , (N − 2))}, are considered outside the closed surface. The algorithm for non-convex closed surfaces is also suitable for the case of multiple non-intersecting closed surfaces, which may contain convex or non-convex closed surfaces. In the 2D example of Fig. 3, the green nodes, which are between the lowest and the highest intersection points of the vertical ray and mesh, are considered inside the mesh. For 3D cases, this routine cast a total of nr = nx × ny rays if the reference plane is XY plane. By casting rays for 1/nz the number of lattice nodes, the rasterization algorithm should have a substantially lower processing cost than the ray casting algorithm.

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In the case of convex body surfaces, there are no problem when the rays intercept the edges or vertices of the mesh because only the minimum and maximum values of the coordinates are considered. On the other hand, for non-convex body surfaces, intersections between ray and edges or vertices are counted only once. So, after sorting the intersection points, the distances between successive intersection points are calculated. If a distance is smaller than a threshold value (l0 was adopted herein), the pair of close intersection points are removed from the list. Rasterization with Three Reference Planes (RST-3). Most of the problems occur when a ray intercepts an edge or a vertex of the mesh without penetrating the region enclosed by the surface. To overcome this issue, rasterization using three orthogonal reference planes (XY , YZ and ZX ) is considered. In this case, a node is inside the closed surface if it is identified as located internally the surface by at least two of those planes.

3.2 Particle Grid from Open Surface As shown in Fig. 4, in case of an open surface represented by a triangle mesh, two situations can be considered: it models a thin shell element or the boundary of a domain.

(a)

(b)

(c)

Fig. 4. Example of particle grids created from an open surface: (a) input mesh of the surface, (b) output particle grid of a thin shell and (c) a domain bounded by the surface.

In the first case (Fig. 4-b), a lattice node is considered to belonging to the thin shell element if it is closer than a threshold distance to a given triangle, edge, or vertex of the mesh. To assure correct modeling of the flow around the thin shell, the thickness of the thin shells model must be the effective radius of the particle-based CFD method plus one particle (re + l0 ). At first, the distances of a lattice node to all the face, vertices and edges of the mesh are calculated. Then, the distance of the node to the surface is the minimum of such values. In the second case (Fig. 4-c), the intersection points between the mesh and the lines of the lattice are calculated and the lattice nodes are identified as above or below the surface and different material IDs are assigned to each side. 3.3 Particles from Lattice Nodes and Assignment of the Material Type After the position of all the lattice nodes are classified as internal, external, etc. in relation to the regions delimited by the surfaces, the pre-processing tool proceeds to assign the material IDs to the nodes to generate the particle grids.

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In case of fluid, all the lattice nodes inside a fluid domain are registered as a fluid particle and assigned a fluid ID. In case of multi-fluid studies, such as water-oil simulations, the fluid domains are defined separately and associated to distinct material IDs. In case of solid bodies, to detect the wall particles, which are in the external layer of the lattice nodes inside the solid, a neighborhood incompleteness criterion can be adopted. Generally, the simplest criterion based on particle number densities, such as the original free-surface detection technique [8], is sufficient. Once detected all the wall particles, the dummy particles are created from all lattice nodes inside the solid and within the effective radius from the wall particles. The remained lattice nodes inside the solid are not necessary for the simulation and are ignored. As the fluids, each solid has its own ID. In case of the thins shells, the lattice nodes within the thin shell are processed like the solids. Boolean operations can also be applied in case of several input surfaces that delimited solid or fluid domains. Generally, the accuracy of the modeling is improved when the lattice of a solid is aligned to its main axes. However, in many fluid-structure interaction cases, the solid’s main axes are not aligned to the global coordinate system, which is usually a reference for the alignment of the fluid domain lattice. Therefore, solid particles grids are normally generated separately following their respective local coordinate axes and rotated to align to the global system to be assembled to the fluid particle grids. Finally, the fluid particles that overlaps the solids are detected based on its distance to any wall particle. If the distance is less than the lattice spacing, the fluid particle is deleted, otherwise if the nearest solid particle is a wall particle, the fluid particle is external to the solid and stay unchanged. If it is a dummy one, the fluid particle is inside the solid and it is deleted.

4 Results In this section, the performance of the pre-processing algorithms and their variants (Sect. 3.1) are evaluated. The solid used in the analysis is a unit sphere (R = 1.0), which is approximated by icosphere (geodesic polyhedron). Meshes of icospheres with number of triangular faces Nt = {20, 80, 320, 1280} were considered. Three different values of distance between particle were considered l0 = {0.10, 0.05, 0.025} to investigate the effect of the resolution of the particle grid on the performance of the routines, which yield cubic lattices with Np = 8.0 · 103 , 6.4 · 104 , 5.12 · 105 nodes, respectively. 4.1 Effectiveness In order to check the effectiveness of the algorithm to generate a particle grid from a closed mesh, the particle grids obtained by using the different algorithms are compared to error free grids that are used as references. The four ray-casting routines were evaluated: three algorithms associated to a single ray in the Z direction (RCT-Z, RCT-U-Z and RCT-C-Z) and the ray-casting with three rays (RCT-3). Two cases of the rasterization were evaluated as well: the rasterization with a single reference plane parallel to XY plane (RST-Z) and the rasterization with three planes (RST-3).

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Table 1. Ratio of misdetected nodes of the ray-casting and rasterization algorithms (in %).

20

40

80

20 80 320 1280 20 80 320 1280 20 80 320 1280

RCT-Z 0.1 0.8 0.2 0.0 0.2 1.3 0.7 0.7 0.1 0.9 0.4 0.4

RCT-U-Z 1.4 0.5 0.1 0.1 0.4 0.4 0.3 0.4 0.2 0.4 0.2 0.2

RCT-C-Z 1.4 0.7 0.1 0.1 0.7 0.6 0.4 0.4 0.3 0.5 0.2 0.3

RCT-3 0.0 0.1 0.0 0.0 0.0 0.2 0.2 0.2 0.0 0.1 0.1 0.1

RST-Z 0.0 0.1 0.0 0.0 1.5 0.1 0.2 0.2 4.6 0.3 0.1 0.1

RST-3 0.0 0.1 0.0 0.0 0.0 0.1 0.1 0.1 0.0 0.1 0.1 0.1

To evaluate the effectiveness, the ratio between the amount of misdetected nodes and the total lattice nodes inside the solid (NP ) is considered. The misdetected nodes comprise both false positives and false negatives. The false positives are nodes outside the mesh wrongly identified as inner nodes and the false negatives are inner nodes wrongly identified as outside nodes. Table 1 shows the ratios of misdetection. For sake of clarity, the cases in which the ratio of misdetected nodes does not exceed an arbitrary threshold of 0.5% were highlighted (green background). Based on such threshold, the ray-casting algorithm with three rays (RCT-3) and rasterization with three planes (RST-3) are more effective ones. Overall, the RST-3 leads to the best results. On the other hand, the ray-casting with a single ray (RCT-Z) was the worst algorithm.

(a)

(b)

(c)

Fig. 5. Particle grids with the nodes inside the icosphere with Nt = 320 triangles and particle resolution ratio of R/l0 = 40 – (a) RCT-X, (b) RCT-3 and (c) RST-3.

Figure 5 shows some examples the particle grids generated from an icosphere with Nt = 320 triangles and particle resolution ratio of R/l0 = 40 by using different algorithms. In Fig. 5-(a), a relatively larger number of misdetections along the ray direction occurs when using ray-casting with a single ray (RCT-X). On the other hand, misdetected particles did not show a correlation with any direction in the rasterization in three planes (RST-3) in Fig. 5-(c).

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4.2 Processing Time For each combination of mesh resolution (Nt ) and particle grid resolution (l0 ), ten particle grids were generated using each algorithm. The hardware used was an Acer Aspire VX5 notebook, with an Intel Core i5 processor with 2.50 GHz, 8.0 GB of RAM and equipped with a NVIDIA GeForce GTX 1050 video card with 4.0 GB of VRAM. The average processing times, in seconds, are presented in Table 2. Table 2. Average processing time of different algorithms (in seconds). R/l0 20

40

80

Nt

RCT-Z

RCT-U-Z

RCT-C-Z

RCT-3

RST-Z

20

0.25

0.32

0.35

1.05

0.06

RST-3 0.19

80

1.10

1.23

1.56

5.01

0.15

0.43

320

4.39

4.74

6.19

19.85

0.49

1.35

1280

17.45

18.38

25.28

79.00

1.83

4.92

20

1.22

1.49

1.43

4.59

0.13

0.52

80

8.35

8.78

9.12

27.25

0.29

0.96

320

33.21

33.84

36.19

108.63

0.82

2.33

1280

132.76

134.76

145.47

432.48

2.92

7.77

20

8.12

9.62

8.90

25.96

0.43

1.83

80

46.89

49.59

50.81

150.73

0.77

3.11

320

186.37

191.09

197.66

594.29

1.84

6.50

1280

736.83

748.61

780.84

2366.43

6.00

20.02

Overall, the faster algorithm is the rasterization in a single plane (RST) for all the combinations of mesh and lattice resolutions. In second place, it is the rasterization in three planes (RST-3), which, as expected, required roughly 3 times the duration of the rasterization in one plane. In general, the processing times of the ray-casting algorithms are substantially higher than those of rasterizations. As the resolution of the mesh and the lattice increase, the ratio between the processing time of the ray-casting and the rasterization (TRCT /TRST ) tends to increase. 4.3 Applications The pre-processing tool was applied in several studies carried out using an in-house developed simulation system based on the MPS method, and three examples are shown in this section. The first one is the simulation of the sudden failure of the Córrego do Feijão dam and its flooded area on 25 January 2019. The pre-processing tool was applied to generate the 3D particle model of the surrounded area of the Córrego do Feijão mine tailings [15] based on the data from Shuttle Radar Topography Mission (SRTM) digital elevation model [16]. The topographic surface was modeled as a solid wall, and the

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particles of fluid were assigned to the lattice nodes inside the domain bounded by a boxshaped closed surface and above the topography surface. Figure 6 provides the mesh, the particle model close to the Córrego do Feijão dam and snapshot of the simulation. The terrain consists of 46216 triangular elements with size of approx. 40 m. The distance between particles is 5 m resulting in 101822 particles to represent the slurry. Figure 6c shows a snapshot of the simulation. After flowing more than 9 km downhill, the tailings wave reached the Paraopeba River.

Fig. 6. Córrego do Feijão mine tailings modeled as (a) triangular mesh, (b) particle model and (c) snapshot of simulation after 2.5 h of the dam failure. The color scale represents elevation of the terrain.

The pre-processing tool was also applied to model the SPB tank used in the experiments to investigate the effects of internal structures on sloshing loads [3, 17]. The internal structures were modeled as thin shell elements described by open surfaces. The particle grid of the tank walls was created from the lattice nodes outside the box that represents the tank walls, and the fluid particles were created using the lattice nodes of domain inside the tank walls and below a surface that represents the fluid free surface. Figure 7 presents the mesh, particle model and snapshot of the simulation. The mesh model of the SPB tank is composed by 165 elements and the particle model has approx. 1.7 million particles with distance of 12.5 cm.

Fig. 7. SPB tank modeled as (a) triangular mesh, (b) particles and (c) snapshot of the simulation at instant t = 18.0 s. The wall and dummy particles are represented by yellow and gray colors, respectively. The color scale represents the pressure magnitude. l0 = 12.5 cm.

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Aiming to investigate the green water effects on a large FLNG platform [18], a particle grid was created using the pre-processing tool. The particle grids of hull and the topside structures were generated separately from closed surfaces. The fluid particle grid was obtained from the intersection between the lattice nodes outside the surfaces of the hull, within the box-shaped computational domain and below the free surface. The mesh, particle model of the FLNG and a snapshot of the simulation are presented in Fig. 8. The mesh of the FLNG is composed by 65856 elements while the particle model has 135436 particles with distance of 2 m. The snapshot shows the results in head seas with maximum wave height for the centenary wave of the worst weather condition of the Santos Basin, Brazil, and maximum operational draught condition of the FLNG.

Fig. 8. FLNG modeled as (a) triangular mesh, (b) section view of the particles and (c) snapshot of the simulation at t = 114.0 s. The wall and dummy particles are represented by yellow and gray colors, respectively. The color scale represents the velocity magnitude. l0 = 2.0 m.

5 Concluding Remarks A pre-processing tool for fully-Lagrangian particle-based CFD simulations was developed in the present work. It generates the initial particle grid from STL files based on relatively simple strategies that identify lattice nodes inside/outside a mesh. Boolean operations can also be applied in case of several input surfaces that delimit solid or fluid domains. Additionally, it also assigns the material IDs to correctly set the types of the particles, which are required to define the proprieties and dynamics to be used in the simulations. Focusing on the identification of the lattice nodes inside a closed surface, ray-casting and the rasterization algorithms were evaluated, as well as their variants. As a result, the most effective algorithm was the rasterization with three planes (RST3). The fastest algorithm was the rasterization with a single plane (RST), followed by the rasterization with three planes (RST-3). The ray-casting algorithm and its variants required substantially higher processing times than the rasterization. In general, by considering both effectiveness and efficiency, the rasterization with three planes (RST-3) is considered the best algorithm for the purpose. Acknowledgments. This work was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001, which provided the doctor degree scholarships to the first author.

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References 1. Bellezi, C.A., Cheng, L.-Y., Okada, T., Arai, M.: Optimized perforated bulkhead for sloshing mitigation and control. Ocean Eng. 187, 106–171 (2019). https://doi.org/10.1016/j.oceaneng. 2019.106171 2. Amaro, R.A., Jr., Cheng, L.-Y., Vieira Rosa, S.: Numerical study on performance of perforated breakwater for green water. J. Waterw. Port Coast. Ocean Eng. 145(6), 04019021 (2019). https://doi.org/10.1061/(asce)ww.1943-5460.0000528 3. Tsukamoto, M.M., Cheng, L.-Y., Kobayakawa, H., Okada, T., Bellezi, C.A.: A numerical study of the effects of bottom and sidewall stiffeners on sloshing behavior considering roll resonant motion. Mar. Struct. 72, 102742 (2020). https://doi.org/10.1016/j.marstruc.2020. 102742 4. Amaro, R.A., Jr., Mellado-Cusicahua, A., Shakibaeinia, A., Cheng, L.-Y.: A fully Lagrangian DEM-MPS mesh-free model for ice-wave dynamics. Cold Reg. Sci. Technol. 186, 103266 (2021). https://doi.org/10.1016/j.coldregions.2021.103266 5. Pereira, L.S., Cheng, L.-Y., Ribeiro, G.H.S., Osello, P.H.S., Motezuki, F.K., Pereira, N.N.: Experimental and numerical studies of sediment removal in double bottom ballast tanks. Mar. Pollut. Bull. 168, 112399 (2021). https://doi.org/10.1016/j.marpolbul.2021.112399 6. Amaro Junior, R.A., Cheng, L.-Y., Osello, P.H.S.: An improvement of rigid bodies contact for particle-based non-smooth walls modeling. Comput. Particle Mech. 6(4), 561–580 (2019). https://doi.org/10.1007/s40571-019-00233-4 7. Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181(3), 375–389 (1977). https://doi.org/10. 1093/mnras/181.3.375 8. Koshizuka, S., Oka, Y.: Moving particle semi implicit method for fragmentation of incompressible fluid. Nucl. Sci. Eng. 123(3), 421–434 (1996) 9. Domínguez, J.M., Crespo, A.J.C., Barreiro, A., Gómez-Gesteira, M., Mayrhofer, A. Development of a new pre-processing tool for SPH models with complex geometries. In: The 6th Intermational SPHERIC Workshop, pp. 117–124 (2015) 10. Zhu, Y., Zhang, C., Yu, Y., Hu, X.: A CAD-compatible body-fitted particle generator for arbitrarily complex geometry and its application to wave-structure interaction. J. Hydrodyn. 33(2), 195–206 (2021). https://doi.org/10.1007/s42241-021-0031-y 11. Fu, L., Ji, Z.: An optimal particle setup method with Centroidal Voronoi Particle dynamics. Comput. Phys. Commun. 234, 72–92 (2019). https://doi.org/10.1016/j.cpc.2018.08.002 12. Diehl, S., Rockefeller, G., Fryer, C.L., Riethmiller, D., Statler, T.S. Generating optimal initial conditions for smoothed particle hydrodynamics simulations. Astron. Soc. Aust. 32 (2015). https://doi.org/10.1017/pasa.2015.50 13. Tsukamoto, M.M., Cheng, L.-Y., Motezuki, F.K.: Fluid interface detection technique based on neighborhood particles centroid deviation (NPCD) for particle methods. Int. J. Numer. Meth. Fluids 82(3), 148–168 (2016). https://doi.org/10.1002/fld.4213 14. Fernandes, D.T., Cheng, L.-Y., Silva, R.C., Favero, E.H., Nishimoto, K.: Geração de grid de partículas a partir de sólidos modelados com malha de triângulos. In: The Proceedings of the XXXVI Iberian Latin American Congress on Computational Methods in Engineering (CILAMCE2015), Rio de Janeiro, Brazil (2015). (in portuguese) 15. Amaro Jr., R.A., Pereira, L.S., Cheng, L.-Y., Shakibaeinia, A.: Polygon wall boundary model in particle-based method Application to Brumadinho tailing dam failure. In: Proceedings of COBEM 2019 (2019). https://doi.org/10.26678/ABCM.COBEM2019.COB2019-1869 16. USGS. United States Geological Survey. (2019) 17. Kobayakawa, H., Kusumoto, H., Toyoda, M.: Numerical simulation of liquid motion in SPB tank. In: The Proceedings of the International Offshore Polar Engineering Conference, vol. 4, pp. 424–430 (2012)

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18. Bellezi, C.A., Fernandes, D.T., Cheng, L.-Y., Tsukamoto, M.M., Nishimoto, K.: Investigation of green water in FPSO by a particle-based numerical offshore tank. In: The Proceedings of the 4th International Conference on Particle-based Methods, pp. 590–601 (2015)

Criteria and Procedures for the Geometric Parametrization of Existing Buildings: The Case Study of the Roof of the Frontón Recoletos in Madrid Andrea Colombo(B)

and Andrea Giordano

Università Degli Studi Di Padova, 35121 Padova, PD, Italy [email protected], [email protected]

Abstract. A comprehensive knowledge of architectural and engineering assets can be implemented involving IT tools that allow, through interoperable 3D modeling, a more immediate visualization and interaction of shapes and structures in virtual space. The conception of such a process then provides for a close integration between creative thinking, scientific knowledge, and operational practice, as well as a series of checks and modifications. Through processes for parametric modeling of a specific case study – the Frontón Recoletos in Madrid – we intend to explore the perspective offered by current software regarding the development of geometric-structural concepts of architectural/engineering works. Keywords: Computational geometry · Representation · Building information modeling

1 Introduction In architectural/engineering studies, the role of the disciplines of Representation today takes on a particularly important character due to an absolute theoretical and instrumental irreplaceability. In fact, if we consider Representation not only as a necessary medium for buildings documentation and communication, but for the management and elaboration of the project, we can recognize a new awareness in terms of autonomy and disciplinary complexity. It is precisely in interdisciplinary research projects that the importance of the graphic-expressive capacity of what has been processed is evident, both with respect to the quality of the images and the invention and management of complex spatial structures. Therefore, it is possible to implement a perfect knowledge of architectural and engineering assets, also with IT tools that allow, through interoperable 3D modeling, a more immediate visualization and interaction of shapes and structures in virtual space. The conception of such a process then provides for a close integration between creative thinking, scientific knowledge, and operational practice, as well as a series of checks and modifications. Through processes for parametric modeling, therefore we intend to explore the perspective offered by current software regarding the development of geometric-structural © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 661–672, 2023. https://doi.org/10.1007/978-3-031-13588-0_57

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concepts of architectural/engineering works. A specific case study was examined: the Frontón Recoletos in Madrid by the engineer Eduardo Torroja and the architect Secundino Zuazo in 1936. Particular attention has been paid to the coverage, whose geometric and constructive peculiarity made it famous worldwide. Engineer Torroja took care of the configurative conception, the structural calculation and the effective construction of the rooftop and its skylights.

2 The Configurative Geometry A correct understanding/interpretation of the configurative geometry of architectural/engineering artefacts attests the improved interest in several aspects of the Representation disciplinary field. In this case, our attention is aimed at the geometric component and the function it performs in the building’s construction process, whether existing or in progress. This component goes beyond the circumscribed range of Elementary Geometry (which deals only with metric properties of figures and spaces), showing, together with the properties of projection, unsuspected aspects of flexibility and dynamism. It is capable not only of becoming inalienable for the representation of complex surfaces and configurations and the related shadows, but also, above all, to be bearers of authentic expressive values, aimed at enriching the individual and collective inheritance of creativity and aesthetic culture [1]. The implicit dynamism of geometry is highlighted by its morphogenetic properties, those properties of a line, straight or curved, which, moving in space according to a definite law, also generates a definite surface (see Fig. 1) [2, 3].

Fig. 1. Helicoid: surface generated by the roto-translation of a straight line.

Therefore, a configurative property is recognized in the geometric structures of architecture, abstract structures as sets of shapes, from which the figurative and formal definitions of spaces arise. In this sense, each shape is taken as an “instant stop” of the ongoing movement of that generating line, maintaining in the image the same dynamism that the real presence expresses and suggests: a structural interpretation of the geometric configuration of surfaces generates the identification and then the graphic construction of the geometric matrix of the spaces. Then a set of lines - straight and curved, flat or

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convex - which are generated by the intersection between different surfaces - flat or curved - that delimit the environments and, at the same time, with the extensions to infinity, visualize the geometric genesis of real configurations, thus reinterpreting the spaces in a dynamic sense. The dynamism in fieri that spaces suggest, favoring not only a configurational and structural interpretation - specifically indicative of what will be the leading structure of an organism -, but above all a relational reading, aimed at understanding the relationships between the parts and again between the parts and the whole. This interpretation, assuming a communicative role of the existing reality, develops an important propositional function within the projective definition. It is therefore interesting to underline how, in a context built on the basis of Western metaphysics, in which matter and spirit are opposed, geometric shapes move across the boundary of the visible and the invisible, of the corporeal and the incorporeal, of the absolute and contingent, of the ideal and the real. In this sense, Marsilio Ficino, in one of his numerous reviews on the incorporeal nature of beauty (which involves color, light and numbers understood as formal principles and order), asked his reader to visualize a building, and to try to neutralize with thought, material nature for a moment, leaving only the internal geometric order visible [4]. Moreover, Alberti seems to repeat the same echo suggesting the creation of perfect shapes of buildings totally separated from material [5]. While we can follow Ficino’s instructions, and dematerialize the shape, it is possible to reverse the process, so that these forms can be made by us with such material precision that our imagination is favored? But the same methodology of analysis and synthesis, aimed at identifying the geometric genesis of the configurations, also allows the construction of architectural details of high figurative complexity. However, by resorting to innovative digital methods, it is possible to reach complete knowledge of architecture - both in the presence of existing works and in the genesis of the project - generating digital models of existing or imagined structures. These critical observations motivate the attempt to exanimate architectural works for which interpretation and configurative classification makes it possible to grasp the semantic values of the surfaces that structure them.

3 The Dematerialized Shape The proposed methodology - far from being a scientific systematization of a formal architecture but with the participation of the idea of an architecture determined by multiple factors - allows us to think about the importance of assimilation, by the designer who approaches it, of those concepts that make the spaces of the architecture not only recognizable, but also imaginable. For centuries, architects and engineers have come to the formal baggage of surfaces that geometric research - sometimes contextually, or independently of the work itself - was devising to configure spaces. Spaces that tend or confirm a classically conceived order - in which the surface is the silent testimony of a tectonic archetype based on axes and symmetry - or disintegrate the compositional rules by deconstructing the architectural language through the juxtaposition or, at the limit, the conflictual screeching of shapes. The dialectical approximation with the past, the organic relationship with the landscape, the decorative choices and the didactic purposes of the project often pass through the semantic filter of surfaces. They become the ideal and visibly - or subliminally - apparent medium to convey the architectural message: even

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today, the dematerialization of the building, made possible by new technologies, still requires surfaces - glass, load-bearing structures of limited thickness, etc. - in order to aspire to the status of an architectural project. In particular, we present an investigation that proposes the knowledge and reconstruction, through careful and precise observation/examination of documentary sources, of a building of great historical and figurative interest: the Frontón Recoletos, created in 1936 by the architect Secundino Zuazo and the engineer Eduardo Torroja, destroyed during the Spanish Civil War (see Fig. 2).

Fig. 2. Frontón Recoletos, E. Torroja. The interior space of the Frontón Recoletos. Source: Colonnetti, G.: La tecnica delle costruzioni, le pareti sottili. Vol. III. Einaudi, Torino (1957).

This building has been virtually reconstructed, where the related elaborations have been created according to the double criterion of visual verisimilitude and geometricstructural analyses. The latter have become indispensable, above all, for a deeper understanding of the intimate reality of spaces and configurations, thus achieving knowledge, more than metric and formal - then quantitative -, also relational and qualitative, aimed at evaluating their aesthetic/artistic value. The elaborations testify to the analyticalcritical process aimed at first investigating, and then underlining, in the double version mentioned and through meaningful three-dimensional images, the rich/complex articulations, the specific figurative/structural values of this architecture, while recovering original integrity and primitive monumental dignity (see Fig. 3).

Fig. 3. Qualitative-conceptual rendering of the interior room of the Frontón Recoletos. Front view (sx) and rear view (dx).

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These themes are developed through digital construction VPL (Visual Programming Language) and BIM (Building Information Modeling): in this way, once the essential requirements of aesthetic sensitivity and geometric competence have been acquired, it is possible to offer the possibility of verifying, through the construction of virtual images, the state of the building returning its appearance not only figurative, but also material and structural. It seems logical to underline that this operative proposal generates a sort of hypertext organized as a “container” of all the knowledge belonging to a topic (in our case a building), also presenting itself as a hermeneutic operation of the artifact.

4 The Essence of the Shape The BIM methodology, as mentioned above, therefore provides for the creation of a digital model - or rather, an information model - capable of faithfully reproducing the physical and functional characteristics of a built object (an architectural work, in this case). In the case in question, being a building already built, it is considered more appropriate to speak of HBIM (Historical Building Information Modeling), that is a branch of BIM closely linked to the digital reconstruction and management of the existing building heritage [6]. However, it is not correct to consider the Frontón Recoletos an existing building, as it was unfortunately demolished following the bombings of the Spanish Civil War, a few years after its construction. Therefore, the purposes of the model are linked to activities focused on digital reconstruction and management of the virtual architectural artefact. In HBIM field, the creation of a Digital Twin of the building cannot be separated from careful research aimed at acquiring the data and geometry. Nevertheless, the knowledge of a historic building - even more so if it is a valuable architectural work - cannot be based exclusively on the analysis of the graphic documents and textual sources that describe it, however thorough and accurate. It is therefore necessary to go beyond the physical and material appearance of the object, going into a deeper investigation, aimed at understanding the intimate essence of the architectural work. Only in this way it is possible to realize how the marvelous roof of the Frontón Recoletos - characterized by the intersection of two asymmetrical barrel vaults, each of which hosting a longitudinal band of triangular mesh skylight - is nothing more than the culmination of a conceptual synthesis, gained through an architectural-structural research path that Torroja has carried out during his multifaceted career as a designer, scientist, lecturer and researcher [7]. The wealth of knowledge and skills accumulated by Torroja are contained in his precious writings and can be found in his works, with interesting analogies, from a morphological and philosophical point of view, with the coverage of the Frontón Recoletos. What first of all draws the observer’s attention is the amazing asymmetrical nature of the roof, at that time signifying an absolute novelty in this kind of structures. In fact, this building component was made up of two contiguous curved surfaces, with a directrix characterized by two asymmetrical semicircles that rested on the extreme pediments of the building. The centers of the sections of the two barrel vaults were located along the same horizontal axis, while the radii that defined the circular sectors of the major and minor vault were respectively 12.20 m and 6.40 m [8]. The dimensional relationship between the vaults was conceived - as for aesthetic reasons related to the proportion of the elements - for functional reasons related to natural

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lighting and distribution of the interior spaces. In fact, the planar surfaces, that made up the playing field, intersected orthogonally with each other according to a “trihedral” organization, whose asymmetrical nature wholly matched the original roof (see Fig. 4).

Fig. 4. Conceptual diagram that compares the asymmetry of the playing field walls with the asymmetry of the directrix of the roof.

Fig. 5. Frontón Recoletos. The point of intersection of the laminar vaults.

In this way, the smaller vault offered shelter and illumination to the spectators located in the upper gallery, while the bigger vault accompanied the flow of the playing field (see Fig. 6a) [9]. In this sense, the asymmetry of the roof was the “legitimate daughter” of the principles promoted by the Modern Movement, bearer of concepts of dynamism, change, and uniqueness of the work [10]. The inequality of the lobes of the roof established a hierarchical relationship between the elements that helped to feed a sensation of perpetual motion, perceivable inside the area. In fact, the thin covering of the Recoletos appeared to the amazed eyes of the spectator like a veil shaken by the wind, which contained in its form the very idea of movement [11]. The second peculiar aspect of the roof concerns the straight line generated by the intersection of the surfaces, along which the maximum architectural tension is produced. Torroja, in the name of that formal purism that he had promulgated in his previous works (see the intradoses of the Mercado de Algeciras and the Hipódromo de la Zarzuela), decided to hide the linear intersection between the two vaulted covering structures.

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This constructive choice was dictated by the desire to express the maximum “geometric sincerity” of the roof, masking as much as possible the structural elements that risked disfiguring its aesthetic appearance (see Fig. 5). The other interesting geometric feature that emerges from the connection of the two vaults is the angle generated by the intersection between the tangents of the inter-axis of the profiles. It is a right angle (see Fig. 6b); from whose value it follows interesting properties. One of these is the sense of aesthetic perfection generated by two lines, whether curved or straight, which meet in a perfectly orthogonal way. From Torroja’s point of view, the right angle, which “like all angles is the consequence of the meeting of two alignments […] has a strong personality, is hard as an incontrovertible reality. It admits neither increase nor decrease, nor does it forgive layout errors” [12]. While embracing the modernist architectural philosophy, Torroja never stopped taking the architecture of the past as an example as guidelines for his compositions. Examples based on concepts of proportions and harmony of shapes which - although not directly perceivable by the observer - helped to convey a harmonious feeling of the internal environment. From the study of the cross section of the Recoletos we can observe, once again, a profound geometric link between the elements and the function for which they were designed. Imagining tracing the circumference inscribed in each vault, it can be clearly seen how these are strictly related, from a geometric point of view, to the floors of the room. In particular, the plane of the playing field appeared to be tangent to the directrix circumference of the main vault, while the extrados of the stands represented the horizontal tangent of the directrix of the minor vault (see Fig. 6c).

Fig. 6. Frontón Recoletos. Asymmetry of the internal hall (a), orthogonal intersection of barrel vaults (b), proportions of the vaults (c).

Basically, was re-proposed from a geometric point of view the concept according to which the main vault should correspond to the court, while the minor vault to the space for spectators. In this way, the roof integrated perfectly with the asymmetry of the existing space, generating a visual and conceptual cohesion of the architectural elements. Another clear reference can be found in the visionary architectures of Étienne-Louis Boullée. In particular, the project for the extension of the National Library is famous, conceived in 1785 and destined to “remain on paper”. The interior space of the reading room was covered by a majestic coffered barrel vault, perfectly inscribed within the room, the keystone of which was replaced by a generous rectangular skylight [13]. Unlike Boullée, Eduardo Torroja had to deal with the practical aspects - of a structural and constructive nature - related to the concrete realization of the work, which particularly influenced the positioning, geometry and size of the skylights. These light bands were then created through the adoption of a tessellation of equilateral triangles divided into

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rows, the union of which generated a reticular system of diagonals and transoms. Being equilateral triangles, the internal angles were equal to each other, with a 60-degree opening. The two skylights, consistent with the asymmetrical dimension of the vaults, had a different opening between them. In particular, the major skylight was tessellated with six rows of triangles, for a total arch length of 7.20 m; while the minor skylight, consisting of five rows, had an arched opening of 6.00 m.

5 Versatility and Dynamism: Parametric Modeling The experimental phase of the work, despite the initial premises, was not limited to the mere reproduction of the articulated geometries of the roof, aimed at recovering at least virtually - the majesty of the interior space of the sports hall. In addition to the development of a three-dimensional model of the entire complex faithful to the project measures, it was therefore decided - using parametric modeling processes to create different dimensional combinations of the roof. For the realization of these purposes, it was necessary to implement a procedure based on the Visual Programming Language. In the case of the Frontón Recoletos, it is therefore no longer just the roof itself that conveys sensations of flexibility and dynamism. In fact, the same methodology applied to it, allows expanding the design possibilities by giving the original form an efficient versatility in spatial terms, to evaluate the ideal geometric configuration based on pre-established criteria. The first step towards the realization of the aforementioned purposes was to distinguish the elements that it was convenient to parameterize from those that could have maintained unchanged values. The opaque surface of the foil was decided to keep it unchanged from a dimensional point of view, in order not to alter the previously mentioned geometric proportions and the harmonic bond that was established between the lobes of the roof, the upper gallery and the playing field. What was considered appropriate to parameterize were the skylights of the vaults, since the triangular tessellation ensures adequate elasticity of the element in geometric terms. In particular, we gain the possibility of extending the opening of the skylights, adding any additional rows of triangles. Beyond that, it is interesting to be able to arbitrarily increase the center distance between the rows of crosspieces that constitute the reticular structure. The geometric constraint imposed - because of the design choices adopted by Torroja - is the maintenance of the equilateral triangle, which allows preserving the 60 grades width of the internal angles. This condition leads us to the definition of two direct and two indirect variables. The first direct variable concerns the total width of the skylight (A), while the second refers to the number of subdivisions of the triangulated rows of the skylight (n). Consequently, the indirect variables “b” and “h” derive, referring respectively to the base and height of the single triangle (see Fig. 7).

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Fig. 7. Definition of the variables for the parameterization of skylights.

From an operational point of view, the script for the geometric genesis of the roof was developed through the workflow represented in the following procedural steps: 1. Input data, directly selectable from the information model; 2. Direct variables, whose values change is reflected in real time directly on the geometry of the element; 3. The actual process of parametric design and modeling; 4. Export of the geometries for subsequent import into the information model; 5. Extrapolation of the information and quantities of the exported geometries. The first code blocks allow us to enter the geometric data to start the script. Starting from a rectangular space - the size of the sports hall (55.00 × 32.50 m) and delimited by four walls of the same height - the segments that originate the canonical barrel vault are taken as source data: the base plane and the generating lines. Secondly, we move on to the determination of the values of the parametric variables, on the basis of a previous definition of some criteria, which are of greater or lesser importance depending on the purpose of the work. In this case, the dominant criterion is undoubtedly the geometric one. This aspect mainly concerns the center distance of the crosspieces. By excessively increasing the value, the profile of the roof in correspondence with the skylights would assume an exaggerated polygonal shape. This would lead to the loss of perception of the sinuous shape of the curve of the vault, so much sought after by Torroja to give the work its best aesthetic expression and maximum fluidity. Once the definition of the dimensional parameters is completed, we move on to the actual parametric modeling. The first phase is focused on the creation of the laminar surface, whose direction is characterized by the intersection of two semicircles. It is therefore necessary, first of all, to derive the asymmetrical direction of the roof. By having the coordinates of the central point and the amplitude value of the angle of each arc of circumference, as well as the extreme points located in correspondence with the internal line of the longitudinal walls, it was therefore possible to trace two semicircles: their intersection generates the directrix of the double barrel vault. (See Fig. 8). In the next step, we afford the division of the newly created arc of circumference into three parts. The first, as well as the shortest, consists of a short-curved section with a length of 40 cm, which for structural reasons prevents the main skylight from setting directly on the end of the smaller vault. The central section is instead represented by the skylight, while the last portion is part of the opaque laminar surface. The main vault is then divided into two main components: the first - consisting of the continuous surface - is composed

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of the longitudinal curved section of 40 cm, the lateral shoulders and the remaining surface; the second, on the other hand, is represented by the actual area that will house the skylight, on which the operations for the creation of the triangular tessellation will take place.

Fig. 8. Generating the asymmetric directrix.

The next step was to create a sequence of radial planes which, intersected with the vaulted surface, generated a series of horizontal lines distributed along the curved surface, the generatrices. We started with the creation of the orthogonal plane at the major vault, within which both the starting point of the main arc of the directrix and the center of the vault are contained. The planes, being divided equally along the curved section, guarantee a constant amplitude of the angles. These values were parameterized by recognizing the width of the total opening angle of the skylight, given by the ratio between the total length of the circumference arc and its radius. The individual amplitudes, equivalent to each other, are obtained by dividing the total angle by the number of rows of equilateral triangles. The radial planes subsequently intersect with the curved surface to generate the horizontal lines that will constitute the guides for the creation of the crosspieces at the end of the process (see Fig. 9).

Fig. 9. Intersection between the radial planes and the curved surface.

The last step for creating adaptive points involves a second intersection, between the horizontal lines just obtained and a series of vertical planes distributed along the generatrix of the roof and orthogonal to it. As in the previous case, it involves intersecting a surface with a plane to obtain a curve. In a second moment, the desired points are obtained from the intersection of the curves. The spacing of the vertical planes is achieved with a parametric procedure like the one that allowed the automatic calculation of the angles for the opening of the radial planes. Starting from the value of the height of the single triangle, derived from the definition of the two direct variables, it is possible,

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using the inverse formula, to deduce the value of the base of the triangle, which will in fact constitute the offset between the vertical reference planes. However, the simple intersection between the vertical planes and the horizontal lines, orthogonal to each other, instead of a triangular mesh would have generated a quadrilateral mesh. To cope with this drawback, it was decided, using the geometry of the equilateral triangle, to create a second list of vertical planes offset from the first half of the base of the triangle. In this way, the second series of vertical planes coincides exactly with the upper vertex of each triangle. The next step was to intersect the horizontal lines, alternately, first with the initial vertical planes and then with the staggered vertical planes, thus obtaining a mesh of adaptive points with adjacent rows regularly alternating between them (see Fig. 10).

Fig. 10. Triangular mesh of adaptive points.

The final step of the modeling is aimed at exporting the geometry of the skylight. The adaptive points were grouped into triplets and connected to each other, in such a way as to obtain triangular surfaces which could subsequently be converted into triangular glazed panels. Subsequently, the triad of segments of each triangle were extrapolated that act as guides for the subsequent laying of the material element, namely the concrete joist. The result of the entire parametric modeling process of the roof is given by a pair of skylights: the first, of greater width, has an arc length ranging from 7.20 m to 8.40 m with a subdivision that ranges from 5 to 7 rows; the second, on the other hand, has an arc length range of 6.00–7.20 m and a variable subdivision between 4 and 6 rows. In this way, each skylight has 6 different solutions, which combined with those of the other skylight generate a total of 36 possible geometric configurations. Below, we propose a couple of specific solutions (see Fig. 11).

Fig. 11. Examples of geometric configurations of the skylight.

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6 Conclusions Consequently, is it possible to recognize the importance of knowledge of configurative geometry for creative thinking, scientific knowledge and operational practice? Is it possible to attribute to geometric knowledge a founding and fundamental role for our disciplinary field? The answers will certainly be affirmative if we understand that this knowledge not only allows the (manual / digital) processing of rigorous and expressive graphics, but is also the basis for the construction of interoperable models. Furthermore, as is equally evident, the flexibility of the geometry guarantees the achievement of a high level of expressive capacity, usable in any other experience of the architect and engineer, whether it is aimed at both documentation and the projection of complex/articulated spaces and structures. Acknowledgments. The study was carried out at the University of Padua and the Universidad Politecnica de Madrid, focusing on the natural lighting and visual comfort inside the Frontón Recoletos hall [14]. We would like to thank Professor María Josefa Cassinello Plaza and the Eduardo Torroja Foundation for the bibliographical material provided, the Engineering Library of the University of Padua and the ETSAM Library in Madrid.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14.

Giordano, A.: La geometria dell’immagine, vol. III. Utet, Torino (2002) Giordano, A.: Cupole, volte e altre superfici. Genesi e forma. Utet, Torino (1999) D’Acunto, G., Giordano, A.: Descrivere lo spazio. In: Progetto (ed.) Padova (2017) Ficino, M.: Commentary on the symposium. In: Panofsky E (ed.) Idea: A Concept in Art Theory, New York, pp. 136–137 (1958) Alberti, L. B.: I dieci libri di architettura. In: Ticozzi, S (ed.) Milano (1833) Adami, A.: Architectural Design and History. Franco Angeli, Milano (2021) Cassinello, M.J.: Museo Eduardo Torroja. Fundación Torroja, Madrid (2016) Chías, P., Abad, T.: Eduardo Torroja. Obras y proyectos. Instituto Eduardo Torroja, Madrid (2005) Torroja, E.: Frontón Recoletos. Informes de la construcción, vol. 137, Madrid (1962) Artieda, V. L., Machin, T. L.: Torroja y Wright: una mirada común. In: Actas Congreso Internacional Arquitectura importada y exportada de España y Portugal, Pamplona (1925– 1975) (2016) López-González, C., Carreiro-Otero, M., García Navarro, J.: Génesis y período vital del Frontón Recoletos. Informes de la Construcción 66(536) (2014) Torroja, E.: Razón y Ser de los tipos estructurales. Instituto de la Construcción y del Cemento. CSIC, Madrid (1957) Brancasi, I.: Architettura e illuminismo. Filosofia e progetti di città nel tardo settecento francese. Firenze University Press, Firenze (2015) Colombo, A.: Luce e forma in architettura: il caso del Frontón Recoletos di Madrid. Università degli Studi di Padova (2021)

Design of Surfaces in Cylindrical Coordinates Using GeoGebra AR Alejandro Isaías Flores-Osorio(B)

and Dennis Alberto Espejo-Peña

Universidad Peruana de Ciencias Aplicadas, Chorrillos, Lima 15067, Peru {pcmaaflo,pccedesp}@upc.edu.pe

Abstract. This research focuses on the application of the definition of a convex set, which allows us to design and build the graph of bounded surfaces described by cylinders whose directrix curve is a polar curve that intersects with a quadric surface or a function defined in two variables and thus generates bounded surfaces to be constructed using the GeoGebra Suite Calculator. Three cases are presented where the importance of the convex set concept in the parameterization of this type of surface is highlighted. In the first case, we establish the surface that we have called the Flores surface and from this surface, we establish the construction of the Anthurium flower, in the second case we design the flower with 8 petals, and in the third case designed the leaf house modeled by Mareines and Patalano, whose structure is shaped like a flower with 6 petals. The construction of these surfaces is carried out using the GeoGebra Suite Calculator application for mobile devices, which through its GeoGebra AR tool allows these surfaces to be extended to an augmented reality environment and in this way visualize, manipulate, and understand the abstraction of these mathematical objects in a dynamic and interactive. Keywords: GeoGebra suite calculator · GeoGebra AR · Convex set · Cylindrical coordinates · Parametric surface · Surface of Flores

1 Introduction Geometry is considered one of the oldest sciences in the world, which is found in nature and in everything that surrounds us, being the subject of numerous artistic manifestations, with special relevance in architecture [1]. Currently, geometry is one of the most complicated subjects because it requires a higher level of abstraction, understanding, and vision than other subjects where geometric intuition is and will always be the most powerful source for understanding many topics [2]. This research aims to present in detail the logical sequence to sketch surfaces that are difficult to graph traditionally, that is, using a pencil and a sheet of paper [5]. These surfaces are based on the definition of convexity presented by [4] in the modeling of surfaces, this definition is extended to present parametric surfaces where the definition of a convex set is embedded when designing surfaces that are the result of the intersection of a cylinder whose directrix curve is a polar curve, with a quadric surface or © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L.-Y. Cheng (Ed.): ICGG 2022, LNDECT 146, pp. 673–684, 2023. https://doi.org/10.1007/978-3-031-13588-0_58

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a function defined in two variables, these bounded surfaces will model natural objects such as flowers, which are useful in a mathematics classroom [3] since in this way the mathematical relationship with the natural environment that surrounds us is established and it also allows mathematical models to be established that allow the design of virtual environments through mathematical definitions. The definition of the convex set allows us to graph bounded surfaces described explicitly and implicitly [4] and extends to the design of new surfaces. To graph these surfaces, the GeoGebra Suite Calculator application for mobile devices will be used, within this application the GeoGebra 3D tool is attached and through the Surface command the connection between the mathematical object and the virtual environment will be established and in this way obtain an unbeatable visualization, perception, manipulation, and understanding of the sketch of surfaces [5], in addition to interacting with the augmented reality offered by GeoGebra AR, to generate a more dynamic environment in the classroom and also improve student performance [6].

2 Basic Definitions 2.1 Parametric Surface According to Stewart (2010) a parametric Surface S is defined by a vector function r(u, v) of two parameters u y v: r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k (u, v) ∈ D

(1)

Then x, y and z, which are the component functions of r, are functions of two variables u and v with domain D. x(u, v) y(u, v) z(u, v)

(2)

They are the parametric equations of the surface. If S is a parametric surface given by the vector function r, then S is traced by the position vector r(u, v) as the point (u, v) moves through the domain D. 2.2 Cylindrical Coordinates Under the Stewart approach, we consider that a point P(x, y, z) in three-dimensional space is represented in cylindrical coordinates by the ordered triple (r, θ, z), where (r, θ ) is a representation in coordinates poles of the projection of the point on the XY and z is the directed distance from (r, θ ) the point P. From cylindrical to rectangular: ⎧ ⎨ x = r cos(θ ) (3) y = r sin(θ ) , 0 ≤ θ ≤ 2π ∧ r > 0 ⎩ z=z From rectangular to cylindrical: ⎧ 2 ⎨ r = x2 + y2 tan(θ ) = yx , x = 0 ⎩ z=z

(4)

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2.3 Convex Set According to Canales (2004), a set is convex if given any two points x, y ∈ , λx + (1 − λ)y ∈ , for all λ ∈ [0, 1]. This last definition will be used to establish the variation of one of the parameters that is established when representing a parameterized surface as indicated in Sect. 2.1.

3 Design of Surfaces in Cylindrical Coordinates In this section an alternative way of establishing the parameterization of surfaces will be presented, performing a manipulation of the values of the variables u and v according to 2.1, using definition 2.3 and thus sketching surfaces that will later be displayed in an environment augmented reality using GeoGebra AR. For this purpose, we present three cases where surfaces are designed that will be part of the intersection between a cylinder whose generating curve is a polar curve and the surface that is generated from the graph of a function defined in two variables or on a quadric surface according to the case. For this purpose, we are going to base ourselves on the work carried out by Flores (2022). For the construction of this type of surface we must take into account the contour of the guideline curve, which for the cases we present is a polar curve and where its usual variation will be considered, that is, from 0 to 2π, depending on the type of surface. polar curve. Next, we present the three cases for the design of surfaces in cylindrical coordinates. 3.1 Case 1 For this purpose, we consider the following cylinder S1 : r = 1 + cos(θ ) and the sphere S2 : x2 + y2 + (z − 3)2 = 4. To graph the Surface S1 we are going to use the following syntax that is going to be established in the option of the 3D Calculator of Calculator Suite GeoGebra (see Fig. 1) Surface((1 + cos(t))cos(t), (1 + 1cos(t))sin(t), k, t, 0, 2π, k, 0, 6)

(5)

To sketch the graph of the intersection between the cylinder and the sphere, the extension of definition 2.3 will be used and exemplified in the construction of the cardioid polar curve. Remember that the surface of the sphere according to definition 2.2 can be √ rewritten as follows S2 : z = 3 ± 4 − r 2 . Then we define the parameterization of the intersection surface of the bottom of the sphere as follows. ⎧ ⎪ ⎨ x = k · (1 + cos(t)) · cos(t) t ∈ [0, 2π ] y = k · (1 + cos(t)) · sin(t) , S3 : (6) ⎪ ⎩ z = 3 − 4 − k(1 + kcos(t))2 k ∈ [0, 1] The parameterization of the upper part of the sphere will be established in the following way. ⎧ ⎪ ⎨ x = k · (1 + cos(t)) · cos(t) t ∈ [0, 2π ] y = k · (1 + cos(t)) · sin(t) , S4 : (7) ⎪ ⎩ z = 3 + 4 − k(1 + kcos(t))2 k ∈ [0, 1]

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Fig. 1. Graphic view of surfaces S1 and S2 from a mobile device. Observe the surface of the region bounded by the cylinder and which is part of the sphere.

Using the GeoGebra Surface command with the settings set to (6) and (7) we get the following syntax S3 = Surface(k · (1 + cos(t)) · cos(t), k · (1 + cos(t)) · sin(t), 2 3 − 4 − k(1 + kcos(t)) , t, 0, 2π, k, 0, 1

(8)

S4 = Surface(k · (1 + cos(t)) · cos(t), k · (1 + cos(t)) · sin(t), 3 + 4 − k(1 + kcos(t))2 , t, 0, 2π, k, 0, 1

(9)

The attached Fig. 3 shows the section bounded by the sphere and the cylinder. We are going to call this surface, surface of Flores (Fig. 2).

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Fig. 2. View in 3D and in augmented reality of the Surface S3 and S4 from a mobile device.

It must be considered that the variation of the variable k is from 0 to 1, and this variable comes from the extension that is established from the definition of convexity and we consider it as an adjustment value that allows the vector function defined in (10) construct the surface S3 . 2 r1 (t, k) = k · (1 + cos(t)) · cos(t), k · (1 + cos(t)) · sin(t), 3 − 4 − k(1 + kcos(t)) (10)

To visualize the behavior of the vector function we build two sliders. The slider t, which is an angle that varies from 0 to 2π and the slider k which varies between 0 and 1. By manipulating the slider t, we observe that the points generated will traverse the entire shape of the sphere bounded by the cardioid polar curve, on the other hand, by manipulating the k slider, it allows plotting points within the parameters of the cardioid curve following the contour of the sphere. We observe the behavior of both sliders in Fig. 3, where the red dots are generated by manipulating the t slider and the blue dots are generated by manipulating the k slider. In a similar way, the parameterization of the surface S4 is explained.

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Fig. 3. Behavior of the vector function r1 when the sliders t and k vary.

Fig. 4. Anthurium Flower, based on the design of the Surface of Flores, seen in augmented reality. Source: Pixabay.com [8]

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Considering only Surface S3 and adding other ruled surfaces based on the work of Flores (2022) and Carbajo (2021), we build a flower, taking the Anthurium flower as a reference, which we can see in Fig. 4. 3.2 Case 2 With what was presented in the previous case, we can establish the outline of surfaces that can in a certain way camouflage themselves via augmented reality with nature. For this case, we present the construction of the intersection surface between the cylinder S4 : r = 2cos(4θ ) and the function f (x, y) = 2 − x2 +y22 +1 . Consider for this case the usual variation of θ , that is to say θ ∈ [0, 2π ], that by substituting and simplifying we will obtain the parametric surface S5 . ⎧ ⎪ ⎨ x = 2k · cos(4t) · cos(t) t ∈ [0, 2π ] S5 : y = 2k · cos(4t) · sin(t) , (11) ⎪ k ∈ [0, 1] 2 ⎩ z =2− 2 (2k·cos(4t)) +1

Using the GeoGebra syntax, surface S5 is defined as follows: S5 = Surface 2k · cos(4t) · cos(t), 2k · cos(4t) · sin(t), 2 −

2 , k, 0, 1, t, 0, 2π 4k 2 cos2 (4t) + 1

(12) Then we proceed to graph the Surface S5 in the 3D graphical view of GeoGebra (see Fig. 5).

Fig. 5. Graphic view of Surface S5 , which is a flower with 8 petals.

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In this case we observe that the convexity is immersed in each of the components of the parametric surface through the variable k. To explain this process, as in the previous case, the vector function defined as follows is going to be established 2 r2 (t, k) = 2k · cos(4t) · cos(t), 2k · cos(4t) · sin(t), 2 − (13) (2k · cos(4t))2 + 1 where in the GeoGebra application we consider the slider t that varies from 0 to 2π and the slider k that varies between 0 and 1. By manipulating the slider t position vectors are generated. We observe that the final point of these position vectors generates a counterclockwise path on the surface established by the function f following the shape of the 8-petal rose polar curve. Now when manipulating the slider k, we observe that the final point of the position vector that is generated for each value of k travels a path on the surface generated by f starting from the pole to the contour of the 8-petal rose polar curve, see Fig. 6.

Fig. 6. Behavior of the vector function r2 when the sliders k (blue color) and t (red color) vary.

Next, we add other elements such as a sphere and ruled surfaces established in the work of Flores-Osorio et al. [4] and Carbajo [3], we obtain a flower that is visualized in the figure (see Fig. 7).

Design of Surfaces in Cylindrical Coordinates Using GeoGebra AR

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Fig. 7. Graphic view of surface S5 , which is a flower with 8 petals. Source: Wikimedia.Org [9].

3.3 Case 3 In this case, the leaf house is modeled, named after the architects Mareines and Patalano, which has a roof in the shape of palm leaves (Fig. 10). To do this, we base ourselves on the surface obtained by intercepting the cylinder whose generating curve is the pink polar curve defined as follows r = 2sin(3θ ), − π6 ≤ θ ≤ π6 , with the function f whose correspondence rule is f (x, y) = 4 − y2 + 0.5 sin(x). Considering these requirements, the parameterization of the surface is presented based on the cylindrical coordinates defined in 2.2. ⎧ x = k · 4cos(3t)cos(t) ⎨ t ∈ [− π6 , π6 ] S6 : , (14) y = k · 4cos(3t)sin(t) ⎩ k ∈ [0, 1] z = 4 − (k · 4cos(3t)sin(t))2 + 0.5sin(k · 4cos(3t)cos(t)) Using the surface command syntax.

Surface k · 4cos(3t)cos(t), k · 4cos(3t)sin(t), 4 − (k · 4cos(3t)sin(t))2 π π + 0.5sin(k · 4cos(3t)cos(t)), t, − , , k, 0, 1 6 6

(15)

We obtain the graph of the Surface S6 that is displayed in Fig. 8. In this case, the definition of a convex set is immersed in establishing the variation of the parameter k entre 0 and 1, which causes the surface of a flower petal to be built on the surface generated by the function f . Likewise, we generate the curves C1 and C2

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A. I. Flores-Osorio and D. A. Espejo-Peña

Fig. 8. GeoGe