Geometric concepts for geometric design 9781315275475, 1315275473

Table of contents :
pt. I. Some linear algebra. 1. Linear systems --
2. Linear spaces --
3. Least squares --
pt. II. Images and projections. --
4. Parallel projections --
5. Moving the object --
6. Perspective drawings --
7. The mapping matrix --
8. Reconstruction --
pt. III. Affine geometry. 9. Affine space --
10. The barycentric calculus --
11. Affine maps --
12. Affine figures --
13. Quadrics in affine spaces --
14. More on affine quadrics --
15. Homothetic pencils --
pt. IV. Euclidean geometry. 16. The euclidean space --
17. Some euclidean figures --
18. Quadrics in euclidean space --
19. Focal properties --
pt. V. Some projective geometry. 20. The projective space --
21. Projective maps --
22. Some projective figures --
23. Projective quadrics --
pt. VI. Some descriptive geometry. 24. Associated projections --
25. Penetrations --
pt. VII. Basic algebraic geometry. 26. Implicit curves and surfaces --
27. Parametric curves and surfaces --
28. Some elimination methods --
29. Implicitization, inversion and intersection --
pt. VIII. Differential geometry. 30. Curves --
31. Curves on surfaces --
32. Surfaces.

Citation preview

Geometric Concepts for Geometric Design

Advisory Board Christopher Brown, University of Rochester Eugene Fiume, University of Toronto Brad Myers, Carnegie Mellon University Daniel Siewiorek, Carnegie Mellon University

Geometric Concepts for Geometric Design Wolfgang Boehm

Technische Universitlit Braunschweig Braunschweig, Germany

Hartmut Prautzsch

Universitat Karlsruhe Karlsruhe, Germany

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

AN A K PETERS BOOK

First published 1994 by AK Peters, Ltd. Published 2018 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW; Suite 300 Boca Raton, FL 33487·2742 © 1994 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works

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Library of Congreu Cataloging-in-Publication

Data

Boehm, Wolfgang, 1928Geometric Concepts for Geometric Design/ Wolfgang Boehm, Hartmut Prautzsch.

p. cm.

Includes bibliographic references and index. ISBN 1-56881-004-0 1. Geometry. 2. Geometry-Data processing. Hartmut. II. Title. QA445.B63 1998b 516-dc20

I. Prautzsch, 93-20666

CIP

About the cover: The Cover picture shows a computer generated shaded image of a chalice based on a drawing by Paolo Ucello (1897-1475). Ucello's hand drawing was the first extant complex geometrical form rendered according to the laws of perspective. (Perspective Study of a Chalice, Drawing, Gabinetto dei Disegni, Florence, ca 1430-1440.)

Contents

Preface

xv

Notation

xvii

I Some Linear Algebra

1

1 Linear Systems 1.1 1.2 1.3 1.4 1.5 1.6 1.7

2

3 5 6 7 9 10 11

Linear Spaces 2.1 2.2 2.3 2.4 2.5 2.6

3

Matrix Notation Matrix Multiplication Gaussian Elimination Gauss-Jordan Algorithm LU-Factorization Cramer’s Rule Notes and Problems

Basis and Dimension Change of Bases Linear Maps Kernel and Fibers Point Spaces Notes and Problems

13 14 16 17 18 20

Least Squares 3.1 3.2 3.3 3.4

Overdetermined Systems Homogeneous Systems Constrained Least Squares Linearization v

21 23 24 25

Contents

vi 3.5 3.6

Underdetermined Systems Notes and Problems

II Im ages and Projections 4

26 27

29

Parallel Projections 4.1 4.2 4.3 4.4 4.5

Pohlke’s Theorem Orthogonal Projections Computing a Parallel Projection Projecting Rays Notes and Problems

31 35 37 39 39

5 Moving the Object 5.1 5.2 5.3 5.4 5.5

6

Euclidean Motions Composite Motions Euler Angles Coordinate Extension Notes and Problems

41 43 45 46 47

Perspective Drawings 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Homogeneous Coordinates Central Projection Moving the Object Vanishing Points Completing a Perspective Drawing Moving the Camera Spatial Perspective Maps Notes and Problems

48 49 51 52 54 55 56 57

7 The Mapping Matrix 7.1 7.2 7.3 7.4 7.5 7.6

Main Theorem Camera Data The Spatial Perspective Vanishing Points of the System Stereo Pairs Notes and Problems

59 61 62 62 67 68

vii

Contents

8

Reconstruction 8.1 8.2 8.3 8.4 8.5

Knowing the Object Straight Lines in the Image Plane Several Images Camera Calibration Notes and Problems

III Affine G eom etry 9

71 72 73 75 76

77

Affine Space 9.1 9.2 9.3 9.4 9.5 9.6

Affine Coordinates Affine Subspaces Hyperplanes Intersection Parallel Bundles Notes and Problems

79 80 82 83 84 85

10 The Barycentric Calculus 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

Barycentric Coordinates Subspaces Affine Independence Hyperplanes Join Volumes A Generalization of Barycentric Coordinates Notes and Problems

86 88 89 91 92 93 95 96

11 Affine Maps 11.1 11.2 11.3 11.4 11.5 11.6 11.7

Barycentric Representation Affine Representation Parallelism and Ratio Fibers Affinities Correspondence of Hyperplanes Notes and Problems

99 101 102 102 104 104 105

Contents

viii

12 Affine Figures 12.1 12.2 12.3 12.4 12.5 12.6 12.7

Triangles Quadrangles Polygons and Curves Conic Sections Axial Affinities Dilatation Notes and Problems

107 109 110 112 114 117 118

13 Quadrics in Affine Spaces 13.1 13.2 13.3 13.4 13.5 13.6 13.7

The Equation of a Quadric Midpoints Singular Points Tangents Tangent Planes Polar Planes Notes and Problems

120 122 124 125 126 127 129

14 More on Affine Quadrics 14.1 14.2 14.3 14.4 14.5 14.6 14.7

Diametric Planes Conjugate Directions Special Affine Coordinates Affine Normal Forms The Types of Quadrics in the Plane The Types of Quadrics in Space Notes and Problems

131 133 134 136 138 139 141

15 Homothetic Pencils 15.1 15.2 15.3 15.4 15.5 15.6

The Equation Asymptotic Cones Homothetic Paraboloids Intersection with a Subspace Parallel Intersections Notes and Problems

143 144 145 146 148 150

Contents

ix

IV Euclidean G eom etry

153

16 The Euclidean Space 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9

The Distance of Points The Dot Product Gram-Schmidt Orthogonalization Cartesian Coordinates The Alternating Product Euclidean Motions Shortest Distances The Steiner Surface in Euclidean Space Notes and Problems

155 156 158 159 160 161 162 163 166

17 Some Euclidean Figures 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8

The Orthocenter The Incircle The Circumcircle Power of a Point Radical Center Orthogonal Spheres Centers of Similitude Notes and Problems

168 169 170 172 173 174 175 177

18 Quadrics in Euclidean Space 18.1 18.2 18.3 18.4 18.5 18.6

Normals Principal Axes Real and Symmetric Matrices Principal Axis Transformation Normal Forms of Euclidean Quadrics Notes and Problems

19 Focal Properties 19.1 The Ellipse 19.2 The Hyperbola 19.3 The Parabola 19.4 Confocal Conic Sections 19.5 Focal Conics 19.6 Focal Distances

179 180 181 182 183 186

188 190 192 193 195 198

Contents

X

19.7 Dupin’s Cyclide 19.8 Notes and Problems

V Some Projective G eom etry

199 202

205

20 The Projective Space 20.1 20.2 20.3 20.4 20.5 20.6 20.7

Homogeneous Coordinates Projective Coordinates The Equations of Planes and Subspaces The Equation of a Point Pencils and Bundles Duality Notes and Problems

207 209 211 212 214 216 219

21 Projective Maps 21.1 21.2 21.3 21.4 21.5 21.6 21.7

Matrix Notation Exceptional Spaces The Dual Map Collineations and Correlations The Crossratio Harmonic Position Notes and Problems

221 223 224 225 227 229 230

22 Some Projective Figures 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9

Complete Quadruples in the Plane Desargues’ Configuration Pappus’ Configuration Conic Sections Pascal’s Theorem Brianchon’s Theorem Rational Bezier Curves Rational Bezier surfaces Notes and Problems

231 234 235 237 238 240 242 244 245

23 Projective Quadrics 23.1 Projective Quadrics 23.2 Tangent Planes

247 248

Contents 23.3 23.4 23.5 23.6 23.7 23.8 23.9

xi The Role of the Ideal Plane Harmonic Points and Polarity Pencils of Quadrics Ranges of Quadrics The Imaginary in Projective Geometry The Steiner Surface Notes and Problems

V I Some D escriptive Geom etry

250 252 253 255 257 260 262

265

24 Associated Projections 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8

Plan and Elevation Side Elevation Special Side Elevations Cross Elevation Curves on Surfaces Canal Surface The Four-Dimensional Space Notes and Problems

267 269 271 273 276 278 280 282

25 Penetrations 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8

Intersections Distinguished Points Double Points The Order Bezout’s Theorem Decompositions Projections Notes and Problems

V II Basic Algebraic G eom etry

284 286 287 289 291 292 294 295

297

26 Implicit Curves and Surfaces 26.1 Plane Algebraic Curves 26.2 Multiple Points

299 300

Contents

xii 26.3 26.4 26.5 26.6 26.7

Euler’s Identity Polar Forms of Curves Algebraic Surfaces Polar Forms of Surfaces Notes and Problems

302 303 305 307 309

27 Parametric Curves and Surfaces 27.1 27.2 27.3 27.4 27.5 27.6 27.7

Rational Curves Changing the Parameter Osculants of a Curve Bezier Curves Splines Osculants of a Surface Notes and Problems

310 312 314 317 319 321 324

28 Some Elimination M ethods 28.1 28.2 28.3 28.4 28.5 28.6 28.7

Sylvester’s Method Cayley’s Method Computing Cayley’s Matrix Dixon’s Method Computing Dixon’s Matrix Triangular Matrices Notes and Problems

327 329 330 331 332 333 335

29 Implicitization, Inversion and Intersection 29.1 29.2 29.3 29.4

Parametric Curves in the Plane Parametric Space Curves Normal Curves Parametric Tensor Product Surfaces

29.5 Param etric Triangular Surfaces

29.6 Intersections 29.7 Notes and Problems

337 340 341 343 345 346 348

Contents

xiii

V III Differential G eom etry

351

30 Curves 30.1 30.2 30.3 30.4 30.5 30.6 30.7

Parametric Curves and Arc Length The Prenet Frame Moving the Frame The Spherical Image Osculating Plane and Sphere Osculating Curves Notes and Problems

353 354 356 357 358 361 362

31 Curves on Surfaces 31.1 31.2 31.3 31.4 31.5 31.6

Parametric Surfaces and Arc Element The Local Frame The Curvature of a Curve Meusnier’s Theorem The Darboux Frame Notes and Problems

366 369 370 371 373 374

32 Surfaces 32.1 32.2 32.3 32.4 32.5 32.6

Dupin’s Indicatrix and Euler’s Theorem Gaussian Curvature and Mean Curvature Conjugate Directions and Asymptotic Lines Ruled Surfaces and Developables Contact of Order r Notes and Problems

376 379 381 382 385 386

Bibliography

389

Index

395

Q

~

Taylor & Francis Taylor & Francis Group http://taylora ndfra ncis.com

Preface This book addresses students, teachers and researchers in mathematics, computer science and engineering who are confronted with geometric prob­ lems, attracted by their beauty, and/or wish to get a deeper geometric background. Its purpose is to give a solid foundation of geometric methods and their underlying principles. It may serve as an introduction to geometry as well as a practical guide to geometric design and modeling and to other applications of geometry. The main idea of this book is to provide an imagination for what happens geometrically and to present tools for describing problems. A problem is often solved simply by finding the right description. The topics presented have been chosen from the many geometric problems the authors have confronted during their work in applied geometry and geometric design. In writing this book we intended to disconnect geometric ideas and methods from special applications, in order to make these ideas clear and to allow the reader to apply the presented material to other problems of a geometric nature. Also, in many situations, a figure can say more than a thousand words. This old Chinese proverb ought to be a guideline in writing a text on geometry. Therefore, figures are crucial throughout this book, while diagrams are an integral part of Chapters 1 and 28.

xv

xvi

Preface

This book owes its inception to lectures given by Boehm at Rensselaer Polytechnic Institute and the Technical University of Braunschweig sev­ eral times between 1986 and 1990. This book has been partly written at Rensselaer, and we are greatly indebted to Harry McLaughlin who has been promoting Applied Geometry at Rensselaer and who together with Joe Ecker initiated their cooperation with the TU Braunschweig. Andreas Johannsen read the first and later drafts of the book very carefully, and we benefitted much from his helpful suggestions. We thank Dr. Michael Kaps and Wolfgang Volker for typing the manuscript; Daniel Bister for proof reading the mathematics and Jeannette Machnis for proofreading the English text; and Mrs. Diane McNulty for her judicial and committed assistance in the cooperation with Rensselaer. Troy, in December 1992

Wolfgang Boehm Hartmut Prautzsch

N otation

The following notation is used throughout this book: Scalars

a,

a, 6,...

Vectors, points, coordinate columns

a, b , . . . , p, q , ...

Extended columns (by an additional coordinate)

k, y , ...

Differences between two points

A x, A y , . . . , Ax, A y ...

Matrices

A, I?,...

Augmented matrices

A, B,...

Vector spaces

V, A , ...

Point spaces

A , V , ...

Orthogonal angles

ti

Parallelism

//

Bold type is used whenever a new term is introduced. R e m a rk : Each chapter starts with an abstract and a short bibliography for further information on the particular subject. The complete references are listed at the end of the book.

xvii

PART ONE Som e Linear A lgebra

Many problems encountered in applied mathematics are linear or can be approximated by linear systems which are, in general, computationally tractable. The corresponding mathematical subdiscipline is called linear algebra. At the heart of linear algebra are techniques, such as Gaussian elimination and the Gauss-Jordan algorithm, for computing solutions of linear systems. The main tool of linear algebra is matrices which help to arrange coefficients and describe operations.

1 Linear System s

Most finite linear systems can be described by matrices, a very useful short­ hand notation which emphasizes the underlying linear structure and the interdependencies between the equations.

L ite ra tu re : Atkinson, Boehm Prautzsch, Conte de Boor

1.1 Matrix Notation A linear system is a set of equations of the form

where the a ’s are given real numbers and the x ’s are unknowns. The array A of the coefficients

4

Part I Some Linear Algebra

is called an m x n matrix. The matrix A contains the element a*,*, in its «th row and fcth column. Similarly, the a* can be written as an m x 1 matrix or m column,

Consequently one has where a* represents the fcth column of A. Note that a scalar a can be viewed as a 1 x 1 matrix. The n x m matrix A1 = [a* fc] , defined by a\ k = pose of A , e.g., for A above

is called the trans­

In particular, the transpose a* of an m column a forms an m row

Often it is helpful to visualize an m x n matrix A or an m column a in block form, i.e., as a rectangle of height m and width n or 1, respectively:

The matrix A is a square matrix if m = n, and it is symmetric if addi­ tionally di^k — &k,i- A square matrix [uiyk] is called upper triangular if Ui£ = 0 for i > k. Similarly a square matrix [/*,*] is called lower triangular if l^k — 0 for i < k. The Kronecker symbol

is used to define the identity matrix as the n x n square matrix I = [£»,*:].

Chapter 1 Linear Systems

5

1.2 M a trix M u ltip lica tio n Let A = [d ij] be an m x Z matrix and B = [bj#] an Zx n matrix. The m x n matrix C = [c^*] with the entries

is called the product A B of A and B, in this order. Note that the width I of A has to match the height Z of B. It is helpful to visualize the product A B = C in block form, as introduced above:

The element is the dot or scalar product of the ith row of A with the fcth column of B. This may be memorized as “row times column”. Using this product the linear system in Section 1.1 can be written more compactly as and visualized by blocks as

where x denotes the n column of the unknown X{. Likewise the scalar product a of two m columns a and b can be written as

Part I Some Linear Algebra

6

Note that the product x a is defined as a matrix multiplication, but Aa is not. It is convenient to define Aa = a A as the matrix of elements a ^ a , i.e., one has In particular, one gets for a = 0 the null column o = xO and the null matrix O = 0 A. A square matrix A is said to be non-singular if its inverse A~x defined by A -1 A = A A ~ X = I exists. Finally, a matrix B is said to be orthonormal if B % B = J.

1.3 G au ssian E lim in a tio n Linear systems are most frequently solved by Gaussian elimination. It is convenient to represent the linear system A x — a by the augmented matrix

Then the linear systems obtained by the following simple operations on [A | a] will have the same solutions: 1 2 3 4

exchanging two rows, multiplying one row by a factor / 0, adding one row to another, exchanging two columns of A while simultaneously exchanging the corresponding unknowns in the column x.

It was Gauss’ idea to use these four simple operations to transform [A|a] into the matrix [B|b], where B is composed of an upper triangular, non­ singular r x r matrix [/, an r x n-r matrix matrix B *, an m - r x n null matrix, an r column b, and an m - r column s, as shown below.

Chapter 1 Linear Systems

7

If s ^ o, there exists no solution. However, if s = o, there exists an n - r parameter family of solutions which can easily be determined from the equivalent system 2?x = b as follows. Assigning arbitrary values to xr+ i , . . . , x n as parameters one can compute x r backward from row r , then x r- i from row r - 1 , . . . , and finally x\ from row 1. Rem ark 1: The number r is called the rank of A, denoted by rank A. Note that r < m and r < n . Rem ark 2: If a = o, the linear system Ax = a is called homogeneous. Then one also has b = o. The homogeneous system has a non-trivial solution if and only if r < n as can be inferred from the equivalent system B x = O. If x is a solution of a homogeneous system, then x • g, where q ^ 0, is also a solution.

1.4 G au ss-Jordan A lg o rith m Gaussian elimination can further be used to construct an explicit represen­ tation for the set of all solutions of the linear system A x = a. W ith the aid of the operations 1, 2, 3, one transforms the matrix [U\B*\b] from above into [I\C* |c*] as illustrated in the following diagram.

r

The general solution of this system is depicted below where —I denotes the negative n-r x n-r identity matrix and t an n —r parameter column.

8

Part I Some Linear Algebra

Note that for r < n the representation of x depends on the sequence of operations performed during Gaussian elimination. Remark 3: The construction can be reversed. Let [C |c] represent the set

of (given) solutions where c is some n column, C an n x m matrix, and t a column of m free parameters. The set represented by [C | c] does not change if the transposed matrix [C lef is modified by Gaussian elimination. Using the operations 1 , . . . ,4, the matrix Cl can be transformed into an s x n matrix [—I\D l] provided rankC — s. This is illustrated below where the superfluous zero rows are discarded. Adding appropriate multiples of rows of [—/I D 1] to c* one obtains a row [o11d 1] as illustrated below.

Now one easily obtains a linear system for which x = c + C't is a solution, namely

Chapter 1 Linear Systems

9

1.5 LU-Factorization Often the matrix A of a linear system is square, i.e., m = n. Such a system is uniquely solvable if and only if A is non-singular or equivalently if rank A = n. A non-singular matrix A can sometimes be factored into a lower-triangular matrix L, whose diagonal entries are all equal to 1, and an upper triangular matrix {/, i.e., A — LU ,

The entries of L and U can successively be computed by means of the ma­ trix multiplication rule for a M , . . . , ahn, a2yi, - .. , a2,n, . . . , an, i , . . . , an.n in this order. At each step there is exactly one unknown or u^k to be determined. If a non-singular matrix A cannot be factored in this way, one can always rearrange the rows of A to obtain a matrix A* which has an LU-factorization. In all cases one can start to compute L and U as if .4 were to be factored and interchange the rows of A during the computation whenever it becomes necessary to avoid dividing by zero. The LU-factorization is another organization of Gaussian elimination and can be used to solve a system Ax. = a. Let [A* |a*] be obtained from [A |a] by a row permutation such that an LU-factorization A* = LU exists. Solving the two triangular systems

by forward and backward substitution respectively, yields the solution for A*x = a* and hence for A x = a.

10

Part I Some Linear Algebra

Rem ark 4: The LU-factorization is useful for solving the system Ax = a repeatedly for a fixed coefficient matrix A and different right hand sides a. In particular, if the right hand sides are the columns of the identity matrix I one obtains the inverse of A . Rem ark 5: If A is symmetric and x*Ax > 0 for all x ^ 0, then A is called positive definite, and a symmetric factorization A = C^C, where C is an upper triangular matrix, is possible without row interchanges. This is called a Cholesky factorization.

1.6 C ram er’s R u le Let A = [a^k] be a square n x n matrix and A ^ the submatrix obtained from A by deleting the ith row and kth column. Then the determinant of A, written det A, is defined by the recursion

for any scalar a. This definition does not depend on the choice of i and is called Laplace expansion along the «th row. The term (—l) t+fcdet A^k is called the cofactor of . The determinant can be used to solve a non-homogeneous linear system Ax = a when A is some non-singular n x n matrix . Let Afc = [ax ... a ... an] be obtained from A = [ai ... a* - - - a„] by replacing the feth column with a. Then Cramer’s rule,

gives the coordinates Xk of the solution. Note that det A ^ 0 whenever A is non-singular. In the case of a homogeneous system Ax = with rank A = n — 1, one can show that

o, where A is an n-1 x n matrix

Chapter 1 Linear Systems

11

provides the solution of the system, where g ^ 0 is a free parameter and A*k is obtained from A by deleting the fcth column. R e m a rk 6: Cramer’s rule is of practical use only for small n.

1.7 N otes and Problems 1 It is possible to improve the numerical stability of Gaussian elimination by row interchanges. 2 One of the numerically most stable algorithms used to solve linear sys­ tems is the so-called Householder algorithm. 3 Let A* and I* be obtained from the matrices A and I of equal height by the same row permutation. Then P = I* can be used to write down the permutation of A, i.e.,

P is called a permutation matrix. It is inverse to its transpose, i.e., P tP = I . 4 Using a permutation matrix P (see Note 3), Gaussian elimination can be summarized as

5 The LU-factorization of A* = P A can be used to compute X = A ~x by solving A*X = P column by column. 6 Most elimination methods for solving linear systems are actually just different organizations of the Gaussian elimination process. They differ only in the ordering of the computation steps. 7 For two n x n matrices A and B one has det A B = det A • det B. 8 Matrix multiplication by hand is best organized by Falk’s scheme as illustrated in Figure 1.1.

12

Part I Some Linear Algebra

Figure 1.1: Falk’s scheme.

9 Falk’s scheme can also be used to procure an LU-factorization or a Cholesky factorization.

A linear or vector space V over I I is a set which is closed under linear combinations with real coefficients. The elements of V are called vectors, the coefficients are scalars. A map from one linear space into another is called a linear map if it preserves linear combinations. The standard vector space is R m.

L ite ra tu re : Greub, Strang, van der Waerden

2.1 Basis and Dimension Let o denote the zero vector, then any r vectors a i , ... ,ar belonging to a vector space V are said to be linearly dependent if there exist scalars x i , . . . , x r not all of which are zero such that

Otherwise a i , . . . , a r are said to be linearly independent. On building the matrix A = [ai .. . ar], one has that a i , . . . , ar are linearly dependent if and only if A x = o has a non-trivial solution. The set of all linear combinations of the given a* forms a linear space, called the span of the a», or span [ai .. . a r]. The space A = span [ai ... a r] is called a subspace of V . The dimension of A , or dim A , is defined as the maximum number of linearly independent vectors in A . Occasionally, n = dim A is given as a superscript, A n

14

Part I Some Linear Algebra

Let a i , . . . , ^ be n linearly independent vectors of an n-dimensional linear space V , and let v be some vector of V . Then these n -h i vectors are linearly dependent, i.e.,

in matrix notation v = Ax.. In this equation the factors x* can be uniquely determined, otherwise the a* would not be linearly independent. Hence A is non-singular. One says that the vectors a i , . . . , an form a basis of V. The SiiXi are called the components of v, while the Xi are referred to as the coordinates of v with respect to the a*. Rem ark 1: It is convenient to denote a vector by the vector of its coor­ dinates x = [x\ ... x n]1. This convention is used throughout this book. Rem ark 2: On choosing some fixed basis of V n every vector of V n corresponds to a unique element of ]Rn , and every linear combination in V n corresponds to the same linear combination in IRn . Therefore it is sufficient to consider H n instead of V n. In particular, the a* from above may be viewed as elements of lRm, m > n.

2.2 C h an ge o f B a ses Let a i , . . . , an and b i , . . . , b n denote two bases of a linear space V . Then the a ’s can be expressed uniquely in terms of the b ’s,

Using matrix notation one gets

Chapter 2 Linear Spaces

15

or more concisely A = BC. As a consequence one has C = B ~ XA, i.e., C = [ci,*] is non-singular since it is the product of non-singular ma­ trices. Let v be some arbitrary vector of V with the representations v = Ax. — B y , i.e.,

It then follows that y = Cx, i.e.,

Note that the a ’s are expressed in terms of the b ’s, but the y's are expressed in terms of the ar’s. Both transformations are called contragredient to each other. The representation a* = f?c* has a simple but important geometric mean­ ing: The column c* of C represents the coordinates of the basis vector a* with respect to the basis b i , . . . , bn. Exam ple 1: For

one has

16

Part I Some Linear Algebra

2.3 Linear M aps Of particular interest are maps which are compatible with the linear struc­ ture of linear spaces. Such maps must preserve linear combinations. Con­ sider two linear spaces A and B with bases a i , ... , a n and b i,... ,bm respectively, and a map

0. From Section 7.2, it follows that

Because of this, and since the 1- and 3-directions of the object system are orthogonal and parallel to the image plane, one has

Figure 7.3: One-point perspective, a 2 > 0 .

T w o -p o in t p e rsp e c tiv e : If there are exactly two finite vanishing points, say a i and a 2, C is of the form

Part II Images and Projections

64

Figure 7.4 illustrates the meaning of the a ’s and a ’s. From Section 7.2, it follows immediately that where and after some algebraic manipulations

As above, the a ’s and a ’s are not independent. Namely, one has

Figure 7.4: Two-point perspective, a i > 0, a 2 > 0.

T h re e -p o in t p e rsp ectiv e: If there are three finite vanishing points, C is of the most general form

Figure 7.5 illustrates the meaning of the a ’s and a ’s. Let e denote the eye. The plane e h a 3 is perpendicular to the planes a ia 2e and a ia 2a 3 and consequently also to the line a ia 2. Therefore the lines h a 3 and a ia 2

Chapter 7 The Mapping Matrix

65

Figure 7.5: Three-point perspective, a* > 0.

are perpendicular. Then because of symmetry, h is the orthocenter of the triangle a ia 2a 3 , i.e., one has

where the A, are the barycentric coordinates of h with respect to the a*, and the are as defined by Figure 7.5. From Section 7.2, it follows that the A* are positive and that

Part II Images and Projections

66

i.e., the a* are not independent. Finally, one gets after some calculations

Exam ple Is An example of a one-point perspective was considered in Section 6.3, where

The camera is positioned at of n i = [0 1 0]1.

= [—a —b —c

1]* and

looks in the direction

Exam ple 2 s One gets a two-point perspective if the projection of Ex­ ample 1 is composed with the motion of Example 2 in Section 5.1. The matrix

is the mapping matrix where s

Pr be some points in B and v some vector of the underlying vector space. Then, the affine, extended or barycentric coordinates of the point p + v are obtained by adding the affine, extended or respectively barycentric coordinates of p and v. Similarly, the affine combination p = p0#o H-------1- Prx r ? where xq H-------\- x r = 1 , holds true (with the same weights xo ,. . . , xr ) whether p, p0, . . . , pr are represented by affine, extended or barycentric coordinates.

10.3 A ffine In d ep en d en ce A family of r + 1 points of A, q0, . . . , qr , is called affinely independent if the r vectors b i = qx —q0, . . . , b r = qr —q0 are linearly independent. One can easily check that this definition does not depend on the choice of q0-

The points q 0? • . . , qr are affinely independent if their extended or barycen­

tric coordinate columns are linearly independent and vice versa. Evidently, any r + 1 affinely independent points of the A n can be extended to a frame of A n . The converse procedure is of particular interest in some applica­ tions: If x r ^ 1, the affine combination

Part III Affine Geometry

90 can be written as

where

Since z$, . . . , zr_i sum to 1, they are the barycentric coordinates of qr , the projection of x from pr onto the plane spanned by p0, . . . , pr_i • This is illustrated in Figure 10.2 .

Figure 10.2: Reducing barycentric coordinates.

Exam ple 5: Applying this reduction to a point

of a plane in different ways one obtains several ratios, which are depicted in Figure 10.3. The triangle in Figure 10.3 can be regarded as a part of the tetrahedron in Figure 10.2.

Chapter 10 The Barycentric Calculus

91

Figure 10.3: Ratios in a triangle.

10.4 H y p erp la n es A linear equation in affine coordinates of x,

defines a hyperplane. Switching to barycentric coordinates one gets the additional coordinate xo, defined by

Hence, the hyperplane can be represented with respect to the barycentric coordinates of x simply by the equation

For A = uo the equation reads

where This equation is homogeneous in the barycentric coordinates of x and is called the symmetric representation of the hyperplane U. A straight line

Part III Affine Geometry

92

x = p ( l —a) + q a intersects U at the point x, which corresponds to a satisfying

In particular, for the intercept of the hyperplane U on the axis through p { and p j , one gets

This is illustrated in Figure 10.4. Note that the quotient may be negative in certain cases and that one or both of the Vj may vanish.

Figure 10.4: Intersections of a plane with the fundamental edges.

Exam ple 6: In barycentric coordinates the plane £3 —1 = 0 in A 3 has the symmetric representation xq 4 - X\ + x • • • >P« are not necessarily affinely independent. However, one can compute a frame of A U B by an affine variant of the Gaussian elimination method applied to the rows p \ , where the pi represent either the extended or barycentric coordinate columns of the p*. After using Gaussian elimination to reduce the rows p\ to a linearly independent set of rows, one must multiply the rows by suitable non-zero factors to obtain extended or barycentric coordinates again. At this stage, some of the rows may represent vectors. However, there always is at least one row which represents a point. One can add this row to the rows representing vectors to obtain a frame for A U B. This procedure is illustrated below, with extended affine coordinates and dim A U B = r.

Note that the number of affinely independent points equals dim A U B + l.

10.6 Volumes The convex hull of a subset B of some affine space A is the smallest subset of A which contains B and which also contains the line segment connecting any two points of B. If B consists of r + 1 affinely independent points q 0, . . . , q r , the convex hull of B is called a simplex. For example, a simplex is a triangle if r = 2 and a tetrahedron if r = 3. Let A be the volume of the simplex spanned by q o , . . . , q r and A* the volume of this simplex, where qi is replaced by

Part III Affine Geometry

94

as illustrated in Figure 10.5. One has ]T) A* = A and, as can be seen from Section 10.2 and Figure 10.5,

Hence, it is possible to compute the ratio of volumes in an affine space. Notice that the two volumes must be of the same dimension and must lie in parallel subspaces.

Figure 10.5: Ratios of volumes.

R e m a rk 4: The frame q0, ... , q r spans an affine subspace B. Therefore one may assume that the q^ are given as r columns with respect to some affine coordinate system of B . Hence, one can apply Cramer’s rule to solve the linear system

for y = [yo

...

f/r]1. As a result one obtains a formula for the volumes and

det

where in the first determinant qf is replace by q, and g is some scaling factor.

Chapter 10 The Barycentric Calculus

95

Exam ple 7: If r = 1, the volumes A 0, A i, and A are distances. In par­ ticular,

as illustrated to the left.

10.7 A G en eralization o f B a ry cen tric C oord in ates For certain applications it is necessary to define barycentric coordinates with respect to more than three points in the plane. This can be accom­ plished in the following way. Consider an n-gon with vertices p x, ... , p n and an arbitrary point p in the plane. As illustrated in Figure 10.6, let A 12 denote the area of the triangle p p xp2, let A 2 denote the area of the triangle p i p 2p 3, etc. Then one has

where on taking indices m odn

and Note th at A of A which is an affine subspace of B with dim A -|- dim T = dim A . The affine subspace T is called a fiber of 3>. All fibers of $ are parallel and have the same dimension. They cover A exactly once.

Figure 11.3: Fibers of an affine map.

E x am p le given by

1:

Let $ : A 3 -

A 3 be

The fiber F of $ containing p is

104

Part III Affine Geometry

In the example above, dim T = 1 and d im $ A = 2 . The family of fibers consists of all affine lines in the direction v = [1 —1 i f .

11.5 A ffin ities An affine map $ of an affine space A into itself is called an affinity; if the map is onto it is called a regular affinity. The matrix C of an affinity $ is square and, if and only if $ is regular, also non-singular. Of particular interest are fixed points or fixed directions of affinities. Pro­ vided that images and preimages are given with respect to the same coordi­ nate systems, fixed points and fixed directions are the respective solutions of the linear systems

A map : A —►A is idempotent if o Idempotent affine maps are called projections. Projections are also characterized by the property that their fibers contain their images. The parallel projections of Figure 4.5 may serve as examples. E x am p le 2 : The affinity of Example 1 is non-regular. It has [1 0 0]* as a fixed point but no fixed directions. E x a m p le 3: One has a translation for C = I but c ^ o. A translation has no fixed points while all directions are fixed.

11.6 C o rresp on d en ce o f H yp erp lan es An affine map $ : A —►B maps points into points and subspaces into sub­ spaces, but, in general, it does not map hyperplanes of A into hyperplanes of B. However, the preimage of a hyperplane is either a hyperplane, the entire space A or does not exist. Namely, let

Chapter 11 Affine Maps

105

represent some hyperplane V of B and y = c + Cx an affine map $ . Then, the preimage of V is obtained as the solution of

where

Figure 11.4: The correspondence of hyperplanes.

Moreover, $ defines a correspondence between certain hyperplanes of B and A with the following geometric properties: If V n $ A is a proper subspace of $ A , then u ^ o and U is uniquely defined by V. It is parallel to the fibers of $ and mapped onto V n $ A . If $ A C V, then u = o and uo = 0, i.e., the preimage of V is the entire space A • If V n $ A is empty, i.e., if V is parallel to $ A , then u = o but u$ ^ 0. A preimage of V does not exist.

11.7 N o te s and P ro b lem s 1

Occasionally it is convenient to define o to be parallel to all directions. Any two affine spaces of the same dimension can be viewed as affine images of each other.

106

Part III Affine Geometry

2

: B —>C be affine maps. Then the composition Let $ : A —►B and o $ : A —>C is also an affine map.

3

Let $ : A —►B be an affine map and ^ : B —►A some arbitrary map. If $ o is the identity on S, then ^ is called a pseudo affine map. An example of a pseudo affine map is shown in Figure 11.5.

Figure 11.5: Pseudoaffine map. 4

Another example of a pseudo affine map can be found in Section 10.7 on the generalization of barycentric coordinates. If the v* ’s span A and are independent, then ^(p) = q(p) is pseudo affine. The map ^ composed with the affine map which maps the v* onto the corresponding p± gives the identity.

5

Relative to a proper subspace S of B the hyperplanes of B fall into one of three categories: Hyperplanes intersecting 5 in a hyperplane of

If t varies, p (t) traces out a polyno­ mial curve of degree n. A polyno­ mial curve with this representation is called a B£zier curve. The Bezier points p* control the curve p (t).

De Casteljau’s Algorithm Every point p(£) of a Bezier curve can be constructed by repeated affine combinations. This algorithm is due to de Castejjau. Let p® = p* and

for r = 1 , . . . , n, then pg = p(t)-

Blossom of a B£zier Curve One may choose a different parame­ ter t at each level r of de Casteljau’s algorithm. Then one has

From the A-frame theorem one can infer that interchanging tr with tr- \ does not change the points p£. Con­ sequently, Pq does not depend on the ordering of the t T. Some authors call Po the blossom of p(£) at t i , . . . , tni see also Chapter 27.

112

Part III Affine Geometry

C haikin’s A lgorithm Repeated corner cutting of a polygon in the ratio 1 : 2 : 1 generates a smooth curve in the limit. This curve connects the midpoints of the poly­ gon’s edges by parabolic arcs so that the polygon’s edges are tangential to the curve, as illustrated to the left.

12.4 Conic Sections Conic sections, i.e., ellipses, hyperbolas, and parabolas, have a number of affine properties; for example, the affine image of an ellipse is an ellipse, that of a hyperbola is a hyperbola, and that of a parabola is a parabola. Every ellipse can be viewed as an affine image of the “unit circle” x 2 + y 2 = 1 . This view can be helpful when constructing points and tan­ gents of an ellipse, e.g., as in the tangent octagon in Figure 12.1.

Figure 12.1: Tangent octagon of an ellipse.

Chapter 12 Affine Figures

113

Every hyperbola can be viewed as an affine image of the “unit hyperbola" x 2 — y2 = 1 or of the simple hyperbola y = 1/x. Therefore the follow­ ing two properties of these hyperbolas are shared by all hyperbolas: The areas bounded by the tangents and asymptotes of the hyperbola are all equal, and each tangent contacts the hyperbola at the midpoint of the seg­ ment connecting the two places in which the tangent and the asymptotes intersect.

Figure 12.2 : Affine image of the unit hyperbola. Every parabola can be viewed as an affine image of the “unit parabola” y = x2, where the y-direction is mapped onto the axis direction. Therefore the following property of the unit parabola is shared by all parabolas: Let p i,p2 be two points of a parabola, s the intersection of the corre­ sponding tangents, and m the midpoint of px and p2. Then the line sm is in the axis direction and meets the parabola at the midpoint of s and m. The tangent at this intersection is parallel to the chord p1? p2. This configuration coincides with de Casteljau’s construction for n — 2 and t = 1/ 2 . Similarly, one can use two special tangents of the unit parabola to verify the following important theorem: The joins of affinely related points of two lines in general position envelope a parabola.

Part III Affine Geometry

114

Figure 12.3: Affine image of the unit parabola.

Rem ark 4: The following property common to all conic sections can be verified in the same way. Consider a family of parallel chords of a conic section. The pairs of tangents at the chord ends meet at points on the line which bisects the chords, as illustrated in Figure 12.4 for an ellipse. Note that a tangent can be viewed as a chord of length 0 .

Figure 12.4: Parallel chords of an ellipse.

12.5 A x ia l A ffin ities If an affinity A n —►A n leaves a hyperplane H fixed pointwise, then the affinity is said to be axial. Let H be spanned by the points p x, ... , p n , and let q0 be the image of some p0 $ 7 i. Then every point p € A n has a representation

Chapter 12 Affine Figures

115

and is mapped into

Consequently, one has

where v = q0 —p0 is called the direction of the affinity. H is called the axis or axial plane of the affinity. All points p with the same (fixed) xo form a hyperplane parallel to H which is translated by vx0>as illustrated in Figure 12.5.

Figure 12.5: Axial affinity.

116

Part III Affine Geometry

E x a m p le 1 : Figure 12.6 shows two triangles in axial affine positions in the plane. This configuration exhibits Desargues* Affine Theorem: If the corresponding vertices of two triangles span three parallel lines, then the corresponding edges intersect in points of a straight line. A proof is given in Section 22.2. Note that one can also allow ideal points.

Figure 12.6: Desargues’ Affine Theorem. E x a m p le 2: Figure 12.7 shows two ellipses in two planes, which are in axial affine position. By selecting some points, one can demonstrate the affinity and vividly describe the relationship between the two shapes: Tan­ gents correspond to tangents, midpoints to midpoints, etc. Note that the intersection of the two planes spanned by the ellipses lies in 7i.

Chapter 12 Affine Figures

117

Figure 12.7: Axial affinity of two ellipses. R e m a rk 5: A pair of axial affine figures in the plane can always be viewed as a figure in space together with its parallel projection into another plane, as indicated in Figures 12.6 and 12.7.

12.6 D ila ta tio n An affinity which leaves all lines through some point c fixed is called a dilatation with center c. A dilatation stretches all directions by the same amount g, i.e., a point p = c -I- v is mapped onto q = c + #v, as illustrated in Figure 12.8. Hence this dilatation is given by

while the underlying linear map is given by

Note that g can be negative and that c can be determined by a pair p , q and the dilatation factor g.

Part III Affine Geometry

118

Figure 12.8: Dilatation. Exam ple 3: Two parallelograms constructed by corner cutting, as in Sec­ tion 12.2 , from the same quadrilateral but with different ratios are centric affine. The center agrees with the intersection of the diagonals of the given quadrangle. Let

12.7 N o te s and P ro b lem s 1

Each diagonal of the quadrangle in the corner cutting construction of Section 12.2 is parallel to two edges of the resulting parallelogram. In proving W ittenbauer’s theorem it is advantageous to use these diagonals as the axes of the affine system.

2

The fixed hyperplane of an axial affinity may be ideal. Then the affinity is a translation.

3 The center of a dilatation may be an ideal point. This affinity, then, is also a translation. 4 For 0 < t < 1 one has B f(t) > 0 . As a consequence the respective segment of a Bezier curve lies in the convex hull of its control points. 5 Figure 12.8 can be regarded as an example of Desargues’ General The­ orem where the corresponding edges of both triangles meet in points of an ideal line.

Chapter 12 Affine Figures 6

119

Originally, Ceva and Menelaus used ratios with signs opposite to the ones above. Consequently the products of the ratios also had different signs.

7 An axial affinity is called a shearing if some point not in Ji and its image lie on a line which is parallel to H. 8 In general, the center of area of a planar n -gon differs from the center of gravity of the n vertices, if n > 3. 9 A regular affinity leaves the ratio of two r-'~olumes in parallel subspaces of dimension r invariant.

13 Quadrics in Affine Spaces

The simplest figures in an affine space besides lines and planes are conics which are the intersection curves of planes and right circular cones. Conics were studied by the Greeks, mainly by Menaichmos (about 350 B.C.) and by Apollonios (200 B .C .), who introduced the names ellipse, hyperbola, and parabola. Conics can be conveniently studied using their quadratic equations, and, without additional effort, the analysis of these quadratic equations can be presented for general quadratic surfaces in any dimension, the so-called quadrics. In this chapter affine concepts of quadrics, such as midpoints, singular points, tangents, asymptotes, and polar planes are discussed.

Literature: Berger, Meserve, Samuel

13.1 T h e E q u ation o f a Q uadric A quadric consists of all points x in an affine space A n satisfying a quadratic equation which can be written as

where C = Cx is a symmetric non-zero n x n matrix. The quadric described by the equation Q(x) = 0 will also be denoted by Q . The equation can be visualized by blocks:

Chapter 13 Quadrics in Affine Spaces

121

The intersection of Q with an affine subspace B of dimension r > 1 is a quadric again or a hyperplane in B, i.e., a subspace of dimension r —1 . To prove this let B be represented by

then substitution yields

which is abbreviated by

If C = B tC B # O, this equation represents a quadric in B. Note th at C is symmetric. On the other hand, if C = O but c ^ o, the equation is linear in y and represents a hyperplane in B. If G — O and c = o, but c ^ 0 , the intersection is empty; if c = 0 the subspace B is completely contained in Q. In the case where B = A , the intersection Q = B ClQ equals Q. Hence, Q(y) — 0 represents Q with respect to a different affine system. Note that C ^ O, i.e., a quadric is a quadric in every affine system.

Figure 13.1: Intersection of a quadric.

122

Part III Affine Geometry

In the case where B is an affine line C represented by

the substitution above results in

or more concisely If a # 0 , this equation represents a quadric in C consisting of two (real or non-real) points which can coalesce to a double point. If a = 0 and /? / 0 , Q represents a single point on £ , and if a = (3 = 0, Q equals the line C or is empty. Rem ark 1: A quadric is understood to be the entire set of all real and all non-real points satisfying a quadratic equation. Consequently, equations differing by more than a factor define different quadrics. Rem ark 2: A quadric is said to be real if all its coefficients are real. However, there need not be any real point on a quadric. In this case the quadric is called a null quadric or, occasionally, an imaginary quadric.

13.2 M id p o in ts A point m = b is called a midpoint of a quadric Q if Q is symmetric with respect to m , i.e., if all straight lines x = m+vA intersect Q symmetrically such that Q( A) = 0 implies Q(-A) = 0. This is the case if and only if P = v* [Cm + c] = 0 for all v or, i.e., if

Chapter 13 Quadrics in Affine Spaces

123

The solution of C m + c = o defines an affine subspace A i of A, which may possibly be empty.

Figure 13.2: Midpoints.

Each direction d solving the corresponding homogeneous system Cd = o is called an axial direction. The axial directions span a Unear space, M . If A i is non-empty, then M is the linear space underlying A i .

Part III Affine Geometry

124

Figure 13.3: Axial directions.

13.3 Singular P o in ts A midpoint s which lies on Q is called a singular point. The midpoint condition implies that and with the additional condition implies that

it also

The solution of both equations defines an affine subspace S of A, which may possibly be empty. In particular, if S ^ 0, one has S = A i.

Chapter 13 Quadrics in Affine Spaces

125

Figure 13.4: Singular points.

Rem ark 3: Let s be a singular point, and let q be some point of Q different from s. Then, since s is a midpoint, the line spanned by s and q intersects Q in a third point not equal to s or q, which means that this line lies completely on Q. Moreover, the join of 5 with some point q on Q but not in 5 is an affine space Q of dimension dim 5 + 1. Every such Q lies on Q, i.e., Q is a cone with center 5 and generators Q. If the center consists of only one point, it is also called vertex.

13.4 T an gen ts A straight line C represented by x — b + vA intersects Q i n q = b i f A = 0 is a root of Q(A) = 0, i.e., if 7 = qtCq + 2ctq + c = 0. Furthermore, it is tangent to Q at q if A = 0 is a double root, i.e., if

In this case, v is called tangential to Q at q, and the line C is said to be a tangent of Q at q. If in addition a = 0, then C lies completely on Q; it is a generator. Note that a line through a singular point is either a tangent or a generator.

126

Part III Affine Geometry

Figure 13.5: Tangents of a quadric.

13.5 T angent P la n es Consider a point q on Q and some point x distinct from q. The difference v = x —q is tangential to Q at q if

Adding Q(q) = qtC'q + 2ctq + c = 0 to this equation one obtains

concisely written as u*x + uo = 0. This equation, in which x varies, represents the equation of a hyperplane, the tangent plane T of Q at q. Note that the tangent plane is not defined at a singular point. This property characterizes singular points, by the way. Rem ark 4: A tangent plane of Q at some point q intersects Q in a quadric Q. Since q is a singular point of Q, the quadric Q is a cone. It may happen, though, that q is the only real point of Q.

Chapter 13 Quadrics in Affine Spaces

127

Figure 13.6: Tangent planes of a quadric.

13.6 P olar P la n es Consider a fixed point x. It lies in the tangent plane of Q at y = q if

If x is not a singular point of Q, this is the equation of a hyperplane, the so-called polar plane X of x with respect to Q. The point x is referred to as the pole of X . Pole and polar plane have the following geometric meaning. The tangent plane of Q at q contains x if and only if q belongs to X , as illustrated in Figure 13.7. Note that the polar plane of some point x of Q coincides with the tangent plane there and is undefined if x is a singular point. The equation of A* is symmetric in x and y. Consequently, if y lies in the polar plane X of x, then x also lies in the polar plane y of y. Such pairs of points are called conjugate with respect to Q.

128

Part III Affine Geometry

Figure 13.7: The polar plane X of x.

E x am p le Is One can find n + 1 points which are pairwise conjugate with respect to a given quadric in A n . A general method to find such points is presented in Section 16.3. Figure 13.8 shows three pairwise conjugate points of an ellipse.

Figure 13.8: Polar triangle of an ellipse.

Chapter 13 Quadrics in Affine Spaces

129

R e m a rk 5: If two points x and y of a subspace B of A are conjugate with respect to a quadric Q in A , then they are also conjugate with respect to the intersection Q n B and vice versa. R e m a rk 6 : In particular, if B is a line x = b +vA, the points correspond­ ing to Ai and A2 are conjugate with respect to Q if

where a ,/?,7 are as in Section 13.1.

13.7 N o te s an d P r o b lem s 1

If det C # 0, there exists exactly one midpoint m ; it may even be singular.

2 The points on a quadric are self-coiyugate. This property characterizes the points on a quadric. 3 A singular point is conjugate to every point. Consequently each polar plane of a quadric contains 1 and dim S = dim'M. — 1. Finally, let r = n — dim M , If M / 0, but 5 = 0, the quadric is called a central quadric. If one chooses a midpoint as the origin of the affine system, and all the a» pair­ wise conjugate with a r + i ,. . . , an span­ ning M , then the quadric, after an appropriate scaling, is represented by

where

137

Chapter 14 More on Affine Quadrics

If S ^ 0, the quadric is called a cone. If one chooses a singular point as the origin of the affine system, and all the a« pairwise conjugate with 8 ,+ !, spanning S, then the quadric, after an appropriate scaling, is represented by

where If M = 0, the quadric is called a paraboloid. If one chooses a point of Q as the origin of the affine system, and all the a* pairwise conjugate with ar+i, ...,an spanning M, a„ £ S, and a i , . . . , a*, tangential to Q at o, then the quadric is represented by

where R e m a rk 5: A quadric with singular axial directions is called a cylinder. R e m a rk 6 : The affine normal form of a quadric is unique up to a permu­ tation of the coordinates. The respective special affine system, however, is not uniquely defined. R e m a rk 7: A special affine system can be constructed using the Euclidean principal axes transformation described in Section 18.5. R e m a rk 8: Midpoints and singular points are affine invariants which classify affine quadrics. This means, e.g., that a central quadric cannot be the affine image of a paraboloid or a cone. Moreover, a central quadric can further be classified by the numbers of positive and negative £*. These numbers, which are given by the numbers of real and non-real intersections

138

Part III Affine Geometry

with the coordinate axes of the special system, are affinely invariant. This classification reflects Sylvester’s theorem. In particular, the quadric

is called an ellipsoid. Similarly, one can classify paraboloids and cones by the numbers of positive and negative £*. The resulting types in A 2 and A 3 are presented in the next two sections.

14.5 T h e T y p es o f Q uadrics in th e P la n e The quadrics of an affine plane are called conics or conic sections. Due to their different normal forms there are 9 different types of conic sections. Six of them contain more than one real point. They are shown in Figure 14.6. The other three types of conics containing no or only one real point are: the imaginary central conic section the imaginary pair of intersecting lines and the imaginary pair of parallel lines

Figure 14.5a: Non-degenerate conic sections.

Chapter 14 More on Affine Quadrics

139

Figure 14.5 b: Reducible conic sections.

14.6 T h e T y p es o f Q uadrics in Space Due to their different normal forms there are 17 different types of quadrics in A 3. One can say that 12 of them contain more real points than a line. They are shown in Figure 14.6.

Figure 14.6 a: Non-degenerate central quadrics.

140

Part III Affine Geometry

Figure 14.6b: Nan-degenerate and degenerate paraboloids.

Figure 14.6c: Cone and degenerate central quadrics.

Chapter 14 More on Affine Quadrics

141

Figure 14.6d: Reducible quadrics. The other 5 types of quadrics are: the imaginary central quadric the imaginary cylinder the imaginary pair of parallel planes the imaginary cone the imaginary pair of intersecting planes

14.7 N o te s an d P ro b lem s 1

A quadric is said to be degenerate if it contains a singular point or a singular axial direction, i.e., if it is a cone or a cylinder.

2

A quadric is reducible if its equation is reducible, i.e., if it can be written as the product of two linear equations. A reducible quadric consists of a pair of hyperplanes.

3 Each non-degenerate quadric in 3-space has two families of generators such that there are two generators through each point. The tangent plane at a point is spanned by the generators through this point. 4 If the two families of lines in Note 3 are real, the quadric is called annular and otherwise it is called oval. 5 The generators of a one-sheet hyperboloid and of a hyperbolic paraboloid are real. Therefore these quadrics are called ruled.

142 6

Part III Affine Geometry

A non-reducible but degenerate quadric in 3-space has two coalescing families of generatrices.

7 The name “cone” for quadrics which contain singular points comes from the property mentioned in Remark 2 in Section 13.3. 8

The condition C = B tCB ^ 0 in Section 13.1 means geometrically that there is at least one direction of U whose diametric plane is not parallel to U.

9 An imaginary quadric can be transformed into a real one by an imagi­ nary affine map and vice versa, see Section 23.7. 10

The equation of a quadric in A n depends linearly on |n ( n + 1) + n + homogeneous coefficients.

11

As a consequence of Note by 5 points.

12

A quadric in general position in space is defined by 9 points.

10

1

a conic section in general position is defined

13 The equation of a conic section defined by the 5 points x i , ... , x 5 can be written as

det

15 H om othetic Pencils

Many properties of quadrics do not depend on the constant terms of their equations. These properties reveal interesting relations among quadrics that differ only in the constant terms of their equations. Such a family of quadrics is called a homothetic pencil, and the family’s definition does not depend on a specific affine coordinate system. In particular, homothetic pencils are useful in analyzing intersections of quadrics with pencils of parallel lines and planes.

L ite ra tu re : Berger, Coxeter, Samuel

15.1 T h e E q u ation The family of quadrics represented by

where C and c are fixed, but c varies is called a homothetic pencil. Since Q(x, c) is linear in c, every point x lies on exactly one member of the pencil. This property has two immediate consequences. The quadrics of such a pencil are pairwise disjoint, and the pencil covers the entire space. Moreover, because of Remark 1 in Section 13.1, a homothetic pencil is determined by any one of its members. It follows from Chapter 13 that the definition of a homothetic pencil does not depend on a particular affine system. Moreover, all quadrics of a

144

Part III Affine Geometry

homothetic pencil have the same midpoints, if there are any; the same diametric plane with respect to some arbitrary direction, and therefore the same pairs of conjugate directions; and if there are any, the same asymptotic and the same axial directions. Furthermore, the polar planes of some fixed pole x with respect to the quadrics of a homothetic pencil form a pencil of parallel planes. One of these planes is the tangent plane at x of the unique quadric containing x. Note that a homothetic pencil can also contain imaginary quadrics.

15.2 A sy m p to tic C ones Consider a homothetic pencil of quadrics with a midpoint m , and let Qo be the unique quadric of the pencil containing m . According to Section 13.3 the midpoint m is a singular point of Qq. Hence, Qo is a cone, called the asymptotic cone of all quadrics of the pencil. Note that the points of M might be the only real points of Qo- Note also that paraboloids do not possess an asymptotic cone. E x a m p le Is On using the normal form, a homothetic pencil of central quadrics is represented by

Figure 15.1 shows the corresponding three types of homothetic pencils in the plane. R e m a rk Is An important property of homothetic central quadrics is that they are centric affine with respect to every m G M . More exactly, the centric affinity y = xg transforms the quadric given by eixf H----- \-erx 2 = c into the one given by

Chapter 15 Homothetic Pencils

145

Figure 15.1: Homothetic pencils of central conic sections.

This property accounts for the name homothetic. Each such centric affinity maps the asymptotic cone onto itself. Note that the three hyperbolas in Figure 15.1 are real members of the same pencil. Moreover, a one-sheet hyperboloid is always homothethic to a hyperboloid of two sheets having the same asymptotic cone. They are related by an imaginary dilatation. E x a m p le 2 : A homothetic pencil of quadrics on an affine line consists of pairs of points which are symmetric with respect to a point m .

15.3 H o m o th etic P arab oloid s If the quadrics of a homothetic pencil are paraboloids, they have no mid­ point and therefore no asymptotic cone. However, analogous to the centric affinity above, one has the important property that homothetic paraboloids are translates of each other. Namely, if the pencil is given in normal form,

one can observe that the members are simple translates of each other in the direction of x n . The direction of x n represents a non-singular axial direction d as mentioned in Section 14.4. Figure 15.2 shows the three types of homothetic paraboloids in A 3.

Part III Affine Geometry

146

Figure 15.2: Homothetic paraboloids.

15.4 In tersectio n w ith a S u b sp ace According to Section 13.1 the intersections of homothetic quadrics with an affine subspace B either forms another homothetic pencil of quadrics or a pencil of parallel subspaces, unless B lies in one of the quadrics itself. The dimension and the position of B determine which of these cases one actually has. Any non-asymptotic line intersects a pencil of quadrics in pairs of points symmetric with respect to some fixed point n and, consequently, is tangent to the quadric of the pencil containing n. This symmetry property leads to a simple construction of the quadric Q\ containing p and homothetic to some given quadric Q: Every non-asymptotic line p + vA intersects Q in two points p + vAi and p + vA2 which could be imaginary or identical. In any case, the point lies on Q i, and if v varies, x traces out the entire quadric Q i . Note that Ai + A2 is always real, although Ax and A2 may be complex.

Chapter 15 Homothetic Pencils

147

Figure 15.3: Intersection with a straight line. Exam ple 3: Consider a one-sheet hyperboloid Q and its asymptotic cone Qo- Each tangent plane Tq of Qo intersects Qo in a double line GoConsequently, T0 intersects Q in two lines Q\ and Q2 parallel to Qo- If To varies on Q0, both lines Q\ and Q2 vary on Q forming the two families of generators of the one-sheet hyperboloid, as illustrated in Figure 15.4

Figure 15.4: Asymptotic cone and generators of a hyperboloid of one sheet.

148

Part III Affine Geometry

15.5 Parallel Intersections The intersections of a quadric with two parallel subspaces are related by means of homothetic pencils. Let B be the subspace defined by x = b + B y , and let B* be its translation defined by x = b + ^ b + B y . Their intersections with the quadric are given by

respectively. Q and Q each define a homothetic pencil. These pencils are identical up to a translation if B tC A b = o, i.e., if A b is conjugate to all directions of B with respect to Q.

Figure 15.5: Parallel intersections of a quadric.

E x a m p le 4: Consider the cone Q given by the equation

The pencil of parallel planes x\ — const intersects Q in a family of conic sections which are central affine, but

Chapter 15 Homothetic Pencils

149 the corresponding pencils are not translates of each other since the x\direction and the X3-direction are not conjugate with respect to Q. Exam ple 5: Consider an ellipsoid Q given by the equation

where 0 < c < b < a and its intersec­ tion with the sphere

If r = 6, the intersection consists of two circles each of which lies in a plane. The planes parallel to each of these planes also intersect Q in cir­ cles. These planes are therefore called planes of circular sections. Exam ple 6 : Consider two parabolas in different planes but with a com­ mon point p and the same axis direc­ tion. Sweeping one parabola paral­ lel in space such that p moves along the second parabola generates a para­ boloid. One obtains the same para­ boloid when the second parabola is swept along the first one, as illus­ trated to the left. Rem ark 2: Planes of circular sections exist for most quadrics. The four points of a quadric where the circles degenerate to a point are the circular or umbilical points of the quadric. Note that one can distinguish spheres and circles from ellipsoids and ellipses only in a Euclidean space.

Part III Affine Geometry

150

15.6 N o te s and P ro b lem s 1

A polynomial of total degree 2 over a triangle represents a segment of a paraboloid if and only if the axes of the three boundary parabolas are parallel.

2

A paraboloid in 3D is defined by three pairwise intersecting parabolas having the same axis direction.

3 The volume bounded by a quadric and a tangent plane of a homoth­ etic quadric does not depend on the position of the tangent plane, as illustrated in Figure 15.6.

Figure 15.6: Constant volumes.

4 The property mentioned in Note 3 is applied in shipbuilding to determine the waterline of a tilted ship. 5 Parallel intersections of a quadric occur in scan-fine algorithms for com­ puter graphics. 6

In Example 5 one also gets planes of circular sections for r = a and r = c. However, these planes are not real.

7 Consider the one-sheet hyperboloid given by

Chapter 15 Homothetic Pencils

151

Its intersection with the sphere x 2 + y2 + z 2 = a2 consists of two circles. 8

A one-sheet hyperboloid has two pencils of circular intersections, yet it does not possess real umbilical points.

9 A homothetic pencil of quadrics can be viewed as the generalization of a pencil of parallel planes.

Q

~

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PART FOUM Euclidean G eom etry

As far as we know, geometry originated from land measurements and area and volume computations made by the early Babylonians and Egyptians in the years approximately from 4000 to 1500 B.C. The word geometry itself is derived from the Greek word for “earth measure.” The Greeks (600 to 300 B.C.) developed the empirical geometry of the Babylonians and Egyptians into a more systematic science which ultimately prepared the ground for Euclid’s outstanding Elements. Euclid, who lived in Alexandria in about 300 B.C., gave a systematic logical organization of what is now called Euclidean geometry. Euclid’s work was so solid th at it took 2000 years before geometries other than Euclidean geometry were discovered. The structure of an affine space does not encompass distances and angles. If distances and angles are defined in an affine space, one refers to it as a Euclidean space. In general, affine maps do not preserve distances and angles. Those that do are the Euclidean motions.

16 The Euclidean Space

The Euclidean space is an affine space where, in addition, the distance between two points is defined in the usual way. This distance between points induces a compatible length of vectors in the underlying linear space and facilitates the introduction of angles. However, a Euclidean space can be introduced more geometrically by means of a gauge ellipsoid.

L ite ra tu re : Berger, Perdoe, Coxeter

16.1 T h e D ista n ce o f P o in ts One can introduce a distance into an affine space by means of a distin­ guished ellipsoid, the so-called gauge quadric. Let G be such a gauge quadric and m its midpoint. Then the distance of two points p and q is defined as the non-negative number 6 such that

where x is a point of G.

Part IV Euclidean Geometry

156

Figure 16.1: Gauge ellipsoid and Euclidean distance. For the sake of simplicity one may assume that m is the origin of the affine coordinate system. Then the equation of G can be brought into the form x*Cx = 1 and one has

An affine space with such a distance is called a Euclidean space. In partic­ ular, the notation £ n means the affine space A n with such a distance. R e m a rk 1 : The gauge quadric is regarded as the unit sphere in £ n . Hence, the members of the homothetic pencil

are spheres of radii g — y/c and center o.

16.2 T h e D o t P r o d u ct In a Euclidean space, two directions v and w are called orthogonal if they are conjugate with respect to the gauge ellipsoid. Furthermore, the scalar or dot product of two vectors v and w is defined by

where x t C x = 1 is the equation of the gauge quadric. The dot product is symmetric and bilinear, i.e., linear in each of its two arguments.

Chapter 16 The Euclidean Space

157

The underlying vector space of £ n is called a Euclidean vector space if it is associated with such a dot product. This Euclidean vector space is denoted by En. The Euclidean length or norm of a vector v G En is defined by

which is also the distance between any two points p and p + v. Multiplying a non-zero vector v by 1/ |v| produces a normalized vector of length 1 . Two vectors u and v are called orthogonal if u • v = 0. Moreover, a family of pairwise orthogonal vectors is called orthonormal if in addition all vectors have unit length. Note that an orthogonal set of non-zero vectors is linearly independent. The dot product can be used to introduce angles between two vectors v and w. Let g be such that v —w g is orthogonal to w. Then the angle

and let P 5 t>e the point on p 3p 4 such that P 1P 5 touches C. Thus one has $i2 + 535 = s 23 + S51 and, by assumption, also the identity above. This implies that 545 -I- 514 = 515 and consequently p 4 = p 5. Note that all six bisectors meet in one point, the center of the circle.

17.3 T h e C ircum circle Another group of classical theorems is concerned with perpendicular bi­ sectors of the sides of a triangle or quadrangle.

Chapter 17 Some Euclidean Figures

171 M id p e rp e n d ic u la r The midperpendicular of two points a and b consists of all points p such that The points p on the midperpendicu­ lar of a and b also satisfy

C irc u m c e n te r a n d C ircu m circle It follows that the three midperpen­ diculars of each triangle meet in a point m called the circumcenter. The circumcenter is the midpoint of the circumcircle. From the identities

etc., it follows that the radius R of the circumcircle is related to the area A of the triangle by

A ngle o f C ircu m feren ce For the angles g, a , r shown in the fig­ ure one has

Hence

As a consequence, the angle of cir­ cumference 7 does not change if c is

172

Part IV Euclidean Geometry moved along the circle, and it changes to 180° —7 if c crosses a or b. B arycentric Coordinates From the equations A: = i? 2sin 27 , . . . , one gets that the barycentric coordi­ nates of m with respect to a, b, c form the ratios sin 2a : sin 2/3 : sin 27 . The angle am b = central angle.

27

is called the

Ptolemy’s Theorem Let pi , . . . , p4 be four points of a plane and let Sik = |pi —pfc| . Then one can show that p1?..., p4, in this order, he on a circle if and only if

Note that all six midperpendiculars meet in one point, the center of the circle.

17.4 Pow er o f a P o in t In £ 71 a sphere is given in Cartesian coordinates by its normalized equation

One can compute the sphere’s its intersection with a straight line C given by x = p -h vA, where vtv = 1 , to obtain a quadratic equation in A,

where a — v4v = 1 and 7 = S'(p). Let Ai and A2 denote the equation’s roots; then by Vieta’s formula A1A2 = 7 . The number 7 = S( p) is called

Chapter 17 Some Euclidean Figures

173

the power of p with respect to S. The power has the following geometric interpretation. Since vtv = 1 the values A form a metric scale on £ , i.e., one has A = |x —p|. Thus 7 = AiA2 is the product of the distances between each of the two intersection points and p. This product does not depend on v. In particular, if C is tangent to 5 , one has Ai = A2. Therefore y /j is the distance between p and the point of contact of any tangent of S containing p. Note that p lies on S if 7 = 0, and p lies inside of S if 7 < 0. R e m a rk 1 : By varying c one obtains a pencil of concentric spheres, where c presents the power of the origin.

Figure 17.1: Power of a point.

17.5 R ad ical C en ter The powers of a point x with respect to two spheres Si and 5 2 are equal if i.e., if x lies in a hyperplane Vn defined by

This hyperplane is called the radical plane of both spheres or the radical axis if the spheres are circles. Note that this plane contains the real or

Part IV Euclidean Geometry

174

non-real intersection of both spheres, where both powers vanish, and that it is perpendicular to the line spanned by the midpoints of both spheres. For the three radical planes of any pair of the three spheres S i , S2, S 3 one has V12 4 - Thz 4 - 7%i = 0 , i.e., the three planes either meet in a line or are parallel. It follows that in £ 3 the six radical planes corresponding to four spheres in general position meet at a point which is called the radical center of the four spheres. Analogously, in £ 2 the three radical axes of three circles meet at a point, the radical center of the three circles, as illustrated in Figure 17.2. Note that this configuration can be viewed as the intersection of a plane with three spheres and their radical planes in £ s .

Figure 17.2: Radical axes and radical center of circles in the plane.

17.6 O rthogon al S p h eres The geometric meaning of the power S(p) implies that each sphere around p with radius y/S(p) meets S orthogonally. Consequently, the midpoint of any sphere which meets two spheres S% and S2 orthogonally lies in the radical plane of Si and S2. There exists exactly one sphere orthogonal

Chapter 17 Some Euclidean Figures

175

to four spheres given in general position in £ 3. Its midpoint m is their radical center; its radius is the root of the power of m . Analogous properties hold for orthogonal circles in the plane, as illustrated in Figure 17.3.

Figure 17.3: Orthogonal circles in the plane. The radii r\ and r 2 of two orthogonal spheres and the distance d of their midpoints are related by Note that a sphere orthogonal to some given sphere S can have an imagi­ nary radius. Its midpoint then lies inside of 5.

17.7 C en ters o f S im ilitu d e Any two spheres S\ and S2 with midpoints m* and radii r* are central affine. Namely, if m i + v lies on S \ , then m 2 + wg lies on S2, where

176

Part IV Euclidean Geometry

6 = ± r 2/ r i . Solving the equation m 2 = c (l centers of the dilatations are

q)

+

one finds the

This fact is illustrated in Figure 17.4 with circles. Note that both centers are real even if the two spheres intersect, if S 2 lies in the interior of S 2 , or if they are concentric. The center c„ is an ideal point only if ri = r 2. Note that c+ and c_ are the vertices of the tangent cones common to both spheres.

Figure 17.4: Both centers of similitude of two circles.

Three circles or spheres define three pairs of centers of similitude. These pairs form the opposite vertices of a complete quadrilateral, cf. Sec­ tion 22.1. Every diagonal of the quadrilateral goes through two midpoints. In £ 3 the four sides of the quadrilateral are the intersections of the four symmetric pairs of planes tangent to all three spheres. These intersections lie in the plane of symmetry.

Chapter 17 Some Euclidean Figures

177

Figure 17.5: The six centers of similitude of three circles.

17.8 N o te s and P ro b lem s 1

The orthocenter, the center of gravity, and the circumcenter of any tri­ angle lie on one line (Euler).

2 The two bisectors of two lines in the plane are orthogonal. 3 In a plane, the midpoints of a family of circles touching two given circles lie on a conic section. 4 The family of spheres in £ 3 having the midpoints and radii of the cir­ cles of the family considered in Note 3 envelope a Dupin’s cyclide, as presented in Section 19.7.

5 The distances

Sik

= |p* —p*| of five points pl5..., p5 on a sphere satisfy

This generalizes Ptolemy’s theorem to spheres.

178 6

Part IV Euclidean Geometry

The area A of a triangle spanned by p i? p 2, p3 satisfies

det

7 The volume A of a tetrahedron spanned by p 1?. . . ,p 4 satisfies

det

8

Multiplying the radii of two spheres by a common factor does not in­ fluence the centers of similitude. In particular, one can regard the mid­ points as spheres of radius 0 .

18 Quadrics in Euclidean Space

Introducing distances and angles into an affine space also has several appli­ cations to quadrics. Normals and principal axes are Euclidean properties of quadrics which are invariant under Euclidean motions, but not under general affine maps. Moreover, one can distinguish quadrics in a Euclidean space by their semi-axes, i.e., by their shapes and sizes.

L ite ra tu re : Berger, Blaschke, Greub

18.1 N orm als Consider a quadric Q given by the equation

where x denotes the Cartesian coordinates and Cx = C. In Section 16.4 it was shown that u represents the normal of the hyperplane u tx + uo = 0 , and one gets the following immediately: The polar plane V of a point p with respect to Q has the equation

Hence, the normal of V is given by

180

Part IV Euclidean Geometry Note that V does not exist if p is a midpoint. The tangent plane T of Q at a point q € Q is the polar plane of q with respect to Q. Therefore

represents the normal of Q at q. Recall that T is undefined if q is a singular point. The diametric plane V of a direction v with respect to Q is defined by

Hence the normal u of V is given by

Recall that V does not exist if v is an axial direction.

18.2 P rin cip a l A x es If the diametric plane V with respect to v is perpendicular to v, the quadric Q is symmetric to V. Then v is called a principal axis direction or a principal axis. In order to procure a principal axis v one can exploit the fact that v is parallel to the normal of its diametric plane, i.e., one has to solve the equation Cv = Av or This matrix equation represents a homogeneous linear system for the prin­ cipal axes v. It has non-trivial solutions only if

Chapter 18 Quadrics in Euclidean Space

181

which is a polynomial of proper degree n, called the characteristic poly­ nomial of C. Its n roots A* are called the eigenvalues of C . For a given eigenvalue A* any non-trivial solution v» of

is called a characteristic vector or eigenvector associated with A*. The linear space spanned by all solutions v* associated with A* is called a characteristic space or eigenspace. Eigenvalues coincide if they are multiple roots of the characteristic polynomial. Note that eigenvalues may be zero. R e m a rk 1 : Because of their geometric definition, the principal axes of a quadric do not depend on the choice of the coordinate system.

18.3 R ea l and S y m m etric M atrices If the matrix C is real and symmetric, eigenvalues and eigenvectors have particularly nice properties: The eigenvectors corresponding to different eigenvalues are orthogonal. The eigenvalues are real. The eigenspace associated with an r-fold eigenvalue has dimension r. In order to prove these facts consider first two arbitrary eigenvectors v x and V2 associated with two different eigenvalues Ai and A2, respectively. One has and and since

one also has

If Ai ^ A2, one has v 2v! =

0

which means that v x and v 2 are orthogonal.

Secondly, assume that Ai = c* + i/3, where f3 ^ 0 is a non-real eigenvalue, and Vi = a + ib is the associated eigenvector. Then A2 = a —*/? also is an eigenvalue, and v 2 = a —zb is the associated eigenvector. Since

182

Part IV Euclidean Geometry

one has Ai = A2. Thus (3 = 0 and Ai is real. Finally, assume that Ai is an r-fold eigenvalue and Vi is an associated eigenvector. Then one can introduce a new orthonormal basis b i , . . . , bn such th at b i is parallel to V i. Thus, one gets

Because of Remark 1, C has the eigenvalues C* and Ai. Hence, Ai is an (r —l)-fold eigenvalue of C *. Since C* is symmetric and real, one can per­ form this reduction step r times thereby obtaining r (linearly independent) orthonormal eigenvectors associated with Ai.

18.4 P rin cip a l A x is T ransform ation The matrix C of a quadratic form vtCrv = 0 in En has n orthogonal eigenvectors, as is shown in the previous section. Thus one can construct an orthonormal basis b i , . . . , b n of En consisting of the eigenvectors of C such that the determinant of B = [bi ... b n] equals +1. The associated Euclidean transformation is called a principal axis transformation. The equation of the transformed quadratic form is w t C w where C = B tCB. Let A denote the diagonal matrix of the eigenvalues A* corresponding to b i , . . . , b n , i.e.,

Consequently one has

Chapter 18 Quadrics in Euclidean Space

183

In words: The principal axis transformation transforms the quadratic form vtCv into the diagonal form w*Aw. Note that the principal axis transfor­ mation is not unique. Rem ark 2: One can compute the eigenspace associated with some eigen­ value by means of the Gauss-Jordan algorithm. Subsequently one can con­ struct an orthonormal basis of the eigenspace by, e.g., the Gram-Schmidt orthonormalization. R em ark 3: In view of Section 13.2 the eigenvectors associated with a zero eigenvalue of C are called proper axis directions of the quadratic form v ^ v = 0.

18.5 N orm a l Form s o f E u clid ean Q uadrics Let xt Crx + 2ctx + c = 0 be the equation of some arbitrary quadric Q in £ n . Zero entries of C correspond to conjugate pairs of coordinate directions. In the sequel, it is shown that one can construct a Cartesian system such that all coordinate directions are conjugate with respect to Q and that the equation of Q takes on one of the following three normal forms

Figure 18.1: Euclidean normal forms.

184

Part IV Euclidean Geometry

Figure 18.1 gives an illustration. Recall from Section 14.4 that the first normal form represents a central quadric, where M. ^ 0 but S = 0; the second normal form represents a cone, where M = S ^ 0; and the third normal form represents a paraboloid, where M = 0 . The underlying Cartesian system is constructed in the following way: First, after a principal axis transformation x = B y , which is discussed in the previous section, one obtains the equation

where Second, one can rid the equation of the linear term by the translation = b + z, where

y

provided that there is a solution b. Subsequently one can multiply the equation of the quadric by a suitable factor in order to procure the normal form as a function of z. However, if Ab + c = o has no solution, the quadric is a paraboloid. Then one constructs the orthonormal basis b i , . . . , b n for the principal axis transformation such that b r+ i , ... , b n span the eigenspace Vo associated with the zero eigenvalue, where b r+ i , ... , b n_i are orthogonal to c. This is possible since c is not an element of Vo. After performing the axis transformation above, one solves the first n — 1 equations of the linear system Ab + c = o and chooses the last coordinate bn of b such that b tAb 4- 2c%b + c = 0 . Then, the translation y = b 4 *z changes the equation Q(y) = 0 into the normal form depending on z and multiplied by some factor. R e m a rk 4: The coefficients fii above represent the non-vanishing eigen­ values of the normal forms. On setting — ± 1 /a? the three normal forms can also be written as

Chapter 18 Quadrics in Euclidean Space

185

The positive numbers ai are called the semi-axes of the respective quadric. An axis is called real if jz* > 0 . Otherwise it is called a non-real or imaginary axis. E x am p le

1:

Consider the quadric in

£3

given by

The three coordinate axes are already parallel to the eigenvectors of C, i.e., = C. Obviously, b = —c is a solution of Ab + c = o. Hence, x —b + y transforms Q into y\ — 1. This example is illustrated in the left part of Figure 18.2. A

E x a m p le

2:

Consider the quadric in

£3

given by

The three coordinate axes are already parallel to the eigenvectors of C, i.e., A = C, but Am + c ^ o for all m . However, the basis vectors associated with the zero eigenvalue, the 2- and 3-direction, are not orthogonal to c. Therefore these vectors are replaced by t >2 = [0 Q —I?]* >

b3

= [0 g q]*

where g = V2/2 .

Then the quadric with respect to this new system is given by

Obviously, b = [1 0 6 f solves the first two equations of Ab + c = o, and the transformation y — b + z changes the equation of Q into z\ —qzz —(1 + q8) = 0 which is the normal form for S = —1/g. This example is illustrated in the right part of Figure 18.2.

Part IV Euclidean Geometry

186

Figure 18.2: Normal form, examples.

18.6 N o te s an d P ro b lem s 1

For n = 2 the angles a between the principal axes and the ar-axis satisfy

This follows from the coordinate transformation x = B y ,

where

becomes diagonal.

2 Homothetic quadrics in a Euclidean space have the same principal axes directions. 3 In £ n homothetic quadrics are represented by their normal form

4 Parallel intersections of a quadric have parallel principal axes directions.

Chapter 18 Quadrics in Euclidean Space

187

Figure 18.3: Parallel intersections. 5 Since the affine normal form can easily be derived from the Euclidean one, the Euclidean method can also be used in affine spaces to construct the affine normal form of a quadric. 6

Note 4 also verifies the existence of the affine normal forms in Sec­ tion 14.4.

7 A matrix C also represents a linear map (p. This means that the eigen­ vectors of C are the vectors mapped onto certain multiples of themselves under , respectively. From the string constructions it follows immediately th a t the isolines meet orthogo­

nally. Similarly, the parabolas of the family

Part IV Euclidean Geometry

194

where A varies, are confocal. The origin is the common focus and the xaxis the common axis. The parabolas of the family have the parametric representation where the isolines Ai = fixed < 0 and A2 = fixed > 0 represent confocal parabolas which meet orthogonally. Besides the position in the plane there is only one family of confocal parabolas, which is shown in the right part of Figure 19.4.

Figure 19.4: Confocal conic sections. R e m a r k Is

Trigonometric functions of tp are known to be rational

quadratic functions of

where u = 0, \ , 1 corresponds to (p = 0°, 90°, 180° and t — 0 ,1 , 0 0 . Using the quadratic Bernstein polynomials

Chapter 19 Focal Properties

195

one has, for example,

W ith the abbreviations Bi(u) = U{ and Bi(v) = V* this transformation can be applied to the parametric representation of confocal ellipses and hyperbolas which in homogeneous coordinates results in

This representation is called a rational Bernstein-B£zier representation. It is evaluated most effectively by the algorithm of de Casteljau in Sec­ tion 12.3.

10.5 Focal Conics A cone which circumscribes a sphere is called a right cone. Using the string constructions of ellipses and hyperbolas it can be shown that the vertices h of all real right cones which contain an ellipse E given by

lie on the hyperbola H defined by

Similarly, the vertices of all real right cones containing H lie on E .

196

Part IV Euclidean Geometry

Figure 19.5: Right cone to an ellipse.

Moreover, it can be shown that every cone circumscribing a quadric Q of the family

is a right cone if its vertex is on E or H . E and H are called the focal conics of Q, and the family of quadrics above is called confocal. In particular, one has for ellipsoids, hyperboloids of one sheet, hyperboloids of two sheets, as illustrated in Figure 19.6. For A > a 2 the members of the family have no real points. The focal conics E and H can be viewed as degenerate members of this family, where A = 0 and A = —b2, respectively.

Chapter 19 Focal Properties

197

Figure 19.6: Confocal central quadrics.

For any given point [x y z]1 there are three members of the fam­ ily containing this point. Furthermore, any two (or three) members Q i(x) —0 , Q 2 M —0 of the family intersect each other orthogonally since at a common point the scalar product of the normals of Q\ and Q 2 equals a multiple of Qi —Q 2 = 0 . R em ark 2: As a consequence of the right cone property above, E and H meet each quadric of the confocal family in its umbilical points (see Sections 15.6 and 32.1). In particular, if the interior of E is regarded as a very flat ellipsoid, the foci of E can be viewed as the umbilical points of this flat ellipsoid, and analogously for H . Note that the foci of E and H fire the vertices of H and E , respectively.

Part IV Euclidean Geometry

198

Rem ark 3: Similarly, the vertices of right cones which are circumscribed about a paraboloid lie on two so-called focal parabolas. The parabolas lie in perpendicular planes, and the focus of each parabola is the vertex of the other one.

19.6 Focal D istances The points e and h of the two focal conic sections E and H above have the parameter representations

respectively. By some standard algebraic manipulations one can compute the distance d between e and h,

Note that d has a sign. One has d > negative branch of H , respectively.

0

ord
# 3 are (homogeneous) projective coordinates of p with re­ spect to pQ) • • •) P 3 and p u . In particular, if r = n, the representation

together with the unit point p u = p 0 ^------ b p n defines the new projective coordinates in V with respect to the new frame p0, . . . , p n; ptt.

20.3 T he Equations o f P lanes and Subspaces A projective hyperplane of V n is an n —1 -dimensional subspace, i.e., the span of n independent points in V n . Hence, the solution of some linear equation represents a projective hyperplane and, conversely, each projective hyper­ plane has such an equation. The homogeneous coefficients u* are called projective hyperplane coordinates. An inhomogeneization shows that a projective hyperplane can be viewed as the union of an affine hyperplane in V n given by

with its ideal points v represented by the solution of [ui .. . wn] v =

0

Let S be the intersection of s + S forms the solution of

Then

1

hyperplanes given by