Shadows Of The Circle: Conic Sections, Optimal Figures And Non-Euclidean Geometry 9789810234188, 981023418X

Table of contents :
Front Cover
Title Page
Copyright Page
Preface
Table of Contents
Chapter 1: An ellipse in the shadow
The ellipse as a plane section of a cylinder
The equation of the ellipse
A parametrization of the ellipse
The ellipse as a locus
Directrix for the ellipse
Geometrical determination of foci and directrices for the ellipse
The tangents of the ellipse
An application to gear wheel movements
Sources for Chapter 1
Chapter 2: With conic sections in the light
The ellipse as a plane section in a cone
Geometric determination of foci and directrices for a conic section
The parabola
The hyperbola
Hyperbolic navigational systems
Conic sections as algebraic curves
Epilogue
Sources for Chapter 2
Chapter 3: Optimal plane figures
Isosceles triangles
Perrons paradox
Some simple geometrical problems without solutions
A fundamental property of the real numbers
Maxima and minima of real-valued functions
The equilateral triangle as optimal figure
The square as optimal figure
The regular polygons as optimal figures
Some limit values for regular polygons
The isoperimetric problem
Epilogue: Elements of the history of the calculus of variations
Sources for Chapter 3
Chapter 4: The Poincare disc model of non-Euclidean geometry
Euclids Elements
The parallel axiom and non-Euclidean geometries
Inversion in a circle
Inversion as a mapping
Orthogonal circles and Euclids Postulate 1 in the hyperbolic plane
The notion of distance in the hyperbolic plane and Euclids Postulate 2
Isometries in the hyperbolic plane
Hyperbolic triangles and n-gons
The Poincare half-plane
Elliptic geometries
Sources for Chapter 4
Exercises
Index

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Sections

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The Truth of Shadows

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