Circles, Spheres and Spherical Geometry

Preface
Contents
1 Inversion and Stereographic Projection
	1.1 Inversion of the Plane
	1.2 Euler\'s Triangle Theorem
	1.3 Steiner Cycles
	1.4 Inversions of 3-Space
	1.5 Soddy\'s Hexlet
	1.6 Stereographic Projection
	1.7 Appendix: Poncelet\'s Porism
	1.8 Exercises
	1.9 Notes
		1.9.1 Notes on Inversion in the Plane and in 3-Space
			Books and Book Chapters
			A Few Applications
		1.9.2 Notes on Euler\'s Triangle Theorem
		1.9.3 Notes on Steiner Chains and Steiner\'s Porism
			Books and Book Chapters
			Some Research Results
		1.9.4 Notes on Soddy\'s Hexlet
			Books and Book Chapters
			Some Research Results
		1.9.5 Notes on the Stereographic Projection
			Some History and Applications
			Books and Book Chapters
			Some Research Results: Differential Geometry
			Cartography and Geodesy
			Classical Geometry, Foundations of Geometry
			Further Topics
		1.9.6 Notes on Poncelet\'s Porism
			Some History
			Books and Book Chapters
			Expository Chapters and Articles
			Some Research Results
2 Bend Formulas
	2.1 Bend Formulas for Steiner Cycles
	2.2 Bend Formula for Soddy\'s Hexlet
	2.3 Descartes\' Circle Theorem
	2.4 Soddy\'s Formula
	2.5 Linked 4-Cycle Pairs
	2.6 Exercises
	2.7 Notes
		2.7.1 Notes on Descartes\' Circle Theorem and Soddy\'s Formula
		2.7.2 Notes on Apollonian Circle Packings
		2.7.3 Notes on Apollonius\' Touching Problem
			Book Chapters and Surveys
			Some Research Results
3 Graphs and Circle-Systems
	3.1 Graphs and Planar Graphs
	3.2 Quadrangulations
	3.3 Orthogonal-Circle Representations
	3.4 3-Connected Graphs and Radial Graphs
	3.5 The Coin Graph Theorem
	3.6 The Theorem of Steinitz
	3.7 Exercises
	3.8 Notes
		3.8.1 Notes on the Coin Graph Theorem
			Book Parts and Surveys
			Some Research Papers
		3.8.2 Notes on the Theorem of Steinitz
			Book Chapters
			Surveys
			Some Research Results and Directions
4 Spherical Geometry I
	4.1 Geodesic Segments
	4.2 Cylindrical Projections
	4.3 Spherical Polygons
	4.4 The Inscribed Angle Theorem
	4.5 The Polar Set
	4.6 Exercises
	4.7 Notes
		4.7.1 Notes on Spherical Geometry
		4.7.2 Notes on Cylindrical Projections and Related Topics
			Complete Books
			Book Chapters
			Survey-Like and Historical Articles
		4.7.3 Notes on Spherical Polygons
			Books, Book Parts, and Surveys
			Some Research Contributions
5 Spherical Geometry II
	5.1 The Cesàro Triangle
	5.2 Edge-Lengths of Cesàro Triangles
	5.3 Spherical Cosine Law and Sine Law
	5.4 The Triangle Comparison Theorem for Spheres
	5.5 Triangles with Two Fixed Edge-Lengths
	5.6 The Isoperimetric Theorem for Quadrilaterals
	5.7 Exercises
	5.8 Notes
		5.8.1 Notes on Spherical Trigonometry and the Respective Cosine and Sine Laws
			Some History
			More Recent Representations
			A Few Related Journal Articles
		5.8.2 Notes on Triangle Comparison Theorems
			Some Historical Remarks
			The Comparison Theorem in Several Books
			Research Papers
6 The Problem of Thirteen Balls
	6.1 The Lemma on Proper Diagonals
	6.2 Major Triangles
	6.3 On a Triangulation of a Quadrilateral
	6.4 The Problem
	6.5 Solution
	6.6 Exercises
	6.7 Notes
		6.7.1 Notes on the Problem of Thirteen Balls
			The Problem Itself and Its Proofs
			First Generalizations: Related Book Chapters and Surveys
			Further Extensions and Variations
			Related Problems
7 Spherical Geometry III
	7.1 Spherical Ellipses
	7.2 Lexell\'s Theorem
	7.3 Equilateral Spherical Triangles
	7.4 Spherical Polygons Inscribed in a Cap
	7.5 Regular Spherical Polygons
	7.6 Appendix
		7.6.1 Orthogonal Projections of a Spherical Ellipse
		7.6.2 Proof of Lemma 7.1
	7.7 Exercises
	7.8 Notes
		7.8.1 Notes on Spherical Ellipses
		7.8.2 Notes on Lexell\'s Theorem
			The Theorem Itself and Different Proofs
			Some History
			Extensions and Modifications
8 Geometric Probability on the Sphere
	8.1 Random Points on a Sphere
	8.2 Random Spherical Triangles
	8.3 Random Quartets
	8.4 Wendel\'s Theorem
	8.5 Crofton\'s Formula
	8.6 Random Points on a Hemisphere
	8.7 Santaló\'s Chord Theorem
	8.8 Exercises
	8.9 Notes
		8.9.1 Notes on Random Points and Triangles on the Sphere
			Book Parts and Surveys
			Some Selected Research Papers
		8.9.2 Notes on Crofton\'s Formulae
			Some History
			Later Basic Books
			Surveys
			Some Selected, Recent Research Papers
9 Intersection Graphs of Spherical Caps
	9.1 Connectivity
	9.2 On the Intersection of a Few Equal Caps
	9.3 Intersection Graphs of Equal Caps
	9.4 Random Spherical Caps
	9.5 Chebyshev\'s Inequality and a Lemma
	9.6 Asymptotic Probability
	9.7 Exercises
	9.8 Notes
		9.8.1 Notes on Intersection Graphs
			Books and Book Parts
			Surveys
		9.8.2 Notes on Chebyshev\'s Inequality
			Book Parts
			Surveys and Expositions
			Selected Research Papers
10 Quartets on a Sphere
	10.1 Metric Spaces
	10.2 The Vertices of a Convex Quadrilateral
	10.3 Unispherical Quartets
	10.4 An Equation for the Radius
	10.5 A Planar Quartet That Is also Spherical
	10.6 Exercises
	10.7 Notes
		10.7.1 Notes on Four-Point Systems
11 Higher Dimensions
	11.1 Figures in High Dimensions
	11.2 Moser\'s Paradox
	11.3 The Volume of a Ball
	11.4 Two Lemmas
	11.5 Kissing a Small Ball
	11.6 Exercises
	11.7 Notes
		11.7.1 Notes on Cubes
			Sections of Cubes
			Projections of Cubes
			Further Topics
		11.7.2 Notes on Simplices
			Books, Book Chapters and Surveys
			Selected Properties of General Simplices
			Selected Results on Special Types of Simplices
		11.7.3 Notes on the Volume of (Unit) Balls
			High Dimensional Phenomena
			Inequalities
			On Ball Volumes (Mainly) in Three Dimensions
			Further Results
12 The Cayley-Menger Determinant
	12.1 The Cayley-Menger Matrix
	12.2 Odd Integral Distances
	12.3 Orthogonal-Sphere Systems
	12.4 Tangent-Sphere Systems
	12.5 A Generalization of Ptolemy\'s Theorem
	12.6 Exercises
	12.7 Notes
		12.7.1 Notes on Cayley-Menger Determinants
			Assignment and Some History
			Overviews in Books, and Surveys
			Simplex Geometry
			Extensions and Modifications
			Applications
13 Casey\'s Theorem
	13.1 Bicolored Sets of Circles
	13.2 Inversion Invariants
	13.3 Isometric Radii Change
	13.4 Proof of Casey\'s Theorem
	13.5 A Theorem on Five Circles
	13.6 An Extension of Casey\'s Theorem
	13.7 Exercises
	13.8 Notes
		13.8.1 Notes on Ptolemy\'s Theorem
			Some History
			Book Chapters and Surveys
			Extensions of Ptolemy\'s Theorem
			Interesting Proof Methods
			Applications
		13.8.2 Notes on Casey\'s Theorem
			The Theorem Itself and Some History
			Extensions, Modifications, and Applications
14 Solutions to the Selected Exercises
Bibliography
Index