| Preface | |
| The Euclidean Plane | |
| Approaches to Euclidean Geometry | p. 1 |
| Isometries | p. 2 |
| Rotations and Reflections | p. 5 |
| The Three Reflections Theorem | p. 9 |
| Orientation-Reversing Isometries | p. 11 |
| Distinctive Features of Euclidean Geometry | p. 14 |
| Discussion | p. 18 |
| Euclidean Surfaces | |
| Euclid on Manifolds | p. 21 |
| The Cylinder | p. 22 |
| The Twisted Cylinder | p. 25 |
| The Torus and the Klein Bottle | p. 26 |
| Quotient Surfaces | p. 29 |
| A Nondiscontinuous Group | p. 33 |
| Euclidean Surfaces | p. 34 |
| Covering a Surface by the Plane | p. 36 |
| The Covering Isometry Group | p. 39 |
| Discussion | p. 41 |
| The Sphere | |
| The Sphere S[superscript 2] in R[superscript 3] | p. 45 |
| Rotations | p. 48 |
| Stereographic Projection | p. 50 |
| Inversion and the Complex Coordinate on the Sphere | p. 52 |
| Reflections and Rotations as Complex Functions | p. 56 |
| The Antipodal Map and the Elliptic Plane | p. 60 |
| Remarks on Groups, Spheres and Projective Spaces | p. 63 |
| The Area of a Triangle | p. 65 |
| The Regular Polyhedra | p. 67 |
| Discussion | p. 69 |
| The Hyperbolic Plane | |
| Negative Curvature and the Half-Plane | p. 75 |
| The Half-Plane Model and the Conformal Disc Model | p. 80 |
| The Three Reflections Theorem | p. 85 |
| Isometries as Complex Functions | p. 88 |
| Geometric Description of Isometries | p. 92 |
| Classification of Isometries | p. 96 |
| The Area of a Triangle | p. 99 |
| The Projective Disc Model | p. 101 |
| Hyperbolic Space | p. 105 |
| Discussion | p. 108 |
| Hyperbolic Surfaces | |
| Hyperbolic Surfaces and the Killing-Hopf Theorem | p. 111 |
| The Pseudosphere | p. 112 |
| The Punctured Sphere | p. 113 |
| Dense Lines on the Punctured Sphere | p. 118 |
| General Construction of Hyperbolic Surfaces from Polygons | p. 122 |
| Geometric Realization of Compact Surfaces | p. 126 |
| Completeness of Compact Geometric Surfaces | p. 129 |
| Compact Hyperbolic Surfaces | p. 130 |
| Discussion | p. 132 |
| Paths and Geodesics | |
| Topological Classification of Surfaces | p. 135 |
| Geometric Classification of Surfaces | p. 138 |
| Paths and Homotopy | p. 140 |
| Lifting Paths and Lifting Homotopies | p. 143 |
| The Fundamental Group | p. 145 |
| Generators and Relations for the Fundamental Group | p. 147 |
| Fundamental Group and Genus | p. 153 |
| Closed Geodesic Paths | p. 154 |
| Classification of Closed Geodesic Paths | p. 156 |
| Discussion | p. 160 |
| Planar and Spherical Tessellations | |
| Symmetric Tessellations | p. 163 |
| Conditions for a Polygon to Be a Fundamental Region | p. 167 |
| The Triangle Tessellations | p. 172 |
| Poincare's Theorem for Compact Polygons | p. 178 |
| Discussion | p. 182 |
| Tessellations of Compact Surfaces | |
| Orbifolds and Desingularizations | p. 185 |
| From Desingularization to Symmetric Tessellation | p. 189 |
| Desingularizations as (Branched) Coverings | p. 190 |
| Some Methods of Desingularization | p. 194 |
| Reduction to a Permutation Problem | p. 196 |
| Solution of the Permutation Problem | p. 198 |
| Discussion | p. 201 |
| References | p. 203 |
| Index | p. 207 |
| Table of Contents provided by Blackwell. All Rights Reserved. |
