Chapter 1．The Euclidean Plane 1
1.1 Approaches to Euclidean Geometry 1
1.2 Isometries 2
1.3 Rotations and Reflections 5
1.4 The Three Reflections Theorem 9
1.5 Orientation-Reversing Isometries 11
1.6 Distinctive Features of Euclidean Geometry 14
1.7 Discussion 18
Chapter 2．Euclidean Surfaces 21
2.1 Euclid on Manifolds 21
2.2 The Cylinder 22
2.3 The Twisted Cylinder 25
2.4 The Torus and the Klein Bottle 26
2.5 Quotient Surfaces 29
2.6 A Nondiscontinuous Group 33
2.7 Euclidean Surfaces 34
2.8 Covering a Surface by the Plane 36
2.9 The Covering Isometry Group 39
2.10 Discussion 41
Chapter 3．The Sphere 45
3.1 The Sphere S2 in R3 45
3.2 Rotations 48
3.3 Stereographic Projection 50
3.4 Inversion and the Complex Coordinate on the Sphere 52
3.5 Reflections and Rotations as Complex Functions 56
3.6 The Antipodal Map and the Elliptic Plane 60
3.7 Remarks on Groups,Spheres and Projective Spaces 63
3.8 The Area of a Triangle 65
3.9 The Regular Polyhedra 67
3.10 Discussion 69
Chapter 4．The Hyperbolic Plane 75
4.1 Negative Curvature and the Half-Plane 75
4.2 The Half-Plane Model and the Conformal Disc Model 80
4.3 The Three Reflections Theorem 85
4.4 Isometries as Complex Functions 88
4.5 Geometric Description of Isometries 92
4.6 Classification of Isometries 96
4.7 The Area of a Triangle 99
4.8 The Projective Disc Model 101
4.9 Hyperbolic Space 105
4.10 Discussion 108
Chapter 5．Hyperbolic Surfaces 111
5.1 Hyperbolic Surfaces and the Killing-Hopf Theorem 111
5.2 The Pseudosphere 112
5.3 The Punctured Sphere 113
5.4 Dense Lines on the Punctured Sphere 118
5.5 General Construction of Hyperbolic Surfaces from Polygons 122
5.6 Geometric Realization of Compact Surfaces 126
5.7 Completeness of Compact Geometric Surfaces 129
5.8 Compact Hyperbolic Surfaces 130
5.9 Discussion 132
Chapter 6．Paths and Geodesics 135
6.1 Topological Classification of Surfaces 135
6.2 Geometric Classification of Surfaces 138
6.3 Paths and Homotopy 140
6.4 Lifting Paths and Lifting Homotopies 143
6.5 The Fundamental Group 145
6.6 Generators and Relations for the Fundamental Group 147
6.7 Fundamental Group and Genus 153
6.8 Closed Geodesic Paths 154
6.9 Classification of Closed Geodesic Paths 156
6.10 Discussion 160
Chapter 7．Planar and Spherical Tessellations 163
7.1 Symmetric Tessellations 163
7.2 Conditions for a Polygon to Be a Fundamental Region 167
7.3 The Triangle Tessellations 172
7.4 Poincaré's Theorem for Compact Polygons 178
7.5 Discussion 182
Chapter 8．Tessellations of Compact Surfaces 185
8.1 Orbifolds and Desingularizations 185
8.2 From Desingularization to Symmetric Tessellation 189
8.3 Desingularizations as(Branched) Coverings 190
8.4 Some Methods of Desingularization 194
8.5 Reduction to a Permutation Problem 196
8.6 Solution of the Permutation Problem 198
8.7 Discussion 201
References 203
Index 207