A Gateway to Modern Geometry: The Poincaré Half-Plane

Front Cover......Page 1
Title......Page 5
Contents......Page 7
Preface......Page 11
1.1. An Introduction to Euclidean Geometry......Page 13
1.2. Excerpts from Euclid\'s Elements......Page 15
1.3. Hilbert\'s Axiomatization (Optional)......Page 29
1.4. Variations on Euclid\'s Fifth......Page 33
2.1. Rigid Motions......Page 37
2.2. Translation, Rotation, Reflection......Page 39
2.3. Glide Reflection......Page 47
2.4. The Main Theorems......Page 50
2.5. Rigid Motions and Absolute Geometry......Page 51
3.1. An Interesting Nonrigid Transformation......Page 53
3.2. Inversions Applied (Optional)......Page 61
4.1. Hyperbolic Length......Page 65
4.2. Hyperbolic Straight Lines......Page 68
4.3. Hyperbolic Angles......Page 71
4.4. Hyperbolic Rigid Motions......Page 72
4.5. Riemannian Geometry (Optional)......Page 74
5.1. Euclid\'s Postulates Revisited......Page 79
5.2. Hyperbolic Geometry......Page 87
5.4. Hyperbolic Rigid Motions......Page 88
6.1. The Angles of the Hyperbolic Triangle......Page 91
6.2. Regular Tessellations (Optional)......Page 96
7.1. The General Definition of Area......Page 101
7.2. The Area of the Hyperbolic Triangle......Page 105
8.1. The Trigonometry of Hyperbolic Line Segments......Page 111
8.2. Hyperbolic Right Triangles......Page 113
8.3. The General Hyperbolic Triangle......Page 116
9.1. Euclidean Rigid Motions......Page 121
9.2. Hyperbolic Rigid Motions......Page 127
9.3. Hyperbolic Flow Diagrams I: Special Cases......Page 135
9.4. Hyperbolic Flow Diagrams II—The General Case......Page 142
9.5. Hyperbolic Rigid Motions: Construction......Page 145
10.1. The Sum of the Angles of the Triangle......Page 149
11.1. Geodesics on the Sphere......Page 155
11.2. Spherical Trigonometry......Page 158
11.3. Spherical Area......Page 162
11.4. A Digression Into Geodesy (Optional)......Page 164
11.5. Elliptic Geometry......Page 166
12.1. Differential Geometry......Page 169
12.2. A Review of Length and Area on Surfaces......Page 175
12.3. Gauss\'s Formula for the Curvature at a Point......Page 180
12.4. Riemannian Geometry Revisited......Page 182
13.1. Conformal Transformations......Page 191
13.2. The Cross-Ratio......Page 192
13.3. The Unit Disk Model and Its Flow Diagrams......Page 196
13.4. Explicit Rigid Motions of the Unit Disk Model......Page 202
13.5. The Riemann Metric of the Unit Disk Model......Page 205
13.6. Regular Tessellations of the Unit Disk Model......Page 208
14.1. The Beltrami-Klein Model......Page 211
15.1. History......Page 223
16. Spheres and Horospheres......Page 231
16.1. Hyperbolic Space and Its Rigid Motions......Page 232
16.2. Hyperbolic Geodesics......Page 235
16.3. The Stereographic Projection......Page 238
16.4. The Geometry of Spheres and Horospheres......Page 243
Appendix A: Proofs of Some of Euclid\'s Propositions......Page 247
Appendix B: Formulas for Hyperbolic Trigonometric Functions......Page 261
Sources, References, and Suggested Readings......Page 263
Index......Page 265
Back Cover......Page 272