Geometry, a metric approach with models

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Preface......Page 8
Computers and Hyperbolic Geometry......Page 11
Contents......Page 12
1.1 Axioms and Models......Page 16
1.2 Sets and Equivalence Relations......Page 18
1.3 Functions......Page 24
2.1 Definition and Models of Incidence Geometry......Page 32
2.2 Metric Geometry......Page 42
2.3 Special Coordinate Systems......Page 52
3.1 An Alternative Description of the Cartesian Plane......Page 57
3.2 Betweenness......Page 62
3.3 Line Segments and Rays......Page 67
3.4 Angles and Riangles......Page 74
4.1 The Plane Separation Axiom......Page 78
4.2 PSA for the Euclidean and Poincard Planes......Page 85
4.3 Pasch Geometries......Page 90
4.4 Interiors and the Crossbar Theorem......Page 96
4.5 Convex Quadrilaterals......Page 101
5.1 The Measure of an Angle......Page 105
5.2 The Moulton Plane......Page 112
5.3 Perpendicularity and Angle Congruence......Page 119
5.4 Euclidean and Poincard Angle Measure (optional)......Page 124
6.1 The Side-Angle-Side Axiom......Page 139
6.2 Basic TNangle Congruence Theorems......Page 146
6.3 The Exterior Angle Theorem and Its Consequences......Page 150
6.4 Right Ttiangles......Page 158
6.5 Circles and Their Thngent Lines......Page 165
6.6 The TWo Circle Theorem (optional)......Page 175
6.7 The Synthetic Approach......Page 180
7.1 The Existence of Parallel Lines......Page 184
7.2 Saccheri Quadrilaterals......Page 193
7.3 The Critical Function......Page 202
8.1 Asymptotic Rays and Thiangles......Page 211
8.2 Angle Sum and the Defect of a Ttiangle......Page 220
8.3 The Distance Between Parallel Lines......Page 229
9.1 Equivalent Forms of EPP......Page 239
9.2 Similarity Theory......Page 245
9.3 Some Classical Theorems of Euclidean Geometry......Page 254
10.1 The Area Function......Page 263
10.2 The Existence of Euclidean Area......Page 271
10.3 The Existence of Hyperbolic Area......Page 279
10.4 Bolyai\'s Theorem......Page 287
11.1 Collineations and Isometrics......Page 300
11.2 The Klein and Poincarb Disk Models (optional)......Page 312
11.3 Reflections and the Mirror Axiom......Page 320
11.4 Pencils and Cycles......Page 328
11.5 Double Reflections and Their Invariant Sets......Page 335
11.6 The Classification of Isometrics......Page 343
11.7 The Isometry Group......Page 351
11.8 The SAS Axiom in  H......Page 356
11.9 The Isometry Groups of E and H......Page 366
Bibliography......Page 374
Index......Page 376
Undergraduate Texts in Mathematics ......Page 386
Back Cover......Page 388