# Geometry

Cambridge University Press, Apr 13, 1999 - Mathematics - 497 pages
This textbook demonstrates the excitement and beauty of geometry. The approach is that of Klein in his Erlangen programme: a geometry is a space together with a set of transformations of that space. The authors explore various geometries: affine, projective, inversive, non-Euclidean and spherical. In each case the key results are explained carefully, and the relationships between the geometries are discussed. This richly illustrated and clearly written text includes full solutions to over 200 problems, and is suitable both for undergraduate courses on geometry and as a resource for self study.

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### Contents

 15 Exercises 42 Affine Geometry 45 21 Geometry and Transformations 46 22 Affine Transformations and Parallel Projections 53 23 Properties of Affine Transformations 66 24 Using the Fundamental Theorem of Affine Geometry 73 25 Affine Transformations and Conies 82 26 Exercises 90
 61 NonEuclidean Geometry 263 62 NonEuclidean Transformations 272 63 Distance in NonEuclidean Geometry 284 64 Geometrical Theorems 297 65 NonEuclidean Tessellations 317 66 Exercises 323 Spherical Geometry 327 71 Spherical Space 328

 Projective Geometry Lines 94 31 Perspective 95 32 The Projective Plane RPē 102 33 Projective Transformations 114 34 Using the Fundamental Theorem 130 35 CrossRatio 136 36 Exercises 147 Projective Geometry Conics 151 41 Projective Conics 152 42 Tangents 165 43 Theorems 177 44 Duality and Projective Conics 195 45 Exercises 197 Inversive Geometry 199 51 Inversion 200 52 Extending the Plane 211 53 Inversive Geometry 228 54 Fundamental Theorem of Inversive Geometry 240 55 Coaxal Families of Circles 246 56 Exercises 257 NonEuclidean Geometry 261
 72 Spherical Transformations 331 73 Spherical Trigonometry 339 74 Spherical Geometry and the Extended Complex Plane 349 75 Exercises 358 The Kleinian View of Geometry 360 82 Projective Reflections 365 83 NonEuclidean Geometry and Projective Geometry 366 84 Spherical Geometry 371 85 Euclidean Geometry and NonEuclidean Geometry 373 A Primer of Group Theory 375 A Primer of Vectors and Vector Spaces 377 Solutions to the Problems 383 Chapter 2 404 Chapter 3 417 Chapter 4 436 Chapter 5 448 Chapter 6 467 Chapter 7 482 Index 495 Copyright

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