## GeometryThis 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 |

495 | |

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### Common terms and phrases

ABCD affine geometry affine transformation Apollonian circle Apollonian family asymptotic axis Ceva's Theorem collinear complex number composite cone conic with equation corresponding cross-ratio d-line d-triangle deduce defined denote Determine the equation diameter direct non-Euclidean transformation ellipse embedding plane equation x2 Euclidean geometry Euclidean transformation example extended line family of circles figure follows formula function Fundamental Theorem Hence homogeneous coordinates hyperbola ideal Points inversive geometry inversive transformation isometry lies line segment linear maps the points matrix associated meet midpoint Mobius transformation non-degenerate projective conic non-Euclidean distance non-Euclidean geometry non-Euclidean length non-Euclidean reflection non-zero obtain origin parabola parallel projection parametric passes perpendicular plane conic Point of intersection Points of RP2 polar Problem projective geometry projective transformation properties prove radius ratios real numbers respectively result right angles rotation of S2 Section sphere strategy Subsection tangent transformation that maps transformation which maps unit circle vector vertices

### Popular passages

Page ix - Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

Page 11 - The curve obtained is an ellipse if 0 < e < 1 , a parabola if e = 1, or a hyperbola if e > 1.

Page 15 - Let M be the foot of the perpendicular from P to the line KR. Let the plane through P, perpendicular to the axis of the cone, cut AA