Geometry

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Cambridge University Press, Apr 13, 1999 - Mathematics - 497 pages
3 Reviews
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
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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

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About the author (1999)

David A. Brannan is Emeritus Professor in the Department of Mathematics and Computing at The Open University, Milton Keynes.

Matthew F. Esplen is a Lecturer in the Department of Mathematics and Statistics at The Open University, Milton Keynes.

Jeremy Gray is Professor of the History of Mathematics and Director of the Centre for the History of the Mathematical Sciences at the Open University in England, and is an Honorary Professor in the Mathematics Department at the University of Warwick. He is the author, co-author, or editor of 14 books on the history of mathematics in the 19th and 20th Centuries and is internationally recognised as an authority on the subject. His book, Ideas of Space, is a standard text on the history of geometry (see competitive literature).

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