Geometry

Cover......Page 1
Title......Page 4
Preface......Page 6
Contents......Page 10
1 The Geometry of Numbers......Page 14
1.1 Natural Numbers......Page 15
1.3 Multiplying Natural Numbers......Page 17
1.4 Square and Triangular Numbers......Page 18
1.6 Zero and Negative Numbers......Page 22
1.7 Rational Numbers or Fractions......Page 23
1.9 Rational Numbers as a Field......Page 27
1.10 Real Numbers......Page 28
Answers to Selected Questions in Chapter 1......Page 37
2.1 Coordinates......Page 42
2.2 Rn, the Space of Coordinates......Page 43
2.3 The Line through Two Points......Page 48
2.4 The Plane Containing Three Points......Page 49
2.5 Distance and Angle......Page 51
2.6 Polar Coordinates......Page 57
2.7 Area......Page 61
2.8 Hyperplanes......Page 64
2.9 Angles between Hyperplanes and NearestPoints to Hyperplanes......Page 68
Answers to Selected Questions in Chapter 2......Page 70
3.1 The Life of Euclid......Page 76
3.2 The Euclidean Axioms for the Plane......Page 77
3.3 Angles and Lines......Page 78
3.4 Some Basic Facts about Triangles......Page 80
3.5 General Polygons......Page 82
3.6 Congruences and Similarities......Page 86
3.7 Isosceles Triangles......Page 89
3.8 Circles......Page 91
3.9 Triangles and their Centres......Page 96
3.10 Metric Properties of Triangles......Page 100
3.11 Three Surprising (and Beautiful) Theorems......Page 102
Answers to Selected Questions in Chapter 3......Page 108
4 The Geometry of Complex Numbers......Page 112
4.1 What is sqrt(-1)?......Page 113
4.2 Modulus and Division......Page 116
4.3 Unimodular Complex Numbers and theUnit Circle......Page 117
4.4 Lines and Circles in the Complex Plane......Page 119
4.5 Manipulating Complex Numbers......Page 120
4.6 Infinity and the Riemann Sphere......Page 124
4.7 Division and Inversion......Page 128
4.8 Mobius Transformations......Page 131
4.9 Cross Ratios......Page 133
4.10 A Formula for the Cross Ratio......Page 138
4.12 Formulre for the nth Roots of Unity......Page 140
4.13 Solving Cubic and Biquadratic Polynomials......Page 143
Answers to Selected Questions in Chapter 4......Page 146
5.1 Points and Coordinates......Page 150
5.2 Scalar Product......Page 154
5.3 Cross Product......Page 155
5.4 The Scalar Triple Product......Page 157
5.5 The Vector Triple Product......Page 159
5.6 Planes......Page 160
5.7 Lines in Space......Page 162
5.8 Isometries of Space......Page 167
5.9 Projections......Page 175
5.10 Polyhedra......Page 179
Answers to Selected Questions in Chapter 5......Page 189
6 Projective Geometry......Page 195
6.1 The Projective Plane......Page 196
6.2 Lines in the Projective Plane......Page 197
6.3 Incidence and Duality......Page 198
6.4 Desargues\' Theorem......Page 200
6.5 Cross Ratios Again......Page 202
6.6 Cross Ratios and Duality......Page 205
6.7 Projectivities and Perspectivities......Page 207
6.8 Quadrilaterals......Page 211
6.9 Projective Transformations......Page 212
6.11 Pappus\' Theorem......Page 214
6.12 Perspective Drawing: Tricks of the Trade......Page 216
6.13 The Fano Plane......Page 219
Answers to Selected Questions in Chapter 6......Page 220
7 Conics and Quadric Surfaces......Page 223
7.1 Conic Sections......Page 224
7.2 The Conic as Quadratic Curve......Page 226
7.3 Focal Properties of Conics......Page 237
7.4 The Motion of the Planets......Page 243
7.5 Quadric Surfaces......Page 250
7.6 The General Quadric Surface......Page 256
Answers to Selected Questions in Chapter 7......Page 259
8 Spherical Geometry......Page 264
8.1 Geodesics......Page 266
8.2 Geodesic Triangles......Page 267
8.3 Latitude and Longitude......Page 271
8.4 Compass Bearings......Page 275
8.5 The Celestial Sphere......Page 280
8.6 Observer\'s Coordinates......Page 283
8.7 Time and Right Ascension......Page 285
Answers to Selected Questions in Chapter 8......Page 292
9.1 Extended Complex Numbers......Page 297
9.2 Multiplying Quaternions......Page 298
9.3 Inverses of Quaternions......Page 299
9.4 Real and Pure Parts of Quaternions......Page 301
9.5 Multiplying Quaternions and LinearTransformations of R4......Page 302
9.6 Octonions......Page 305
9.7 Vector Products in R7......Page 307
9.8 Octonions and Associativity......Page 309
9.9 Hexadecanions?......Page 310
Answers to Selected Questions in Chapter 9......Page 312
Bibliography......Page 314
Index......Page 316