Non-Euclidean Geometry

(Mathematical Association of America Textbooks)

H. S. M. Coxeter

The Mathematical Association of America | 1998 | 355 páginas | rar - pdf | 11 Mb

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This is a reissue of Professor Coxeter's classic text on non-Euclidean geometry. It begins with a historical introductory chapter, and then devotes three chapters to surveying real projective geometry, and three to elliptic geometry. After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases of a more general 'descriptive geometry'. This is essential reading for anybody with an interest in geometry.

CONTENTS
I. THE HISTORICAL DEVELOPMENT OF NON-EUCLIDEAN GEOMETRY
1.1 Euclid 1
1.2 Saccheri and Lambert 5
1.3 Gauss, Wächter, Schweikart, Taurinus 7
1.4 Lobatschewsky 8
1.5 Bolyai 10
1.6 Riemann 11
1.7 Klein 13
II. REAL PROJECTIVE GEOMETRY: FOUNDATIONS
2.1 Definitions and axioms 16
2.2 Models 23
2.3 The principle of duality 26
2.4 Harmonic sets 28
2.5 Sense 31
2.6 Triangular and tetrahedral regions 34
2.7 Ordered correspondences 35
2.8 One-dimensional projectivities 40
2.9 Involutions 44
III. REAL PROJECTIVE GEOMETRY: POLARITIES, CONICS AND QUADRICS
3.1 Two-dimensional projectivities 48
3.2 Polarities in the plane 52
3.3 Conies 55
3.4 Projectivities on a conic 59
3.5 The fixed points of a collineation 61
3.6 Cones and reguli 62
3.7 Three-dimensional projectivities 63
3.8 Polarities in space 65
IV. HOMOGENEOUS COORDINATES
4.1 The von Staudt-Hessenberg calculus of points 7
4.2 One-dimensional projectivities 74
4.3 Coordinates in one and two dimensions 76
4.4 Collineations and coordinate transformations 8
4.5 Polarities 85
4.6 Coordinates in three dimensions 87
4.7 Three-dimensional projectivities 90
4.8 Line coordinates for the generators of a quadric 9
4.9 Complex projective geometry 94
V. ELLIPTIC GEOMETRY IN ONE DIMENSION
5.1 Elliptic geometry in general 95
5.2 Models 96
5.3 Reflections and translations 97
5.4 Congruence 100
5.5 Continuous translation 101
5.6 The length of a segment 103
5.7 Distance in terms of cross ratio 104
5.8 Alternative treatment using the complex line 10
VI. ELLIPTIC GEOMETRY IN TWO DIMENSIONS
6.1 Spherical and elliptic geometry 109
6.2 Reflection 110
6.3 Rotations and angles Ill
6.4 Congruence 113
6.5 Circles 115
6.6 Composition of rotations 118
6.7 Formulae for distance and angle 120
6.8 Rotations and quaternions 122
6.9 Alternative treatment using the complex plane 126
VII. ELLIPTIC GEOMETRY IN THREE DIMENSIONS
7.1 Congruent transformations 128
7.2 Clifford parallels 133
7.3 The Stephanos-Cartan representation of rotations by points 136
7.4 Right translations and left translations 138
7.5 Right parallels and left parallels 141
7.6 Study's representation of lines by pairs of points 146
7.7 Clifford translations and quaternions 148
7.8 Study's coordinates for a line 151
7.9 Complex space 153
VIII. DESCRIPTTE GEOMETRY
8.1 Klein's projective model for hyperbolic geometry 157
8.2 Geometry in a convex region 159
8.3 Veblen's axioms of order 161
8.4 Order in a pencil 162
8.5 The geometry of lines and planes through a fixed point . . 164
8.6 Generalized bundles and pencils 165
8.7 Ideal points and lines 171
8.8 Verifying the projective axioms 172
8.9 Parallelism 174
IX EUCLIDEAN AND HYPERBOLIC GEOMETRY
9.1 The introduction of congruence 179
9.2 Perpendicular lines and planes 181
9.3 Improper bundles and pencils 184
9.4 The absolute polarity 185
9.5 The Euclidean case 186
9.6 The hyperbolic case 187
9.7 The Absolute 192
9.8 The geometry of a bundle 197
X. HYPERBOLIC GEOMETRY IN TWO DIMENSIONS
10.1 Ideal elements 199
10.2 Angle-bisectors 200
10.3 Congruent transformations 201
10.4 Some famous constructions 204
10.5 An alternative expression for distance 206
10.6 The angle of parallelism 207
10.7 Distance and angle in terms of poles and polars 208
10.8 Canonical coordinates 209
10.9 Euclidean geometry as a limiting case 211
XI CIRCLES AND TRIANGLES
11.1 Various definitions for a circle 213
11.2 The circle as a special conic 215
11.3 Spheres 218
11.4 The in- and ex-circles of a triangle 220
11.5 The circum-circles and centroids 221
11.6 The polar triangle and the orthocentre 223
XII. THE USE OF A GENERAL TRIANGLE OF REFERENCE
12.1 Formulae for distance and angle 224
12.2 The general circle 226
12.3 Tangential equations 228
12.4 Circum-circles and centroids 229
12.5 In- and ex-circles 231
12.6 The orthocentre 231
12.7 Elliptic trigonometry 232
12.8 The radii 235
12.9 Hyperbolic trigonometry 237
XIII. AREA
13.1 Equivalent regions 241
13.2 The choice of a unit 241
13.3 The area of a triangle in elliptic geometry 242
13.4 Area in hyperbolic geometry 243
13.5 The extension to three dimensions 247
13.6 The differential of distance 248
13.7 Arcs and areas of circles 249
13.8 Two surfaces which can be developed on the Euclidean plane 251
XIV. EUCLIDEAN MODELS
14.1 The meaning of "elliptic" and "hyperbolic" 252
14.2 Beltrami's model 252
14.3 The differential of distance 254
14.4 Gnomonic projection 255
14.5 Development on surfaces of constant curvature 256
14.6 Klein's conformai model of the elliptic plane 258
14.7 Klein's conformai model of the hyperbolic plane 260
14.8 Poincaré's model of the hyperbolic plane 263
14.9 Conformai models of non-Euclidean space 264
XV. CONCLUDING REMARKS
15.1 HjelmsleVs mid-line 267
15.2 The Napier chain 273
15.3 The Engel chain 277
15.4 Normalized canonical coordinates 281
15.5 Curvature 283
15.6 Quadratic forms 284
15.7 The volume of a tetrahedron 285
15.8 A brief historical survey of construction problems . . . . 289
15.9 Inversive distance and the angle of parallelism 292
APPENDIX: ANGLES AND ARCS IN THE HYPERBOLIC PLANE 299
BIBLIOGRAPHY 317
INDEX 327