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Coxeter H.S.M. — Non-Euclidean Geometry
Coxeter H.S.M. — Non-Euclidean Geometry

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Название: Non-Euclidean Geometry

Автор: Coxeter H.S.M.

Аннотация:

Throughout most of this book, non-Euclidean geometries in spaces of two or three dimensions are treated as specializations of real projective geometry in terms of a simple set of axioms concerning points, lines, planes, incidence, order and continuity, with no mention of the measurement of distances or angles. This synthetic development is followed by the introduction of homogeneous coordinates, beginning with Von Staudt's idea of regarding points as entities that can be added or multiplied. Transformations that preserve incidence are called colineations. They lead in a natural way to elliptic isometries or "congruent transformations". Following a recommendation by Bertrand Russell, continuity is described in terms of order. Elliptic and hyperbolic geometries are derived from real projective geometry by specializing an elliptic or hyperbolic polarity which transforms points into lines (in two dimensions) or planes (in three dimensions) and vice versa.

An unusual feature of the book is its use of the general linear transformation of coordinates to derive the formulas of elliptic and hyperbolic trigonometry. The area of a triangle is related to the sum of its angles by means of an ingenious idea of Gauss. This treatment can be enjoyed by anyone who is familiar with algebra up to the elements of group theory.


Язык: en

Рубрика: Математика/Геометрия и топология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 6-th edition

Год издания: 1998

Количество страниц: 175

Добавлена в каталог: 12.11.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abscissa      74 104 107 206
Absolute      ix 126 153—157 195 199 224 252
Absolute geometry      11 17 26 179—185
Absolute involution      96 104 198
Absolute points      106
Absolute polarity      95 110 128 157 181 185 193 211 212 224 252 296
Acute segment      103
Acute-angle hypothesis      5
Addition of points      71
Affine geometry      18 24 31 35 77 96 159 178
Altitudes of a triangle      220 223 231 233 289
Analytical geometry      71—94 121—127 132 148—153 156 209—212 224—231 253—265 281—285
Angle descriptive      163 180
Angle metrical      6 14 107 112 127 129 132 207—211 225 260
Angle of parallelism      9 189 208 239 277 295—296 300 305 312
Angle-bisector      200 220
Angle-sum of a triangle      2 7 191 243 263
Apollonius of Perga      48
Appendix      299 316
ARC      249 303 307
Area      6 18 241 253
Area of a circle      8 250
Area of a horocyclic sector      243 250 258
Area of a triangle      7 9 243—247 257
Area of the elliptic plane      242 250
Areal coordinates      237
Aristaeus      48
Associated reguli      63 69 272
Associative law      72
Astral geometry      7
Astronomy      9
Asymptotic approach of parallel lines      6 210 257 301
Axial pencil      21; see also Pencil generalized
Axiom      16
Axiom Pasch's      20 (2.115) 162
Axiom Playfair's      3 186
Axioms of congruence      180
Axioms of continuity      23 36 162 174 194
Axioms of descriptive geometry      161
Axioms of incidence      20
Axioms of parallelism      186 187 197
Axioms of plane hyperbolic geometry      199
Axioms of plane inversive geometry      297
Axioms of plane projective geometry      28
Axioms of separation      22 174
Axis of a circle      116 213
Axis of a pencil of planes      21
Axis of a projectivity      60
Axis of a rotation      130
Axis of a translation      136 202
Axis of an equidistant curve      213
Baker, H.E.      18 28 166—171 232—233
Baldus, Richard      204 213 252
Bellavitis, Giusto      59
Beltrami, Eugenio      252 255—258 260 262 264
Bernays, Paul      24
BETWEEN      18 23 159
Bibliography      317—325
Bilinear relation      75
Binary expansion      101
Birectangle, isosceles      5 190 241 308—309
Bisection      100 200 268 297
Bolyai's parallel construction      204
Bolyai, Farkas      2 10 301
Bolyai, Janos      10 17 157 179 204 215 220 250 277 298 300
Bonola, Roberto      3 134 171 199 210 211 215 252 302 315
Boole, George      122
Brianchon, C.J.      59 62 200
Bundle (of lines and planes)      14 21
Bundle generalized      166—171 184 197 213 218
Bundle of parallels      24 177 184 187 195
Bundles, correlated      219
Canonical form for a conic      87 209
Canonical form for a polarity      86 91 249
Canonical form for a quadric      92
Canonical form for an involution      76
Carroll, Lewis      301
Carslaw, H.S.      176 188 199 206 210 215 241 244 249 250 259 267
Cartan, Elie      89 94 137
Cartesian coordinates      24 211 253—256 258—260 262 312
Categorical      160 186 196
Cayley, Arthur      13 109 122 125 126 149 157 182 196 226 266
Central projection      256
Centre of a (flat) pencil      20
Centre of a bundle      21
Centre of a circle      1 213
Centre of a projectivity      60 203
Centre of a rotation      111 202
Centroid      31 221 230 236 239
Chasles, Michel      53 106 217
circle      1 2 14 115—117 122 181 213—218 221—222 226—231 237 252 259—264; Area Circumference Coaxal
Circular points at infinity      106
Circum-centres      221 223 230
Circum-radius R      230 236 239
Circumferences      8 11 250
Circumscribed circles (or Circum-circles)      221 230 237
class      16
Clifford parallel      134 151 154
Clifford parallelogram      142
Clifford surface      143—145 148 153 155 251
Clifford translation      135—142 149 155
Clifford, W.K.      128 270
Coaxal circles      261
Coaxal circles, Clifford surfaces      145
Coaxial planes      21
Collinear      20 80
Collineation      49 60—63 82 90
Colours      ix
Colunar triangles      220 233
Common parallel      205
Common perpendicular      133 147 158 191 204 272
Complex geometry Euclidean      106 261
Complex geometry non-Euclidean      207 212 217
Complex geometry projective      94 106 126 153 207
Complex number      94 107 258 262
Complex, linear      69 92
Composition of rotations      118—120 125
Cone, quadric      26 62 178
Cone, right circular      14 48
Conformal mapping      258—265 295
Congruence (relation)      18 95 100 103 106 115 132 179 201 241
Congruence, linear      92 151 155
Congruent transformation      95 113 126 130—132 139—141 150 154 179 201—203 245; Opposite
conic      26 48 56—62 87 199 215
Conjugate lines      52 65 86 252
Conjugate planes      65
Conjugate points      52 65 86 95 96 106
Conjugate quaternions      123
Consistency of axioms      4 10 23 97 197 255
Construction for a common parallel      205
Construction for a common perpendicular      191
Construction for parallels to a given line      204 289—291
Contact      see Double Quadruple Ring Third
Continuity      23 74 101 162 174 187 194 201
Convex region      159 162
Coolidge, J.L.      117 134 151 229 248
Coordinate transformation      76 83
coordinates      see Areal Cartesian Homogeneous Non-homogeneous Pliicker Study Trilinear
Coplanar      20
Correlation      51 65 85 91 219
Correspondence      16 52 215; Ordered
cosh      206
Coxeter, H.S.M.      9 201 275 288 293—295
Craats, Jan van de      314
Cremona, Luigi      59
Cross (angle)      112
Cross ratio      76 84 90 105 107 120 126—132 259 262 299
Curvature of space      11 212
Curvature, Gaussian (or Specific)      12 144 211 257
Curve      248
CYCLE      213 309
Cyclic order on a line      18
de la Hire, Philippe      29
de Sitter, Willem      209
Dedekind, Richard      19 74 101 162
Defect, angular      6 246 301
Degenerate involution      71
Degenerate polarity      186 212
Dehn, Max      166—173 179 187 194
Desargues, Girard      14 28 172
Descriptive geometry      159—178
Developable surfaces      144 220 251
Diagonal triangle      27 56
Diameter      213 252
DICTIONARY      14 24—26 97 266
Differential geometry      12 15 247—251 255—257 260 265 283
Dihedral angle      129 163
dimensions      21
Direct collineation      64
Direct congruent transformation      132 140 141 146 154 202
Direct correspondence      36
Direct projectivity      43
Directed angle      112
Directrices of a congruence      93 155
Distance      225; see also Elliptic Hyperbolic
Distributive law      72
Dodgson, C.L.      301
Donkin, W.E.      109 119 122 136—138
Donnay, J.D.H.      304
Double contact      127 217 260
Double perspectivity      34
Double point      36
Double rotation      131 141 150
Doubly oriented line      34 137
Doubly-asymptotic triangle      188 210 245 301
Duality      19 26 51 62 165
Dummy suffix convention      83
Duplication of the cube      48
Elements of Euclid      1 115 181—184 188 191
ellipse      18 48 252—254
Elliptic (linear) congruence      93 151 155
Elliptic distance      14 103 117 120—122 126 132 254—255
Elliptic geometry      4 13—15 95—156 207 220—223 242 252—255 260 264—266
Elliptic involution      45—47 96 104 252
Elliptic line      55 96—108 112 253
Elliptic point      55
Elliptic polarity      54 66 86 96 110 157 252
Elliptic projectivity      43 75 203
Elliptic trigonometry      232—238 272
Engel, Friedrich      6 277
Enriques, Federigo      29 36—45 54—56 61—63 101
Equidistant curve      213—217 222 227 260 309 312
Equidistant surface      218
Equivalent regions      241
Erlanger Programm      17 115
Escribed circles (or Ex-circles)      221 231 239
Euclid      1 18 48 175 181 197; Postulates
Euclidean geometry      6 9 17 96 99 106 117 144 178 186—187 197 202 220 251—266 297—298
Euclidean geometry as a limiting case      8 212 235 237 242
Euler, Leonhard      109
Ewald, Gunther      297
Factor of proportionality      82
Fibonacci numbers      293
Finite but unbounded space      11 301
Fixed points of a collineation      61
Flat pencil (of lines)      6 20 158 175
Flat pencil generalized      199 213 261
Forder, H.G.      112 186 187 195 272 299
Forms, primitive      21
Four-dimensional space      12 26
Fractional linear transformation      75
Fundamental theorem of projective geometry      42 76
Gauss, C.F.      2 7 8 144 157 175 176 190 215 234 247 257 275
Genealogy of geometries      19
Generators of a quadric      62 68 94
Generators of the Absolute      153—156
Geodesies      144 256—257
Geographical sphere      298
Glide-reflection      201—203 267
Gnomonic projection      256
Goursat, Edouard      150
Greuzer, S.L.      292
Group      16 103 115 138 140 145
H(AB, CD)      28
Half-plane      163 263 299
Half-space      163 178 302
Halsted G.B.      267
Hamilton, Sir W.R.      109 119 122
Hamlet      ix
Harmonic conjugate      28 39 42 45 76 81 107
Harmonic homology      50 64 65 81 116 203 228
Harmonic set of lines      30 51
Harmonic set of points      28 40 51
Heffter, Lothar      18 84 90
Hesse, L.O.      53 223
Hessenberg, Gerhard      71
Hexagon      59 201
Hilbert, David      24 179 180 191 194 195 205 241 301
Hippocrates of Chios      1 48
Hire, Philippe de la      29
history      1—15 48 226 277 289—291
Hobson, E.W.      254
Holgate, T.F.      39
Homogeneous coordinates      25 76—94 121 124—127 132 148—153 156 209—212 224—233 236 240 253
Homogeneous space      15 256
Homography      265 266
Homology      see Harmonic
Horocycle      9 213 214 217—222 227 239 244 250 257 290 303 308
Horosphere      7 9 218—220 251 280 311
Hyper-space      12 14 26 94 160 265
Hyper-sphere      12 264
Hyperbola      18 48 252—254
Hyperbolic      ix
Hyperbolic cosine      206
Hyperbolic distance      157 201 206—211 254—255 294—295
Hyperbolic functions      206 254 292—296
Hyperbolic geometry      4—11 157 178 187—210 213—223 243—247 252—258 260—266 300
Hyperbolic involution      45 76 252
Hyperbolic line      55 201 253
Hyperbolic point      55
Hyperbolic polarity      54 56 86 110 157 252
Hyperbolic projectivity      43 75 203
Hyperbolic trigonometry      220 237—240 277—282
Hypercycle      see Equidistant curve
Hyperosculation      218
Ideal line      171 185 199 209
Ideal plane      173 185
Ideal point      158 171—178 185 199 209 300; Ultra-infinite
Imaginary geometry      10 158
Improper bundle      166 174 184
In-centre      200 221 231 236
In-radius r      221 231 235 237 239
incidence      18 20 80 171 203
Infinitesimal      9 212 235 238 298
infinity      ix see line Plane
Inscribed circle (or In-circle)      221 231
Interior of a conic      56 87
Interior of an interval      23 36—38 41
Intermediacy      159
Intersection condition for lines      89 152 271
interval      23 36—38 41 161
Invariant      17 77;
Inverse correspondence      17
inversion      259 263 276 297
Inversive distance      265 292 296 299 313
Involution      44 61 76 203 252; Degenerate Elliptic Hyperbolic
Involutory collineation      50
Isosceles birectangle      5 190 241 308—309
Isotropic space      15
Johnson, R.A.      112 221
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