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| Semple J.G., Kneebone G.T. — Algebraic projective geometry |
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| Предметный указатель |
Absolute conic in euclidean geometry 255
Absolute conic in non-euclidean geometry 97
Absolute invariant 194
Absolute points 34 93
Absolute points, in trilinears 128 (Ex. 21)
Absolute quadric 356
Affine geometry 1-dimensional 64
Affine geometry 2-dimensional 91
Affine geometry 3-dimensional 254
Algebraic correspondences 176
Algebraic curves and envelopes 102
Allowable coordinate representations 41
Angle, modulus and Laguerre definition 93
Apolarity 184
Associated points, in space 339
Associated points, in the plane 321
Asymptotes 119 313
Auxiliary circle 127 (Ex. 20)
Axes, of a conic 121 169
Axes, of a quadric 284
Axes, of planes for a range of quadrics 336
Axes, of points for a pencil of quadrics 330
Biaxial collmeation 350
Bifocal plane-pair 269 278
Birational transformation 230
Bitangents of a/plane quartic 344
Brianchon's theorem 143
Cayley's notation 104
Centre locus of a pencil or range of conies 164
Centre, of a conic 119
Characteristic roots and vectors 211 286 365 16)
Chasles's Theorem 20 133
Circular points 32
Class of an envelope 102
Clifford parallels 357
Coaxal circles 158 165
Collinear fields and stare 254
Collinear fields and stare, reduced equations 259 (Ex. 12)
Collineations (plane) 205
Collineations (plane), a geometrical haracterization 238 (Ex. 6)
Collineations (plane), cyclic 218—221
Collineations (plane), involutory 218
Collineations (plane), permutable 239 (Ex. 12)
Collineations (plane), real 216
Collineations (plane), which leave a conic invariant 221 240
Collineations (space) 348
Collineations (space), involutory 353
Collineations (space), which leave a quadric invariant 354 364
Collineations (space), which leave a twisted cubic invariant 364 (Ex. 5)
Complex points 12
Complexes (of lines) 367 371
Concomitants 200
Confocal quadrics 336
Confocal quadrics, complex of normals 376
Confoeal conies 166 174
Congruence transformations 358
Congruences (of lines) 367 377
conic 19 27 103
Conic, as a one-dimensional projective domain 132
Conic, axes 121
Conic, canonical parametric representation 131
Conic, centre 119
Conic, conjugate diameters 120
Conic, degenerate forms 112 117
Conic, director circle 123
Conic, discriminant 112
Conic, foci 123 294
Conic, freedom 105
Conic, normals 150
Conic, projective generation 136
Conic, proper 116
Conic, self-duality 115
Conic, space dual of a quadric cone 269
Conic, special forms of equation 129
Conies, apolar 188
Conies, harmonically circumscribed or inscribed 148
Conies, invariants of a pair 194
Conies, nets of conies 170
Conies, pencils and ranges 156
Conies, reciprocation of one conic into another 148
Conies, representation by points of  106 388
Conies, triangularly circumscribed or inscribed 147
Conjugate lines for a quadric 271
Conjugate points, for a conio 108
Conjugate points,, for a correlation 225
Coordinates, allowable systems 41
Coordinates, areal 25
Coordinates, dual (tangential) 79 244
Coordinates, homogeneous and non-homogeneous 40 41
Coordinates, homogeneous cartesian 14 22
Coordinates, homogeneous, projective 23
Coordinates, Plucker (or Graasmann) 246
Coordinates, tetrahedral 260 (Ex. 22)
Coordinates, trilinear 25
Correlations (plane) 224
Correlations (plane), generating (2, 2) correspondences on a conic 240 (Ex. 19)
Correlations (space) 359 364
Covariants and contra variants 199
Cremona transformations 230
Critical points of a correspondence 177
Cross axis, of related ranges on a conic 141
Cross axis, of related ranges on lines 89
Cross ratio 17 26 45
Cross ratio, of pairs of points on a conic 133
Cross ratio, six cross ratios of four numbers 47
Cross ratio, symmetric function of six cross ratios 69 (Ex. 18)
Cubic curves (plane), nodal 240 (Exx. 25 26)
Cubic curves (plane), projections of twisted cubic 299 325
Cubic curves (plane), property of non-singular cubic 346 (Ex. 10)
Cubic developable 310
Cubic surfaces 317
Cubic surfaces 4-nodal surface and cubic scroll 322 326 22)
Cubic surfaces, cubic ruled surface in  384 394 3) 397
Cubic symmetroid 389
Curves in higher space, rational normal  384 395
Curves in higher space, rational normal  and projections 396 (Exx. 11—12)
Cylinder 283
Dandelin's Theorem 39 (Ex. 18)
Desargues, involution theorem for a pencil of conies 161
Desargues, theorem for tetrahedra in perspective 250
Desargues, Theorem for triangles in perspective 80 251
Desargues, Theorem for triangles in perspective, and failure in limiting case 99 (Ex. 14)
Descartes 7
Developable 268
Developable surface 379
Developable, cubic 310
Diameters, of conies 119
Diameters, of conies, of quadrics 283
Direct and opposite transformations 354
Director circle 123 127 165
Directrix of parabola 140
Disk quadric 265
Distance 96
Distance, non-euclidean 97 357
Duality 21
Duality, principle 79 248
Elation 54
ellipse 119
Ellipsoid 282
Elliptic paraboloid 283
Equianharmonic tetrads 48
Euclidean geometry 1-dimensional 66
Euclidean geometry 2-dimensional 93
Euclidean geometry 3-dimensional 255
Euclidean transformations 256—257
False position, method of 182
Fifth associated plane 384
Focal conies 337
Foci and directrices of a conic 38 123
Fregier point 153 (Ex. 16)
| Fregier point, extension for a quadric 293 (Ex. 20)
Fundamental points (of a transformation) 232
Generators 273
Gordan's Theorem 198
Grassmann coordinates 246
Group 4 9 10
Group, projective 41
Harmonic construction 82
Harmonic locus and envelope 181
Harmonic pole and polar, for a tetrahedron 249
Harmonic pole and polar, for a triangle 78
Harmonic relation 18 48
Hesse, quadrilateral 185
Hesse, Theorem for a conic 150
Homaloidal net 232
Homographic correspondences 28 50
Homographic correspondences of period 3 or 4 68 (Ex. 9)
Homographic correspondences, canonical forms 55—57
Homographic correspondences, cyclic 58
Homographic correspondences, elliptic, hyperbolic, or parabolic 67
Homographic correspondences, generation of elliptic type 101 (Ex. 25)
Homographic correspondences, on a conic 137
Homographic correspondences, representation as product of involutions 61
Homographic correspondences, representation in Argand plane 69 (Ex. 19)
Homographic ranges and pencils 82 86
Homographies, permutable 62
Homologies (plane) 214
Homologies (plane), general equations 238 (Ex. 3)
Homologies (plane), transforming one conic into another 223
Homologies (space) 351
Hyperbola 119
Hyperbola of Apollonius 151
Hyperboloid, hyperbolic paraboloid 282
Ideal elements 12 92
Incidence constructions with a conic 142
Inpolar relation 188 306
Invariants 191
inversion 39 (Ex. 20) 236 291
Involution 55
Involution, on a conic 138
Involution, on a twisted cubic 301
Involution, quaternary 305
Involution, ternary 304
Isotropic lines 93
Isotropic lines, planes 255
Jacobian curve, of a net of conies 171
Jacobian curve, of a net of conies, of a net of quadrics 342
Joachimsthars equation 107 265
Klein, Erlangen Programme 6
Laguerre's formula 95
Length 97
Line at infinity 13 92
Line systems 367
Line-coordinates 22 75 244 366
Linear dependence of points 73 242
Linear transformations, basic theorem 398
Mobius net 207
Net, of conies 170
Net, of quadrics 339
Non-euclidean geometry 1-dimensional 70 (Ex. 21)
Non-euclidean geometry 2-dimensional 97 223
Non-euclidean geometry 3-dimensional 356
Normals, to a conic 150
Normals, to a quadric 293 (Exx. 16 18) 338 376
Null polarity 311 361 364
Order, of a Cremona transformation 232
Order, of a plane curve 102
Order, of a twisted cubic 296
Orthocentre 99 (Ex. 15) 172
Orthogonality see “Perpendicularity”
Outpolar relation 188 306
Pappus's Theorem 81
parabola 119
Parabolic cylinder 283
Parallels 13 92 254
Parallels, right and left 357
Pascal's theorem 143
Pencil, of conics 156
Pencil, of lines 83
Pencil, of quadrics 327
Perpendicularity 34 93 255
Perpendicularity, non-euclidean 97 357
Perspective (relation), conies 333
Perspective (relation), perspective ranges and pencils 86
Perspective (relation), triangles 81
Perspectives (transformations) see “Homologies”
Perspectives (transformations), perspectivity 86
Perspectives (transformations), skew perspective 350
Points at infinity 13 92
Polar k-points and k-sides 184
Polar lines for a quadric 271
Polarity, for a conic 108
Polarity, for a quadric 266
Polarity, for a twisted cubic 311
Polarization 198
Poncelet 5
Poncelet, Porism 147 181
Position ratio 10 44
Primal 383 388
Prime 387
Principal axes, transformation to 285
Projection 15
Projection, in  383
Projection, in  388
Projection, of a quadric 278
Projection, of a twisted cubic 299
Projective equivalence 15
Projective geometry 10
Projective geometry 1-dimensional 42
Projective geometry 2-dimensional 72
Projective geometry 3-dimensional 241
Projective geometry, general definition 40
Projective geometry, subordinate 76 134 247
Projective hierarchy 6 9
Projective parameter 25 43
Projective transformation 29
Pseudo-invariant 194
Quadrangle 18 88
Quadratic complex 372
Quadratic forms, reduction to sum of squares 400
Quadratic transformation 233
Quadratic transformation, by incidence construction 259 (Ex. 13)
Quadric 263
Quadric systems, centre locus of a net 346 (Ex. 11)
Quadric systems, net 339
Quadric systems, pencil 327
Quadric systems, range 335
Quadric, centre, diameter, axes 283
Quadric, circular sections 284
Quadric, cone and disk quadric 264 277
Quadric, curves on the surface 281
Quadric, focal conies 337
Quadric, foci 338 347
Quadric, generators 273
Quadric, projection 278
Quadric, reduction to principal axes 285
Quadric, surface of revolution 285 293 346
Quadrilateral, complete 19
Quadrilateral, complete, Hesse 185
Quadrilateral, complete, metrical properties 203 (Ex. 14)
Radial expansion 217
Radial expansion, product of two expansions 239 (Ex. 10)
Range, of conies 156
Range, of conies, of quadrics 335
Rational curve 132
Reciprocal transformation 231
Reciprocation 230
Reciprocation, of one conic into another 148
Rectangular hyperbola 122
Rectangular hyperbola, orthocentric property 172 (Exx. 1 2)
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