Introduction ................................................... xi
General conventions ............................................ xv
1 Projective spaces and linear varieties ....................... 1
1.1 How projective spaces arise ............................. 1
1.2 Projective spaces ....................................... 5
1.3 Linear varieties ........................................ 7
1.4 Incidence of linear varieties ........................... 9
1.5 Linear independence of points .......................... 13
1.6 Projectivities ......................................... 16
1.7 Projective invariance .................................. 18
1.8 Pappus' and Desargues' theorems ........................ 21
1.9 Projection, section and perspectivity .................. 25
1.10 What is projective geometry about? ..................... 28
1.11 Exercises .............................................. 30
2 Projective coordinates and cross ratio ...................... 34
2.1 Projective references .................................. 34
2.2 Projective coordinates ................................. 36
2.3 Change of coordinates .................................. 39
2.4 Absolute coordinate .................................... 41
2.5 Parametric equations ................................... 43
2.6 Implicit equations ..................................... 45
2.7 Incidence with coordinates ............................. 47
2.8 Determination and matrices of a projectivity ........... 52
2.9 Cross ratio ............................................ 56
2.10 Harmonic sets .......................................... 62
2.11 Projective classification .............................. 66
2.12 Exercises .............................................. 67
3 Affine geometry ............................................. 72
3.1 Recalling basic facts of affine geometry ............... 72
3.2 The projective closure of an affine space .............. 75
3.3 Affine and projective coordinates ...................... 78
3.4 Affine linear varieties ................................ 81
3.5 Affine transformations, affine ratio ................... 85
3.6 Affine geometry in the projective frame ................ 89
3.7 The Erlangen Program ................................... 94
3.8 Exercises .............................................. 96
4 Duality .................................................... 100
4.1 The space of hyperplanes .............................. 100
4.2 Bundles of hyperplanes ................................ 101
4.3 The principle of duality .............................. 106
4.4 Hyperplane coordinates ................................ 111
4.5 The dual of a projectivity ............................ 117
4.6 Biduality ............................................. 119
4.7 Duals of linear varieties and bunches ................. 121
4.8 Exercises ............................................. 122
5 Projective transformations ................................. 124
5.1 Complex extension of a real projective space .......... 124
5.2 Equations of projectivities between lines ............. 136
5.3 Projectivities between distinct lines ................. 137
5.4 Pairs of points on projective lines ................... 143
5.5 Projectivities of a line .............................. 145
5.6 Involutions ........................................... 153
5.7 Fixed points of collineations ......................... 160
5.8 Correlations .......................................... 173
5.9 Projectivities and perspectivities .................... 180
5.10 Singular projectivities ............................... 182
5.11 Exercises ............................................. 187
6 Quadric hypersurfaces ...................................... 193
6.1 The notion of quadric ................................. 193
6.2 Quadrics of Pi ........................................ 203
6.3 Tangent lines ......................................... 204
6.4 Conjugation ........................................... 207
6.5 Non-degenerate quadrics. Polarity ..................... 218
6.6 Non-degenerate quadric envelopes ...................... 224
6.7 Degenerate quadrics. Cones ............................ 227
6.8 Degenerate quadric envelopes .......................... 234
6.9 The absolute quadric .................................. 238
6.10 Exercises ............................................. 249
7 Classification and properties of quadrics .................. 256
7.1 Projective reduced equations of quadrics .............. 256
7.2 Projective classification of quadrics ................. 260
7.3 Determining quadrics by their sets of points .......... 268
7.4 Interior and exterior of quadrics ..................... 272
7.5 Quadrics of affine spaces ............................. 277
7.6 Affine reduced equations of quadrics .................. 280
7.7 Affine classification of quadrics ..................... 284
7.8 Affine elements of quadrics ........................... 295
7.9 Exercises ............................................. 302
8 Further properties of quadrics ............................. 309
8.1 Projective generation of conies ....................... 309
8.2 Projective structure on a conic ....................... 315
8.3 Lines on quadrics ..................................... 328
8.4 Lines of P3 ........................................... 339
8.5 Exercises ............................................. 347
9 Projective spaces of quadrics .............................. 356
9.1 Effective divisors on projective lines ................ 356
9.2 Rational curves ....................................... 360
9.3 Linear systems of quadrics ............................ 365
9.4 Independence of linear conditions on quadrics ......... 372
9.5 Pencils of quadrics ................................... 375
9.6 Pencils of conies ..................................... 385
9.7 Desargues' theorem on pencils of quadrics ............. 390
9.8 Spaces of quadric envelopes, ranges ................... 392
9.9 Apolarity ............................................. 397
9.10 Pencils and polarity .................................. 399
9.11 Rational normal curves of  n ......................... 404
9.12 Twisted cubics ........................................ 420
9.13 Exercises ............................................. 427
10 Metric geometry of quadrics ............................... 438
10.1 Circles and spheres .................................. 438
10.2 Metric properties of conies .......................... 442
10.3 Focal properties of conies ........................... 447
10.4 Metric properties of three-space quadrics ............ 457
10.5 Metric reduced equations of quadrics ................. 467
10.6 Metric invariants of quadrics ........................ 474
10.7 Metric classification of quadrics .................... 483
10.8 Exercises ............................................ 486
11 Three projective classifications ........................... 493
11.1 Polynomial matrices ................................... 493
11.2 Classification of polynomial matrices ................. 498
11.3 Projective equivalence of collineations ............... 505
11.4 Classification of collineations of complex
projective spaces ..................................... 508
11.5 Classification of collineations of real projective
spaces ................................................ 515
11.6 Projective classification of pencils of quadrics ...... 520
11.7 Projective classification of correlations ............. 538
11.8 Square roots of regular matrices ...................... 548
11.9 Exercises ............................................. 550
A Perspective (for artists) .................................. 553
A.1 Basic setting and affine matter ....................... 554
A.2 Orthogonality and angles .............................. 558
A.3 Exercises ............................................. 565
В Models of non-Euclidean geometries ......................... 568
B.l Euclidean and non-Euclidean geometries ................ 568
B.2 The models ............................................ 571
B.3 Hyperbolic distance ................................... 575
B.4 Elliptic distance ..................................... 579
B.5 Betweenness ........................................... 582
B.6 Angles between lines .................................. 583
B.7 Circles and similar curves ............................ 588
B.8 Transformations ....................................... 593
B.9 Exercises ............................................. 599
Bibliography .................................................. 603
Symbols ....................................................... 605
Index ......................................................... 607
|
