Perspectives on Projective Geometry

by Richter-Gebert, Jurgen

• ISBN13: 9783642172854
• ISBN10: 3642172857
• Format: Hardcover
• Copyright: 2011-03-01
• Publisher: Springer Nature

Summary

Projective geometry is one of the most fundamental and at the same time most beautiful branches of geometry. It can be considered the common foundation of many other geometric disciplines like Euclidean geometry, hyperbolic and elliptic geometry or even relativistic space-time geometry. This book offers a comprehensive introduction to this fascinating field and its applications. It in particular explains how metric concepts may be best understood in projective terms. One of the major themes that appears throughout this book is the beauty of the interplay of geometry, algebra and combinatorics. This book can especially be used as a guide that explains how geometric objects and operations may be most elegantly expressed in algebraic terms, making it a valuable resource for mathematicians as well as for computer scientists and physicists. The book is based on the author's experience in implementing geometric software and includes hundreds of high quality illustrations.

Table of Contents

Pappos's Theorem: Nine Proofs and Three Variations	p. 3
Pappos's Theorem and Projective Geometry	p. 4
Euclidean Versions of Pappos's Theorem	p. 6
Projective Proofs of Pappos's Theorem	p. 13
Conics	p. 19
More Conics	p. 22
Complex Numbers and Circles	p. 24
Finally	p. 29
Projective Geometry
Projective Planes	p. 35
Drawings and Perspectives	p. 36
The Axioms	p. 40
The Smallest Projective Plane	p. 43
Homogeneous Coordinates	p. 47
A Spatial Point of View	p. 47
The Real Projective Plane with Homogeneous Coordinates	p. 49
Joins and Meets	p. 52
Parallelism	p. 55
Duality	p. 56
Projective Transformations	p. 58
Finite Projective Planes	p. 64
Lines and Cross-Ratios	p. 67
Coordinates on a Line	p. 68
The Real Projective Line	p. 69
Cross-Ratios (a First Encounter)	p. 72
Elementary Properties of the Cross-Ratio	p. 74
Calculating with Points on Lines	p. 79
Harmonic Points	p. 80
Projective Scales	p. 82
From Geometry to Real Numbers	p. 83
The Fundamental Theorem	p. 86
A Note on Other Fields	p. 88
Von Staudt's Original Constructions	p. 89
Pappos's Theorem	p. 91
Determinants	p. 93
A ôDeterminantalö Point of View	p. 94
A Few Useful Formulas	p. 95
Pliicker's ¿	p. 96
Invariant Properties	p. 99
Grassmann-Plücker relations	p. 102
More on Bracket Algebra	p. 109
From Points to Determinants	p. 109
… and Back	p. 112
A Glimpse of Invariant Theory	p. 115
Projectively Invariant Functions	p. 120
The Bracket Algebra	p. 121
Working and Playing with Geometry
Quadrilateral Sets and Liftings	p. 129
Points on a Line	p. 129
Quadrilateral Sets	p. 131
Symmetry and Generalizations of Quadrilateral Sets	p. 134
Quadrilateral Sets and von Staudt	p. 136
Slope Conditions	p. 137
Involutions and Quadrilateral Sets	p. 139
Conics and Their Duals	p. 145
The Equation of a Conic	p. 145
Polars and Tangents	p. 149
Dual Quadratic Forms	p. 154
How Conics Transform	p. 156
Degenerate Conics	p. 157
Primal-Dual Pairs	p. 159
Coniecs and Perspectivity	p. 167
Conic through Five Points	p. 167
Conics and Cross-Ratios	p. 170
Perspective Generation of Conics	p. 172
Transformations and Conics	p. 175
Hesse's ôÜbertragungsprinzipö	p. 179
Pascal's and Brianchon's Theorems	p. 184
Harmonic points on a conic	p. 185
Calculating with Conics	p. 189
Splitting a Degenerate Conic	p. 190
The Necessity of ôIfö Operations	p. 193
Intersecting a Conic and a Line	p. 194
Intersecting Two Conics	p. 196
The Role of Complex Numbers	p. 199
One Tangent and Four Points	p. 202
Projective d-space	p. 209
Elements at Infinity	p. 210
Homogeneous Coordinates and Transformations	p. 211
Points and Planes in 3-Space	p. 213
Lines in 3-Space	p. 216
Joins and Meets: A Universal System	p. 219
… And How to Use It	p. 222
Diagram Techniques	p. 227
From Points, Lines, and Matrices to Tensors	p. 228
A Few Fine Points	p. 231
Tensor Diagrams	p. 232
How Transformations Work	p. 234
The ¿-tensor	p. 236
¿-Tensors	p. 237
The ¿-¿ Rule	p. 239
Transforming ¿-Tensors	p. 241
Invariants of Line and Point Configurations	p. 245
Working with diagrams	p. 247
The Simplest Property: A Trace Condition	p. 248
Pascal's Theorem	p. 250
Closed ¿-Cycles	p. 252
Conics, Quadratic Forms, and Tangents	p. 256
Diagrams in RP3	p. 259
The ¿-¿-rule in Rank 4	p. 262
Co- and Contravariant Lines in Rank 4	p. 263
Tensors versus Plücker Coordinates	p. 265
Configurations, Theorems, and Bracket Expressions	p. 269
Desargues's Theorem	p. 270
Binomial Proofs	p. 272
Chains and Cycles of Cross-Ratios	p. 277
Ceva and Menelaus	p. 279
Gluing Ceva and Menelaus Configurations	p. 285
Furthermore	p. 291
Measurements
Complex Numbers: A Primer	p. 297
Historical Background	p. 298
The Fundamental Theorem	p. 301
Geometry of Complex Numbers	p. 302
Euler's Formula	p. 304
Complex Conjugation	p. 307
The Complex Projective Line	p. 311
CP1	p. 311
Testing Geometric Properties	p. 312
Projective Transformations	p. 315
Inversions and Möbius Reflections	p. 320
Grassmann-Plücker relations	p. 322
Intersection Angles	p. 324
Stereographic Projection	p. 326
Euclidean Geometry	p. 329
The points I and J	p. 330
Cocircularity	p. 331
The Robustness of the Cross-Ratio	p. 333
Transformations	p. 334
Translating Theorems	p. 338
More Geometric Properties	p. 339
Laguerre's Formula	p. 342
Distances	p. 345
Euclidean Structures from a Projective Perspective	p. 349
Mirror Images	p. 350
Angle Bisectors	p. 351
Center of a Circle	p. 354
Constructing the Foci of a Conic	p. 356
Constructing a Conic by Foci	p. 360
Triangle Theorems	p. 362
Hybrid Thinking	p. 368
Cayley-Klein Geometries	p. 375
I and J Revisited	p. 376
Measurements in Cayley-Klein Geometries	p. 377
Nondegenerate Measurements along a Line	p. 379
Degenerate Measurements along a Line	p. 386
A Planar Cayley-Klein Geometry	p. 389
A Census of Cayley-Klein Geometries	p. 393
Coarser and Finer Classifications	p. 398
Measurements and Transformations	p. 399
Measurements vs. Oriented Measurements	p. 400
Transformations	p. 401
Getting Rid of X and Y	p. 407
Comparing Measurements	p. 408
Reflections and Pole/Polar Pairs	p. 413
From Reflections to Rotations	p. 419
Cayley-Klein Geometries at Work	p. 423
Orthogonality	p. 424
Constructive versus Implicit Representations	p. 427
Commonalities and Differences	p. 429
Midpoints and Angle Bisectors	p. 431
Trigonometry	p. 437
Circles and Cycles	p. 443
Circles via Distances	p. 444
Relation to the Fundamental Conic	p. 446
Centers at Infinity	p. 448
Organizing Principles	p. 450
Cycles in Galilean Geometry	p. 459
Non-Euclidean Geometry: A Historical Interlude	p. 465
The Inner Geometry of a Space	p. 466
Euclid's Postulates	p. 468
Gauss, Bolyai, and Lobachevsky	p. 470
Beltrami and Klein	p. 474
The Beltrami-Klein Model	p. 476
Poincaré	p. 479
Hyperbolic Geometry	p. 483
The Staging Ground	p. 483
Hyperbolic Transformations	p. 485
Angles and Boundaries	p. 487
The Poincaré Disk	p. 489
CP1 Transformations and the Poincaré Disk	p. 496
Angles and Distances in the Poincaré Disk	p. 501
Selected Topics in Hyperbolic Geometry	p. 505
Circles and Cycles in the Poincaré Disk	p. 505
Area and Angle Defect	p. 509
Thalea and Pythagoras	p. 514
Constructing Regular n-Gons	p. 517
Symmetry Groups	p. 519
What We Did Not Touch	p. 525
Algebraic Projective Geometry	p. 525
Projective Geometry and Discrete Mathematics	p. 531
Projective Geometry and Quantum Theory	p. 538
Dynamic Projective Geometry	p. 546
References	p. 557
Index	p. 563
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