Perspectives on Projective Geometry : A Guided Tour Through Real and Complex Geometry

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. in Particular, It 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 between 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. 1 Pappos's Theorem: Nine Proofs And Three Variations -- 2 Projective Planes -- 3 Homogeneous Coordinates -- 4 Lines And Cross-ratios -- 5 Calculating With Points On Lines -- 6 Determinants -- 7 More On Bracket Algebra -- 8 Quadrilateral Sets And Liftings -- 9 Conics And Their Duals -- 10 Conics And Perspectivity -- 11 Calculating With Conics -- 12 Projective $d$-space -- 13 Diagram Techniques -- 14 Working With Diagrams -- 15 Configurations, Theorems, And Bracket Expressions -- 16 Complex Numbers: A Primer -- 17 The Complex Projective Line -- 18 Euclidean Geometry -- 19 Euclidean Structures From A Projective Perspective -- 20 Cayley-klein Geometries -- 21 Measurements And Transformations -- 22 Cayley-klein Geometries At Work -- 23 Circles And Cycles -- 24 Non-euclidean Geometry: A Historical Interlude -- 25 Hyperbolic Geometry -- 26 Selected Topics In Hyperbolic Geometry -- 27 What We Did Not Touch -- References -- Index. By Jürgen Richter-gebert. Includes Bibliographical References (p. 557-562) And Index. Cover 1 Perspectives on Projective Geometry 4 ISBN 9783642172854 5 About This Book 6 Contents 18 Overture -1 1 Pappos's Theorem: Nine Proofs and Three Variations 25 1.1 Pappos's Theorem and Projective Geometry 26 1.2 Euclidean Versions of Pappos's Theorem 28 1.3 Projective Proofs of Pappos's Theorem 35 1.4 Conics 41 1.5 More Conics 44 1.6 Complex Numbers and Circles 46 1.7 Finally 51 Part I Projective Geometry 55 2 Projective Planes 57 2.1 Drawings and Perspectives 58 2.2 The Axioms 62 2.3 The Smallest Projective Plane 65 3 Homogeneous Coordinates 69 3.1 A Spatial Point of View 69 3.2 The Real Projective Plane with HomogeneousCoordinates 71 3.3 Joins and Meets 74 3.4 Parallelism 77 3.5 Duality 78 3.6 Projective Transformations 80 3.7 Finite Projective Planes 86 4 Lines and Cross-Ratios 89 4.1 Coordinates on a Line 90 4.2 The Real Projective Line 91 4.3 Cross-Ratios (a First Encounter) 94 4.4 Elementary Properties of the Cross-Ratio 96 5 Calculating with Points on Lines 101 5.1 Harmonic Points 102 5.2 Projective Scales 104 5.3 From Geometry to Real Numbers 105 5.4 The Fundamental Theorem 108 5.5 A Note on Other Fields 110 5.6 Von Staudt's Original Constructions 111 5.7 Pappos's Theorem 113 6 Determinants 115 6.1 A ``Determinantal'' Point of View 116 6.2 A Few Useful Formulas 117 6.3 Plücker's 118 6.4 Invariant Properties 121 6.5 Grassmann-Plücker relations 124 7 More on Bracket Algebra 131 7.1 From Points to Determinants... 131 7.2 ...and Back 134 7.3 A Glimpse of Invariant Theory 137 7.4 Projectively Invariant Functions 142 7.5 The Bracket Algebra 143 Part II Working and Playing with Geometry 147 8 Quadrilateral Sets and Liftings 151 8.1 Points on a Line 151 8.2 Quadrilateral Sets 153 8.3 Symmetry and Generalizations of Quadrilateral Sets 156 8.4 Quadrilateral Sets and von Staudt 158 8.5 Slope Conditions 159 8.6 Involutions and Quadrilateral Sets 161 9 Conics and Their Duals 167 9.1 The Equation of a Conic 167 9.2 Polars and Tangents 171 9.3 Dual Quadratic Forms 176 9.4 How Conics Transform 178 9.5 Degenerate Conics 179 9.6 Primal-Dual Pairs 181 10 Conics and Perspectivity 189 10.1 Conic through Five Points 189 10.2 Conics and Cross-Ratios 192 10.3 Perspective Generation of Conics 194 10.4 Transformations and Conics 197 10.5 Hesse's ``Übertragungsprinzip'' 201 10.6 Pascal's and Brianchon's Theorems 206 10.7 Harmonic points on a conic 207 11 Calculating with Conics 211 11.1 Splitting a Degenerate Conic 212 11.2 The Necessity of ``If'' Operations 215 11.3 Intersecting a Conic and a Line 216 11.4 Intersecting Two Conics 218 11.5 The Role of Complex Numbers 221 11.6 One Tangent and Four Points 224 12 Projective d-space 231 12.1 Elements at Infinity 232 12.2 Homogeneous Coordinates and Transformations 233 12.3 Points and Planes in 3-Space 235 12.4 Lines in 3-Space 238 12.5 Joins and Meets: A Universal System ... 241 12.6 ... And How to Use It 244 13 Diagram Techniques 249 13.1 From Points, Lines, and Matrices to Tensors 250 13.2 A Few Fine Points 253 13.3 Tensor Diagrams 254 13.4 How Transformations Work 256 13.5 The -tensor 258 13.6 -Tensors 259 13.7 The - Rule 261 13.8 Transforming -Tensors 263 13.9 Invariants of Line and Point Configurations 267 14 Working with diagrams 269 14.1 The Simplest Property: A Trace Condition 270 14.2 Pascal's Theorem 272 14.3 Closed -Cycles 274 14.4 Conics, Quadratic Forms, and Tangents 278 14.5 Diagrams in RP3 281 14.6 The –rule in Rank 4 284 14.7 Co- and Contravariant Lines in Rank 4 285 14.8 Tensors versus Plücker Coordinates 287 15 Configurations, Theorems, and Bracket Expressions 291 15.1 Desargues's Theorem 292 15.2 Binomial Proofs 294 15.3 Chains and Cycles of Cross-Ratios 299 15.4 Ceva and Menelaus 301 15.5 Gluing Ceva and Menelaus Configurations 307 15.6 Furthermore ... 313 Part III Measurements 315 16 Complex Numbers: A Primer 319 16.1 Historical Background 320 16.2 The Fundamental Theorem 323 16.3 Geometry of Complex Numbers 324 16.4 Euler's Formula 326 16.5 Complex Conjugation 329 17 The Complex Projective Line 333 17.1 CP1 333 17.2 Testing Geometric Properties 334 17.3 Projective Transformations 337 17.4 Inversions and Möbius Reflections 342 17.5 Grassmann-Plücker relations 344 17.6 Intersection Angles 346 17.7 Stereographic Projection 348 18 Euclidean Geometry 351 18.1 The points I and J 352 18.2 Cocircularity 353 18.3 The Robustness of the Cross-Ratio 355 18.4 Transformations 356 18.5 Translating Theorems 360 18.6 More Geometric Properties 361 18.7 Laguerre's Formula 364 18.8 Distances 367 19 Euclidean Structures from a Projective Perspective 371 19.1 Mirror Images 372 19.2 Angle Bisectors 373 19.3 Center of a Circle 376 19.4 Constructing the Foci of a Conic 378 19.5 Constructing a Conic by Foci 382 19.6 Triangle Theorems 384 19.7 Hybrid Thinking 390 20 Cayley-Klein Geometries 397 20.1 I and J Revisited 398 20.2 Measurements in Cayley-Klein Geometries 399 20.3 Nondegenerate Measurements along a Line 401 20.4 Degenerate Measurements along a Line 408 20.5 A Planar Cayley-Klein Geometry 411 20.6 A Census of Cayley-Klein Geometries 415 20.7 Coarser and Finer Classifications 420 21 Measurements and Transformations 421 21.1 Measurements vs. Oriented Measurements 422 21.2 Transformations 423 21.3 Getting Rid of X and Y 429 21.4 Comparing Measurements 430 21.5 Reflections and Pole/Polar Pairs 435 21.6 From Reflections to Rotations 441 22 Cayley-Klein Geometries at Work 445 22.1 Orthogonality 446 22.2 Constructive versus Implicit Representations 449 22.3 Commonalities and Differences 451 22.4 Midpoints and Angle Bisectors 453 22.5 Trigonometry 459 23 Circles and Cycles 465 23.1 Circles via Distances 466 23.2 Relation to the Fundamental Conic 468 23.3 Centers at Infinity 470 23.4 Organizing Principles 472 23.5 Cycles in Galilean Geometry 481 24 Non-Euclidean Geometry: A Historical Interlude 487 24.1 The Inner Geometry of a Space 488 24.2 Euclid's Postulates 490 24.3 Gauss, Bolyai, and Lobachevsky 492 24.4 Beltrami and Klein 496 24.5 The Beltrami-Klein Model 498 24.6 Poincaré 501 25 Hyperbolic Geometry 505 25.1 The Staging Ground 505 25.2 Hyperbolic Transformations 507 25.3 Angles and Boundaries 509 25.4 The Poincaré Disk 511 25.5 CP1 Transformations and the Poincaré Disk 518 25.6 Angles and Distances in the Poincaré Disk 523 26 Selected Topics in Hyperbolic Geometry 527 26.1 Circles and Cycles in the Poincaré Disk 527 26.2 Area and Angle Defect 531 26.3 Thales and Pythagoras 536 26.4 Constructing Regular n-Gons 539 26.5 Symmetry Groups 541 27 What We Did Not Touch 547 27.1 Algebraic Projective Geometry 547 27.2 Projective Geometry and Discrete Mathematics 553 27.3 Projective Geometry and Quantum Theory 560 27.4 Dynamic Projective Geometry 568 References 579 Index 585 3642172857,9783642172854 Springer 2011 Front Matter....Pages i-xxii Pappos’s Theorem: Nine Proofs and Three Variations....Pages 3-31 Front Matter....Pages 33-33 Projective Planes....Pages 35-46 Homogeneous Coordinates....Pages 47-66 Lines and Cross-Ratios....Pages 67-78 Calculating with Points on Lines....Pages 79-92 Determinants....Pages 93-107 More on Bracket Algebra....Pages 109-123 Front Matter....Pages 125-127 Quadrilateral Sets and Liftings....Pages 129-143 Conics and Their Duals....Pages 145-166 Conics and Perspectivity....Pages 167-187 Calculating with Conics....Pages 189-207 Projective d -space....Pages 209-225 Diagram Techniques....Pages 227-246 Working with diagrams....Pages 247-267 Configurations, Theorems, and Bracket Expressions....Pages 269-292 Front Matter....Pages 293-296 Complex Numbers: A Primer....Pages 297-309 The Complex Projective Line....Pages 311-327 Euclidean Geometry....Pages 329-347 Euclidean Structures from a Projective Perspective....Pages 349-373 Cayley-Klein Geometries....Pages 375-398 Front Matter....Pages 293-296 Measurements and Transformations....Pages 399-422 Cayley-Klein Geometries at Work....Pages 423-442 Circles and Cycles....Pages 443-464 Non-Euclidean Geometry: A Historical Interlude....Pages 465-481 Hyperbolic Geometry....Pages 483-503 Selected Topics in Hyperbolic Geometry....Pages 505-523 What We Did Not Touch....Pages 525-555 Back Matter....Pages 557-571