Projective Geometry, Volume 2

Oswald Veblen, John Wesley Young

Ginn, 1918 - Geometry, Projective

Contents

• FOUNDATIONS SECTION PAGE 1 Plan of the chapter: 1
• Assumption K: 3
• Double points of projectivities: 5
• Complex geometry: 6
• Imaginary elements adjoined to a real space: 7
• Harmonic sequence: 9
• Assumption H: 11
• Order in a net of rationality: 13
• 10 Cuts in a net of rationality: 14
• 11 Assumption of continuity: 16
• 12 Chains in general: 21
• 13 Consistency categoricalness and independence of the assumptions: 23
• 14 Foundations of the complex geometry: 29
• 15 Ordered projective spaces: 32
• 16 Modular projective spaces: 33
• Recapitulation: 36
• CHAPTER II: 37
• The two senseclasses on a line: 40
• Sense in any onedimensional form: 43
• Separation of point pairs: 44
• Segments and intervals: 45
• Linear regions: 47
• Algebraic criteria of sense: 49
• Pairs of lines and of planes: 50
• The triangle and the tetrahedron: 52
• Algebraic criteria of separation Cross ratios of points in space: 55
• Euclidean spaces: 58
• Assumptions for a Euclidean space: 59
• Sense in a Euclidean plane: 61
• 31 Sense in Euclidean spaces: 63
• 32 Sense in a projective space: 64
• Intuitional description of the projective plane: 67
• THE AFFINE GROUP IN THE PLANE SECTION PAGE: 70
• Euclidean plane and the affine group: 71
• Parallel lines 37 Ellipse hyperbola parabola: 73
• The group of translations: 74
• Selfconjugate subgroups Congruence: 78
• Congruence of parallel point pairs: 80
• Metric properties of conics: 81
• Vectors: 83
• Ratios of collinear vectors: 85
• Theorems of Menelaus Ceva and Carnot: 89
• Point reflections: 93
• Extension of the definition of congruence: 94
• The homothetic group: 95
• Equivalence of ordered point triads 49 Measure of ordered point triads: 99
• The equiaffine group: 105
• 51 Algebraic formula for measure Barycentric coördinates: 106
• 52 Line reflections: 111
• 53 Algebraic formulas for line reflections: 115
• Subgroups of the affine group 70 71 72 73 74 78 80 81 82 85 89 92 94 95 96 99 105 106 109 115: 116
• CHAPTER IV: 119
• Orthogonal lines: 121
• Displacements and symmetries Congruence: 123
• Pairs of orthogonal line reflections: 126
• The group of displacements: 131
• Circles: 133
• Congruent and similar triangles: 134
• Algebraic formulas for certain parabolic metric groups: 135
• Introduction of order relations: 138
• The real plane: 141
• Intersectional properties of circles: 142
• The Euclidean geometry A set of assumptions: 144
• Distance: 149
• Area 69 The measure of angles: 151
• The complex plane: 155
• Pencils of circles: 161
• Measure of line pairs: 163
• Generalization by projection: 169
• CHAPTER V: 170
• Interior and exterior of a conic: 174
• Double points of projectivities: 177
• Rulerandcompass constructions: 180
• Conjugate imaginary elements: 182
• 79 Projective affine and Euclidean classification of conics: 186
• Foci of the ellipse and hyperbola: 189
• Focus and axis of a parabola: 193
• Eccentricity of a conic: 196
• Synoptic remarks on conic sections: 199
• Focal properties of collineations: 201
• Homogeneous quadratic equations in three variables: 202
• Nonhomogeneous quadratic equations in two variables: 208
• Euclidean classification of point conics: 210
• Classification of line conics: 212
• 89 Polar systems: 215
• CHAPTER VI: 219
• Correspondence between the complex line and the real Euclidean plane: 222
• The inversion group in the real Euclidean plane: 225
• Generalization by inversion: 231
• Inversions in the complex Euclidean plane: 235
• Correspondence between the real Euclidean plane and a complex pencil of lines: 238
• The real inversion plane: 241
• Order relations in the real inversion plane: 244
• Types of circular transformations: 246
• Chains and antiprojectivities: 250
• Tetracyclic coördinates: 253
• Involutoric collineations: 257
• The projective group of a quadric: 259
• Real quadrics: 262
• The complex inversion plane: 264
• Function plane inversion plane and projective plane: 268
• Projectivities of onedimensional forms in general: 271
• 107 Projectivities of a quadric: 273
• 108 Products of pairs of involutoric projectivities: 277
• Conjugate imaginary lines of the second kind: 281
• The principle of transference: 284
• CHAPTER VII: 287
• Vectors equivalence of point triads etc: 288
• The parabolic metric group Orthogonal lines and planes: 293
• Orthogonal plane reflections: 295
• Displacements and symmetries Congruence: 297
• Euclidean geometry of three dimensions: 301
• 117 Generalization to n dimensions: 304
• Equations of the affine and Euclidean groups: 305
• 119: 311
• 120: 315
• Resolution of a displacement into orthogonal line reflections: 317
• Rotation translation twist: 321
• 123: 325
• Correspondence between the rotations and the points of space: 328
• Algebra of matrices: 333
• 126: 335
• Quaternions: 337
• Quaternions and the onedimensional projective group: 339
• 129: 342
• Parameter representation of displacements: 344
• 131: 350
• Orthogonal lines displacements and congruence: 352
• 134: 354
• Types of hyperbolic displacements: 355
• Interpretation of hyperbolic geometry in the inversion plane: 357
• 135: 360
• Angular measure: 362
• Distance: 364
• 138: 365
• 139 Differential of arc: 366
• 140: 369
• Elliptic plane geometry Definition: 371
• 142: 373
• 144: 375
• Parameter representation of elliptic displacements: 377
• Parameter representation of hyperbolic displacements: 380
• 147: 385
• 149: 388
• Boundary of a convex region: 392
• 151: 395
• The tetrahedron: 397
• Generalization to n dimensions: 400
• 157: 401
• Connected sets regions etc: 404
• Continuous families of sets of points: 405
• Continuous families of transformations: 406
• Affine theorems on sense: 407
• Elementary transformations on a Euclidean line: 409
• Elementary transformations in the Euclidean plane and space: 411
• Sense in a convex region: 413
• Euclidean theorems on sense: 414
• 163: 416
• Senseclasses in projective spaces: 418
• 167: 419
• Elementary transformations in a projective plane: 421
• Elementary transformations in a projective space: 423
• 168 Sense in overlapping convex regions: 424
• 169 Oriented points in a plane: 425
• 170 Pencils of rays: 429
• 171 Pencils of segments and directions: 433
• 172 Bundles of rays segments and directions: 435
• 173 One and twosided regions: 436
• Senseclasses on a sphere: 437
• Direct and opposite collineations in space: 438
• Right and lefthanded figures: 441
• Right and lefthanded reguli congruences and complexes: 443
• 179 Elementary transformations of triads of lines: 446
• 180 Doubly oriented lines: 447
• 181 More general theory of sense: 451
• Broken lines and polygons: 454
• A theorem on simple polygons: 457
• Polygons in a plane: 458
• Subdivision of a plane by lines: 460
• The modular equations and matrices: 464
• Regions determined by a polygon: 467
• Polygonal regions and polyhedra: 473
• 189 Subdivision of space by planes: 475
• The matrices II II and I: 477
• The rank of H: 479
• SECTION PAGE 192 Polygons in space: 480
• Odd and even polyhedra: 482
• Regions bounded by a polyhedron: 483
• The matrices E and E for the projective plane: 484
• Odd and even polygons in the projective plane: 489
• One and twosided polygonal regions: 490
• One and twosided polyhedra O: 493
• Orientation of space: 496
• INDEX: 501

Bibliographic information

• Title: Projective Geometry, Volume 2; Projective Geometry, John Wesley Young
• Authors: Oswald Veblen, John Wesley Young
• Publisher: Ginn, 1918
• Original from: the University of California
• Digitized: Nov 21, 2008