Projective Geometry, Volume 2

Oswald Veblen, John Wesley Young

Ginn, 1918 - Geometry, Projective - 345 pages

Contents

• FOUNDATIONS SKCTION PAOE 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 34 The geometry corresponding to a given group of transformations: 70
• Euclidean plane and the affine group: 71
• Parallel lines: 72
• Ellipse hyperbola parabola: 73
• The group of translations: 74
• Selfconjugate subgroups Congruence: 78
• Congruence of parallel point pairs: 80
• Metric properties of conies: 81
• Vectors: 82
• Ratios of collinear vectors: 85
• Theorems of Menelaus Ceva and Carnot: 89
• Point reflections: 92
• Extension of the definition of congruence: 94
• The homothetic group: 95
• Equivalence of ordered point triads: 96
• Measure of ordered point triads: 99
• The equiaffine group: 105
• 51 Algebraic formula for measure Barycentric coordinates: 106
• 52 Line reflections: 109
• 53 Algebraic formulas for line reflections: 115
• Subgroups of the affine group: 116
• CHAPTER IV: 119
• Orthogonal lines: 120
• Displacements and symmetries Congruence: 123
• Pairs of orthogonal line reflections: 126
• The group of displacements: 129
• Circles: 131
• Congruent and similar triangles: 134
• Algebraic formulas for certain parabolic metric groups: 135
• Introduction of order relations: 138
• The real plane: 140
• Intersectional properties of circles: 142
• The Euclidean geometry A set of assumptions: 144
• Distance: 147
• Area: 149
• The measure of angles: 151
• The complex plane: 154
• Pencils of circles: 157
• Measure of line pairs: 163
• Generalization by projection: 167
• ORDINAL AND METRIC PROPERTIES OF CONICS SECTION FAQK 74 Onedimensional projectivities: 170
• Interior and exterior of a conic: 174
• Double points of projectivities: 177
• Rulerandcoinpass constructions: 180
• Conjugate imaginary elements: 182
• Projective affine and Euclidean classification of conies: 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 conies: 210
• Classification of line conies: 212
• 89 Polar systems: 216
• 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 288: 238
• The real inversion plane: 241
• Order relations in the real inversion plane: 244
• Types of circular transformations: 246
• Chains and antiprojectivities: 250
• Tetracyclic coordinates: 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
• Distance area volume angular measure: 311
• The sphere and other quadrics 815: 315
• Resolution of a displacement into orthogonal line reflections 817: 317
• Rotation translation twist: 321
• Properties of displacements: 325
• Correspondence between the rotations and the points of space: 328
• Algebra of matrices 833: 333
• Rotations of an imaginary sphere: 335
• Quaternions: 337
• Quaternions and the onedimensional projective group: 339
• 129 Representation of rotations and onedimensional projectivities by points: 342
• Parameter representation of displacements: 344
• CHAPTER VIII: 350
• Orthogonal lines displacements and congruence: 352
• Types of hyperbolic displacements: 355
• Interpretation of hyperbolic geometry in the inversion plane 857: 357
• Significance and history of nonEuclidean geometry: 360
• Angular measure: 362
• Distance: 364
• Algebraic formulas for distance and angle 865: 365
• 139 Differential of arc: 366
• Hyperbolic geometry of three dimensions: 369
• Elliptic plane geometry Definition: 371
• Elliptic geometry of three dimensions 873: 373
• Double elliptic geometry: 375
• Parameter representation of elliptic displacements: 377
• Parameter representation of hyperbolic displacements: 380
• CHAPTER IX: 385
• Further theorems on convex regions: 388
• Boundary of a convex region: 392
• Triangular regions: 395
• The tetrahedron: 397
• Generalization to n dimensions: 400
• Curves: 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
• Positive and negative displacements: 416
• Senseclasses in projective spaces: 418
• Elementary transformations on a projective line: 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
• Subdivision of space by planes: 475
• The matrices IIV Bt and Hs: 477
• The rank of Ht: 479
• SECTION PAGE 192 Polygons in space: 480
• Odd and even polyhedra: 482
• Regions bounded by a polyhedron: 483
• The matrices El and Et for the projective plane: 484
• Odd and even polygons in the projective plane: 489
• One and twosided polygonal regions: 490
• One and twosided polyhedra: 493
• Orientation of space: 496
• INDEX 601: 501

Bibliographic information

• Title: Projective Geometry, Volume 2; Projective Geometry, Oswald Veblen
• Authors: Oswald Veblen, John Wesley Young
• Publisher: Ginn, 1918
• Original from: the University of California
• Digitized: Nov 21, 2008
• Length: 345 pages