Projective Geometry, Volume 2

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Contents

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 co÷rdinates
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 co÷rdinates
253
Involutoric collineations
257
The projective group of a quadric
259
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
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