Projective Geometry, Volume 2 (Google eBook)

Front Cover
Ginn and Company, 1918 - Geometry, Projective - 511 pages
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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 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
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
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