The Real Projective Plane, Volume 1

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Springer Science & Business Media, Jan 25, 1993 - Mathematics - 222 pages
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Along with many small improvements, this revised edition contains van Yzeren's new proof of Pascal's theorem (§1.7) and, in Chapter 2, an improved treatment of order and sense. The Sylvester-Gallai theorem, instead of being introduced as a curiosity, is now used as an essential step in the theory of harmonic separation (§3.34). This makes the logi cal development self-contained: the footnotes involving the References (pp. 214-216) are for comparison with earlier treatments, and to give credit where it is due, not to fill gaps in the argument. H.S.M.C. November 1992 v Preface to the Second Edition Why should one study the real plane? To this question, put by those who advocate the complex plane, or geometry over a general field, I would reply that the real plane is an easy first step. Most of the prop erties are closely analogous, and the real field has the advantage of intuitive accessibility. Moreover, real geometry is exactly what is needed for the projective approach to non· Euclidean geometry. Instead of introducing the affine and Euclidean metrics as in Chapters 8 and 9, we could just as well take the locus of 'points at infinity' to be a conic, or replace the absolute involution by an absolute polarity.
  

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Contents

A Comparison of Various Kinds of Geometry
1
13 Central projection
2
14 The line at infinity
4
15 Perspective triangles
6
16 The directed angle or cross
8
17 Hexagramma mysticum
9
18 An outline of subsequent work
10
Incidence
12
73 Construction for a projectivity on a conic
96
74 Construction for the invariant points of a given hyperbolic projectivity
98
75 Involution on a conic
99
76 A generalization of Steiners construction
102
77 Trilinear polarity
103
Affine Geometry
105
82 Intermediacy
106
83 Congruence
107

22 The axioms of incidence
14
23 The principle of duality
15
24 Quadrangle and quadrilateral
17
25 Harmonic conjugacy
18
26 Ranges and pencils
21
28 The invariance and symmetry of the harmonic relation
23
Order and Continuity
25
32 Sense
27
33 The SylvesterGallai theorem
29
34 Ordered correspondence
30
35 Continuity
34
37 Order in a pencil
37
OneDimensional Projectivities
39
42 The fundamental theorem of projective geometry
41
43 Pappuss theorem
43
44 Classification of projectivities
45
45 Periodic projectivities
48
47 Quadrangular set of six points
52
48 Projective pencils
54
TwoDimensional Projectivities
55
52 Perspective collineation
57
53 Involutory collineation
59
54 Correlation
61
55 Polarity
62
56 Polar and selfpolar triangles
66
57 The selfpolarity of the Desargues configuration
68
58 Pencil and range of polarities
70
59 Degenerate polarities
71
Conics
73
62 Elliptic and hyperbolic polarities
74
63 How a hyperbolic polarity determines a conic
76
64 Conjugate points and conjugate lines
78
65 Two possible definitions for a conic
80
66 Construction for the conic through five given points
83
67 Two triangles inscribed in a conic
85
68 Pencils of conics
87
Projectivities on a Conic
92
72 Pascal and Brianchon
94
84 Distance
109
85 Translation and dilatation
113
86 Area
114
87 Classification of conies
117
88 Conjugate diameters
119
89 Asymptotes
121
Euclidean Geometry
126
92 Circles
128
93 Axes of a conic
131
94 Congruent segments
133
95 Congruent angles
134
96 Congruent transformations
138
Continuity
147
102 Proving Archimedess axiom
148
103 Proving the line to be perfect
149
104 The fundamental theorem of projective geometry
152
105 Proving Dedekinds axiom
153
The Introduction of Coordinates
155
111 Addition of points
156
112 Multiplication of points
158
113 Rational points Defining
161
115 The onedimensional continuum
163
116 Homogeneous coordinates
165
The Use of Coordinates
169
122 Analytic geometry
171
123 Verifying the axioms of incidence
173
124 Verifying the axioms of order and continuity
174
125 The general collineation
178
126 The general polarity
181
127 Conies
185
affine and areal coordinates
191
Certesian and trilinear coordinates
193
The Complex Projective Plane
200
How to Use Mathematica
202
Bibliography
214
Index
217
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