Projective Geometry

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McGraw-Hill book Company, Incorporated, 1917 - Geometry, Projective - 215 pages
 

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Contents

Complete Figures in a Plane
12
Simple and Complete Figures in Space
13
CHAPTER III
16
Notation
19
Desargues TheoremTheorem 1
22
HARMONIC RANGES AND HARMONIC PENCILS 21 Harmonic Ranges
25
Effect of Order
26
Construction of Harmonic RangesTheorem III
27
Harmonic Pencils
28
Harmonic Conjugate of an Ideal Point
30
Crossratio of a Harmonic Range
31
Harmonic Mean Between Two Numbers
32
Geometric Mean Between Two Numbers
33
The Circle of Apollonius
34
CHAPTER V
37
Definition of Projective Relationship
38
Two Pairs of Points Each Harmonically Separated by a Third PairTheorem V
39
Converse of Theorem V
40
Consequences of Theorem V and Its ConverseTheorem VI
41
SuperpositionSelfcorresponding Elements
44
Von Staudts Fundamental TheoremTheorem VII
46
Consequences of Theorem VII
47
Determination of Projective RelationshipTheorem VIII
50
CHAPTER VI
55
Fundamental Properties of the Curve and the Envelope
57
Construction of Curves and Envelopes
62
Relations Existing among the Elementary Forms
64
Totality of Elementary Forms
66
CHAPTER VII
68
Converse Theorems
69
Application of Theorems IX
71
Degenerate Cases of Theorems IX
72
The Pentagon Theorem and Its Dual
73
Application of the Pentagon Theorem and Its Dual
74
The Quadrangle Theorem and Its Dual
76
The Principle of Continuity
77
Generation of Particular Conics and Envelopes
78
Cones and Sheaves of Planes of Second Class
79
CHAPTER VIII
81
Special Positions of Pole and Polar Line
83
Construction of Poles and Polar Lines
84
Conjugate Points and Conjugate Lines with Respect to a ConicTheorems XI
86
Consequences of Theorems XI
87
Polar Figures with Respect to a Fixed Conic
89
Selfpolar Figures
91
CHAPTER IX
94
Application of the Harmonic Properties of Poles and Polar Lines
95
The Axes of a Conic
97
The Vertices of a Conic
99
Algebraic Equations of the Conics
100
Diametral Planes and Axes of Cylinders
104
CHAPTER X
105
Sections of a Surface of Second Order
106
Tangent Lines and Tangent Planes
107
The Center of a Projectivity on a Conic
118
Double Elements of a Projectivity on any Form
119
Classification of Proj activities on a Form
120
Cyclic Projectivities
121
CHAPTER XII
127
Hyperbolic Involutions
128
Parabolic Involutions
129
Involutions on a Straight Line
130
Involutions on a Sheaf of Rays of First Class
132
Involutions Determined by a Complete Quadrangle or a Complete Quadrilateral
133
Desargues Theorem and Its Dual
134
Involutions Determined by a Fixed Conic
135
Imaginary Points on a Straight Line
137
Imaginary Lines in a Plane
138
Imaginary Planes
139
Imaginary Elements on any Form
141
CHAPTER XIII
146
Directrices and Focal Radii
149
Fundamental TheoremTheorem XIV
150
CHAPTER XIV
159
Orthogonally Correlated Bundles
160
Definition of Collineation
161
Determination of Projective Relationship
163
Projectivities in a Form of Second Kind
165
Construction of a Perspectivity in a Plane
166
The Invariant of a Perspectivity in a Plane
167
The Harmonic Perspectivity or Involution in a Plane
168
The Affinity
170
Corresponding Conies in Affinately Related Planes
172
The Area of a Parabolic Segment
173
The Similitude
176
Inverse Points Radical Axis
177
The Congruence
178
Collineation in the Plane Selfcorresponding Elements
179
CHAPTER XV
183
Construction of a Polarity in a Plane
184
Selfconjugate Points in a Polarity
185
Classification of Polarities in a Plane
187
The Polarity in a Bundle
189
The Orthogonal Polarity
190
The Absolute Polarity
192
Common Elements in a Double Polarity
193
Double Conjugate Elements
194
Confocal Elements in a Double Polarity
196
Application to Cones of the Second Order
199
Cyclic Planes and Focal Axes of Cones
201
Quadric Transformations
202
Perspective Quadric Transformations
203
Inversion with Respect to a Circle
204
Properties of Positive Inversions
205
Circular Transformations
207
General Note
208
Index
211
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