Projective Geometry, Volume 1

Front Cover
 

Contents

9
22
The remaining assumptions of extension for a space of three dimensions
24
The principle of duality
27
The theorems of alignment for a space of n dimensions
29
CHAPTER II
34
Projection section perspectivity
35
The complete npoint
36
Configurations
39
The Desargues configuration
41
Perspective tetrahedra
43
The quadranglequadrilateral configuration
46
The fundamental theorem on quadrangular sets
47
Additional remarks concerning the Desargues configuration
51
CHAPTER III
55
Perspectivity and projectivity
56
The projectivity of onedimensional primitive forms
59
SECTION PAGE 24 General theory of correspondence Symbolic treatment
64
The notion of a group
66
Groups of correspondences Invariant elements and figures
67
Group properties of projectivities
68
Projective transformations of twodimensional forms
71
Projective collineations of threedimensional forms
75
CHAPTER IV
79
34
80
36
82
Nets of rationality on a line
84
Nets of rationality in the plane
86
38
87
39
88
Nets of rationality in space
89
43
91
The fundamental theorem of projectivity
93
44
98
Construction of projectivities on onedimensional forms
100
Involutions
102
Axis and center of homology
103
47
104
Types of collineations in the plane
106
CHAPTER V
109
51
110
Tangents Points of contact
112
The tangents to a point conic form a line conic
116
The polar system of a conic
120
Degenerate conics
126
Desarguess theorem on conics
127
Pencils and ranges of conics Order of contact
128
CHAPTER VI
141
Multiplication of points
144
The commutative law for multiplication
148
The abstract concept of a number system Isomorphism
149
Nonhomogeneous co÷rdinates
150
Condition that a point be on a line
172
Homogeneous co÷rdinates in the plane
174
The line on two points The point on two lines
180
Pencils of points and lines Projectivity
181
The equation of a conic
185
Linear transformations in a plane
187
Collineations between two different planes
190
Homogeneous co÷rdinates in space
194
Linear transformations in space
199
Finite spaces
201
CHAPTER VIII
205
Projective projectivities
208
Groups of projectivities on a line
209
Projective transformations between conics
212
Projectivities on a conic
217
Involutions
221
Involutions associated with a given projectivity
225
Harmonic transformations
230
Scale on a conic
231
Parametric representation of a conic
234
CHAPTER IX
236
The intersection of a given line with a given conic
240
Improper elements Proposition K2
241
Problems of the second degree
245
Invariants of linear and quadratic binary forms
251
Proposition Kn
254
Taylors theorem Polar forms
255
SECTION PAGE 90 Invariants and covariants of binary forms
257
Ternary and quaternary forms and their invariants
258
Proof of Proposition Kn
260
CHAPTER X
262
Analytic representation of a correlation between two planes
266
General projective group Representation by matrices
268
Double points and double lines of a collineation in a plane
271
Double pairs of a correlation
278
Fundamental conic of a polarity in a plane
282
Poles and polars with respect to a conic Tangents
284
Various definitions of conics
285
Pairs of conics
287
Problems of the third and fourth degrees
294
CHAPTER XI
298
The polar system of a regulus
300
Projective conics
304
Linear dependence of lines
311
The linear congruence
312
The linear complex
319
The PlŘcker line co÷rdinates
327
Linear families of lines
329
Interpretation of line co÷rdinates as point co÷rdinates in S5
331
INDEX
335

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Page 95 - If a projectivity leaves each of three distinct points of a line invariant, it leaves every point of the line invariant.
Page 4 - VI contains at least seven elements. Now, making use of Assumption VII, we find that the m-classes thus far obtained contain only the elements mentioned. The m-classes CD and AEF have an element in common (by Assumption III) which cannot be A or E, and must therefore (by Assumption VII) be F. Similarly, ACG and the m-class DE have the element G in common. The seven elements A, B, C, D...
Page 3 - S and m-class, and one undefined relation, "belonging to a class. The undefined terms, moreover, are entirely devoid of content except such as is implied in the assumptions. Now the first question to ask regarding a set of assumptions is : Are they logically consistent? In the example above, of a...
Page 3 - S is interpreted to mean the digits 0, 1, 2, 3, 4, 5, 6 and the ra-classes to mean the columns in the following table : 0123456 (1) 1234560 3456012 This interpretation is a concrete representation of our assumptions. Every proposition derived from the assumptions must be true of this system of triples. Hence none of the assumptions can be logically inconsistent with the rest ; otherwise contradictory statements would be true of this system of triples. Thus, in general, a set of assumptions is said...
Page 2 - ... this latter procedure makes it possible to give a variety of interpretations to the undefined elements, and so to exhibit an identity of structure in different concrete settings. We shall now demonstrate six theorems, some of which may be regarded as trite consequences of our assumptions.
Page 3 - This interpretation is a concrete representation of our assumptions. Every proposition derived from the assumptions must be true of this system of triples. Hence none of the assumptions can be logically inconsistent with the rest; otherwise contradictory statements would be true of this system of triples. Thus, in general, a set of assumptions is said to be consistent if a single concrete representation of the assumptions can be given.* Knowing our assumptions to be consistent, we may proceed to...
Page 7 - C, and the m-classes to consist of the pairs AB, BC, CA, then it is clear that Assumptions I, II, III, IV, VI, VII are true of this class S, and therefore that any logical consequence of them is true with this interpretation. Assumption V, however, is false for this class, and cannot, therefore, be a logical consequence of the other assumptions. In like manner, other examples can be constructed to show that each of the Assumptions I-VII is independent of the remaining ones.
Page 27 - If A, B, C are points not all on the same line, and D and E (D ^ E) are points such that B, C, D are on a line and C, A, E are on a line, then there is a point F such that A, B, F are on a line and also D, E, F are on a line.
Page 5 - III, which is also true except when the lines are parallel. How this exception can be removed we will discuss in the next section, so that we may also regard the ordinary plane geometry as a representation of Assumptions I-VI.
Page 46 - Prf (fig. 18). We observe first that the diagonal triangle is perspective with each of the four triangles formed by a set of three of the vertices of the quadrangle, the center of perspectivity being in each case the fourth vertex. This gives rise to four axes of perspectivity (Theorem 1), one corresponding to each vertex of the quadrangle.* These four lines clearly form the sides of a complete quadrilateral whose diagonal triangle is Du, Dw Dl,.

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