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| Авторизация |
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| Verdina J. — Projective Geometry and Point Tranformations |
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| Предметный указатель |
Abelian group 5
Affine mapping 55
Algebraic curves 170
Algebraic hypersurface, equation 13
Algebraic hypersurface, order 13
Algebraic varieties 13
Angle between two surfaces 160
Asymptotes 96
Axial pencil 23
Axis of coordinates 2
Base points of a pencil of conies 107
Bezout's theorem 170
Bilinear form 88
Birational transformation 127 132
Bitangent conics 189
Bundle of lines 23
Bundle of planes 23
Canonic equations 65
Characteristic invariant 65
Characteristic invariant of the homothety 77
Characteristic roots 60
Circular inversion 133 See
Circular points at infinity 18
Cofactor 166
Collinear points 15
Collineation between homogeneous spaces 6
Collineation between projective planes 47
Collineation between projective planes, any two corresponding lines intersect at the axis 69
Collineation between projective planes, any two corresponding points are collinear with the center 68
Collineation between projective planes, definition 48
Collineation between projective planes, determined by four pairs of corresponding points 54
Collineation between projective planes, determined by three pairs of corresponding lines and a pair of corresponding points 56 57
Collineation between projective planes, verification that a given collineation is a special homology 66
Collineation between projective spaces 47 80 82
Collineation between projective spaces, as a product of two polarities 103 104
Collineation between projective spaces, between homogeneous spaces 6
Collineation between projective spaces, definition 80
Collineation between projective spaces, determined as a product of two polarities 85
Collineation between projective spaces, equations 81
Collineation between projective spaces, tangent to another collineation 153
Collineations between superposed planes, characteristic equation 60
Collineations between superposed planes, characteristic roots 60
Collineations between superposed planes, determination of its fixed points and fixed lines 84
Collineations between superposed planes, determined by two fixed points and two points which correspond doubly 72
Collineations between superposed planes, fixed points 60
Commutative group 5
Complete plane quadrangle 52
Complete plane quadrangle, diagonal points 52
Complete plane quadrangle, diagonal triangle 52
Complete plane quadrangle, opposite sides 52
Complete plane quadrangle, sides 52
Complete plane quadrangle, vertices 52
Complete quadrilateral 51
Complex number 179
Complex number, conjugate of 179
Complex point 17
Configuration 23
Conformal transformation 160
conic 87
Conic, as a pair of lines 88
Conic, base points 107
Conic, bitangent 189
Conic, degenerate 88 90
Conic, determined by five points 89
Conic, discriminant 88
Conic, equation 87
Conic, equations of a conic satisfying given conditions 190
Conic, fundamental 101
Conic, its intersection with a line 90
Conic, through five given points 190
Conic, through four points and tangent to a given line 191
Conics, iperosculating 189
Conics, osculating 189
Conics, pencil 106
Conics, pencil of conies bitangent to a given conic 193
Conics, pencil of conies tangent to a given conic 193
Conjugate points with respect to a conic 98
Connection 111
Connection between two planes 118
Connection between two planes, algebraic 119
Connection between two planes, orders 119
Contact of order n between curves 115
Contact of order n between curves, analytic condition 115
Contact, invariance under a point transformation 121
Contact, projective character 116
Contacts between plane curves 115
Coordinates, homogeneous 3
Coordinates, homogeneous projective 7
Coordinates, nonhomogeneous 3
Correlation between projective planes 47 77
Correlation between projective planes, definition 77
Correlation between projective planes, determined by four points mapping into four lines 79
Correlation between projective planes, equation 77
Correlation between projective planes, involutory 78
Correlation between projective planes, represented by a bilinear equation 78
Correlation between projective planes, verification that a given correlation is involutory 79
Correlation between projective spaces 82
Correlation between projective spaces, definition 82
Correlation between projective spaces, equations 83
Correlation between projective spaces, incidence quadric 83
Correlation between projective spaces, null correlation 83 86
Correlation between projective spaces, polarity 83
Correlation between projective spaces, represented by a bilinear equation 83
Correspondence, identity 4
Correspondence, product 4
Cramer's rule 171
Cremona transformation 127 128
Cremona transformation, between r-dimensional spaces 160
Cremona transformation, between three-dimensional spaces 159
Cross ratio 30 See
Cubic surface as a transform of a plane 158
Cubic transformation 157
Cubic transformation, equations 157
Cubic transformation, singular points 158
Cubic transformation, transform of a plane 158
Curve 2 161
Curve, double point 112
Curve, isolated point 112
Curve, multiple points 111
Curve, node 112
Curve, simple point 111
Curve, triple point 112
Curve-cusp 112
Curves, algebraic 170
Curves, contacts 114
Curves, osculating at a point 115
Curves, tangent at a point 115
Curves, touching at a point 115
Cyclic points 18
Degenerate conic 88 90
Degenerate conic, a conic is degenerate if its discriminant is equal to zero 91
Degenerate conic, method to determine the two distinct lines if the discriminant rank is 2 92
Degenerate conic, the two lines which compose a degenerate conic are coincident if and only if the rank of the discriminant is equal to 1 92
Degenerate pencil of conics 187
Degenerate pencil of conics, there are three degenerate conies in a pencil 187
Desargues, involution theorem 108
Desargues, theorem on the homological triangle 68
Determinant, cofactor 166
Determinant, defined 165
Determinant, emi-symmetric 167
Determinant, Laplace development 166
Determinant, minor 166
Determinant, properties 167
Determinant, skew-symmetric 167
Diagonal lines 52
Diagonal triangle 52
Distance 162
Double point of a curve 112
Double ratio 30
Double ratio of four lines 30
| Double ratio of four numbers 30
Double ratio of four points 29
Double ratio, it is invariant under a projectivity 32
Double ratio, its permutation effect 34
Dual definitions 24
Duality 23
Duality, plane 25
Duality, space 24
Equations, homogeneous 169
Equations, parametric 2
Euclidean space 162
Euler's theorem 184
Field 178
Field, complex 178
Finite points 16
Fixed lines under the homology 60 61
Fixed points under a collineation, their determination 60
Fixed points under a projectivity, their determination 38
Fixed points under the homology 60
Flat pencil 23
Forms, geometric 23
Fourth harmonic 30
Function 181
Function, complex 181
Function, homogeneous 183
Function, inverse 182
Function, one-to-one 182
Function, real 181
Fundamental conic 101
Fundamental points 47
Fundamental pyramid 3
Fundamental theorem 33 53
Fundamental triangle 47
Geometric forms 23
Geometry, Riemannian 164
Graphic properties 22
Group 4
Group, commutative (abelian) 5
Group, continuous 6
Group, dimension 6
Group, finite 5
Group, infinite 5
Group, order 5
Group, permutable elements 5
Harmonic homology 70
Harmonic range 30
Homogeneous coordinates 16
Homogeneous function 183
Homogeneous function, example 183
Homogeneous Plueckerian coordinates 11
Homogeneous, equations 169
Homography 6 See
Homology 60
Homology, axis 60
Homology, canonic equation 65
Homology, characteristic invariant 65
Homology, construction of the corresponding line to a given line if the center, the axis, and two corresponding points are given 74
Homology, construction of the corresponding point to a given point if the center, the axis, and two corresponding points are given 74
Homology, determined by its center, axis, and characteristic 73
Homology, harmonic 70
Homoloidal net of conies 130
Homothety 76
Homothety, its characteristic invariant 77
Hyperplane 2
Hyperplanes of projective spaces 10
Hyperplanes of the coordinates 2
Hypersurfac, algebraic 13
Hypersurfac, irreducible 13
Hypersurfac, reducible 13
Hypersurface 161
Ideal points 16
Identity 4
Imaginary lines 17
Imaginary points 17
Incidence conic 101
Incidence quadric 83
Incidence quadric, equations 83 84
Indeterminate tangent line to a conic 96
Intrinsic metric geometry 164
Invariant, characteristic 65
Inversion in the complex plane 130
Inversion in the complex plane, singular points 130
Inversion in the real plane 133
Inversion in the real plane, circle of inversion 134
Inversion in the real plane, exceptional point 135
Inversion in the real plane, pole 133
Inversion in the real plane, power 133
Inversion in the real plane, transformation of plane curves 135
Inversion in the real space 159
Inversion in the real space, center of inversion 160
Involution 40
Involution, center 43
Involution, conjugate points 41
Involution, determined by its center and power 44
Involution, double points 41
Involution, metric property 43
Involution, power 44
Involutory correlation 78
Iperosculating conies 189
Irreducible algebraic hypersurface 13
Isotropic line 18
Jacobian 184
Knotted parabola 112
Line 12
Line at infinity 16
Line coordinates 20
Line, complex 2
Line, definition 1
Line, homogeneous 3
Line, real 2
Linear subspaces, equations 11
Linear subspaces, line 12
Linear subspaces, plane 12
Linearly dependent points 8
Mapping 181
Mappings, product 183
Matrix 168
Matrix, rank 168
Metric properties 22
Motions 161
Motions, definition 161
Motions, equations 161
Multiple point of a curve 111
Multiple roots of an equation 170
Net of conies 130
Nondegenerate conic 93
Nondegenerate conic tangent to the x-axis 94
Nondegenerate conic, definition 93
Nondegenerate conic, determination of the asymptotes 96
Nondegenerate conic, property 93
Nonhomogeneous Plueckerian coordinates 11
One-dimensional form 23
One-to-one correspondence 183
Origin 1
Osculating circle of a conic 195
Pappus theorem 52
Parabola, knotted 112
Parameters 197
Parametric representation of rational algebraic curves 123
Parametric representation of the ellipse 123
Parametric representation of the hyperbola 124
Parametric representation of the parabola 124
Pencil of conies 106
Pencil of conies, bitangent to a given conic 193
Pencil of conies, degenerate 187
Pencil of conies, iperosculating a given conic 195
Pencil of conies, osculating a given conic 194
Pencil of conies, tangent to a given conic at one of its points 193
Pencil of lines 23 174
Plane 12
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