Higher Geometry: An Introduction to Advanced Methods in Analytic GeometryGinn, 1922 - Всего страниц: 423 |
Содержание
GENERAL CONCEPTS AND ONEDIMENSIONAL GEOMETRY CHAPTER I GENERAL CONCEPTS SECTION PAGE 1 Coördinatės | 1 |
The principle of duality | 2 |
Infinity | 3 |
Transformations | 4 |
Groups | 6 |
RANGES AND PENCILS 7 Cartesian coördinate of a point on a line | 8 |
Change of coördinates | 9 |
Coördinate of a line of a pencil | 11 |
Onedimensional extents of planes | 210 |
Locus of an equation in plane coördinates | 215 |
Change of coördinates | 218 |
SURFACES OF SECOND ORDER AND OF SECOND CLASS SECTION 90 Surfaces of second order | 220 |
Singular points | 221 |
Poles and polars | 222 |
Classification of surfaces of second order | 224 |
Surfaces of second order in Cartesian coördinates | 227 |
Coördinate of a plane of a pencil | 12 |
PROJECTIVITY 12 The linear transformation | 13 |
The cross ratio | 16 |
Harmonic sets | 18 |
Projection | 20 |
Perspective figures | 21 |
Other onedimensional extents | 23 |
TWODIMENSIONAL GEOMETRY CHAPTER IV POINT AND LINE COÖRDINATES IN A PLANE 18 Homogeneous Cartesian point coördinates | 27 |
The circle points at infinity | 30 |
The conic | 32 |
Trilinear point coördinates | 34 |
Points on a line | 35 |
The linear equation in point coördinates | 36 |
Lines of a pencil | 37 |
Line coördinates in a plane | 38 |
Pencil of lines and the linear equation in line coördinates | 39 |
Dualistic relations | 40 |
Change of coördinates | 41 |
Certain straightline configurations | 44 |
Curves in point coördinates | 50 |
Curves in line coördinates | 53 |
CURVES OF SECOND ORDER AND SECOND CLAS 33 Singular points of a curve of second order | 58 |
Poles and polars with respect to a curve of second order | 59 |
Classification of curves of second order | 65 |
Singular lines of a curve of second class | 67 |
Classification of curves of second class | 68 |
Poles and polars with respect to a curve of second class | 70 |
Projective properties of conics | 72 |
LINEAR TRANSFORMATIONS 40 Collineations | 78 |
Types of nonsingular collineations | 83 |
Correlations | 88 |
Pairs of conics | 95 |
The projective group | 100 |
The metrical group | 101 |
Angle and the circle points at infinity | 105 |
PROJECTIVE MEASUREMENT 47 General principles | 107 |
The hyperbolic case | 110 |
The elliptic case | 115 |
The parabolic case | 117 |
CONTACT TRANSFORMATIONS IN THE PLANE 51 Pointpoint transformations | 120 |
Quadric inversion | 121 |
Inversion | 124 |
Pointcurve transformations | 127 |
The pedal transformation | 131 |
The line element | 133 |
SECTION PAGE 57 Special tetracyclical coördinates | 138 |
Distance between two points | 139 |
The circle | 140 |
Relation between tetracyclical and Cartesian coördinates | 142 |
Orthogonal circles | 144 |
Pencils of circles | 146 |
The general tetracyclical coördinates | 150 |
Orthogonal coördinates | 153 |
The linear transformation | 154 |
The metrical transformation | 155 |
Inversion | 156 |
The linear group | 159 |
Duals of tetracyclical coördinates | 161 |
A SPECIAL SYSTEM OF COÖRDINATES 70 The coördinate system | 164 |
The straight line and the equilateral hyperbola | 166 |
The bilinear equation | 167 |
The bilinear transformation | 169 |
THREEDIMENSIONAL GEOMETRY CHAPTER XI CIRCLE COÖRDINATES 74 Elementary circle coördinates | 171 |
The quadratic circle complex | 173 |
Higher circle coördinates | 177 |
POINT AND PLANE COÖRDINATES 77 Cartesian point coördinates | 180 |
Distance | 181 |
The straight line | 182 |
The plane | 185 |
Direction and angle | 188 |
Quadriplanar point coördinates | 193 |
Straight line and plane | 194 |
Plane coördinates | 197 |
Onedimensional extents of points | 200 |
Locus of an equation in point coördinates | 205 |
Surfaces of second order referred to rectangular axes | 229 |
Rulings on surfaces of second order | 232 |
Surfaces of second class | 235 |
Poles and polars | 238 |
TRANSFORMATIONS 100 Collineations | 240 |
Types of nonsingular collineations | 241 |
Correlations | 246 |
The projective and the metrical groups | 249 |
Projective geometry on a quadric surface | 250 |
Projective measurement | 253 |
Clifford parallels | 255 |
Contact transformations | 258 |
Pointpoint transformations | 260 |
Pointsurface transformations | 262 |
Pointcurve transformations | 263 |
THE SPHERE IN CARTESIAN COÖRDINATES 111 Pencils of spheres | 266 |
Bundles of spheres | 268 |
Complexes of spheres | 269 |
Inversion | 270 |
Dupins cyclide | 274 |
Cyclides | 279 |
PENTASPHERICAL COÖRDINATES 117 Specialized coördinates | 282 |
The sphere | 284 |
Angle between spheres | 286 |
The power of a point with respect to a sphere | 287 |
General orthogonal coördinates | 288 |
The linear transformation | 291 |
Relation between pentaspherical and Cartesian coördinates | 293 |
Tangent circles and spheres | 295 |
Cyclides in pentaspherical coördinates | 297 |
GEOMETRY OF FOUR AND HIGHER DIMENSIONS CHAPTER XVII LINE COÖRDINATES IN THREE DIMENSIONAL SPACE 127 The Plücker... | 301 |
Dualistic definition | 303 |
Intersecting lines | 304 |
General line coördinates | 305 |
Pencils and bundles of lines | 306 |
Complexes congruences series | 308 |
The linear line complex | 310 |
Conjugate lines | 314 |
Complexes in point coördinates | 316 |
Complexes in Cartesian coördinates | 317 |
The bilinear equation in point coördinates | 321 |
The linear line congruence | 322 |
The cylindroid | 323 |
The linear line series | 324 |
The quadratic line complex | 328 |
Singular surface of the quadratic complex | 331 |
Plückers complex surfaces | 334 |
The 2 2 congruence | 335 |
Line congruences in general | 336 |
SPHERE COÖRDINATES 146 Elementary sphere coördinates | 341 |
Higher sphere coördinates | 343 |
Angle between spheres | 344 |
The linear complex of oriented spheres | 346 |
Linear congruence of oriented spheres | 348 |
Linear series of oriented spheres | 349 |
Pencils and bundles of tangent spheres | 350 |
Quadratic complex of oriented spheres | 353 |
Duality of line and sphere geometry | 357 |
FOURDIMENSIONAL POINT COÖRDINATES | 362 |
Definitions 156 Intersections | 365 |
Euclidean space of four dimensions | 368 |
Parallelism | 370 |
Perpendicularity | 373 |
Minimum lines planes and hyperplanes | 378 |
Hypersurfaces of second order | 382 |
Duality between line geometry in three dimensions and point geometry in four dimensions | 384 |
GEOMETRY OF N DIMENSIONS 163 Projective space | 388 |
Intersection of linear spaces | 390 |
The quadratic hypersurface | 392 |
Intersection of a quadric by hyperplanes | 396 |
Linear spaces on a quadric | 401 |
Stereographic projection of a quadric in S upon | 407 |
Application to line geometry | 410 |
Metrical space of n dimensions | 413 |
Авторские права | |
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