Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry

Frederick Shenstone Woods

Ginn, 1922 - Geometry, Analytic - 423 pages

Contents

• GENERAL CONCEPTS AND ONEDIMENSIONAL GEOMETRY CHAPTER I GENERAL CONCEPTS SECTION PAGE 1 Coordinates: 1
• The principle of duality: 2
• Infinity: 3
• Transformations: 4
• Groups: 6
• RANGES AND PENCILS 7 Cartesian coordinate of a point on a line: 8
• Change of coordinates: 9
• Coordinate of a line of a pencil: 11
• Coordinate 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 COORDINATES IN A PLANE 18 Homogeneous Cartesian point coordinates: 27
• The circle points at infinity: 30
• The conic: 32
• Trilinear point coordinates: 34
• Points on a line: 35
• SECTION PAGE 24 The linear equation in point coordinates: 36
• Lines of a pencil: 37
• Line coordinates in a plane: 38
• Pencil of lines and the linear equation in line coordinates: 39
• Dualistic relations: 40
• Change of coordinates: 41
• Certain straightline configurations: 44
• Curves in point coordinates: 50
• Curves in line coordinates: 53
• CURVES OF SECOND ORDER AND SECOND CLASS 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: 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 Poinlpoint transformations: 120
• Quadric inversion: 121
• Inversion: 124
• Pointrcurve transformations: 127
• The pedal transformation: 131
• The line element: 133
• TETRACYCLICAL COORDINATES SECTION PAGE 57 Special tetracyclical coordinates: 138
• Distance between two points: 139
• The circle: 140
• Relation between tetracyclical and Cartesian coordinates: 142
• Orthogonal circles: 144
• Pencils of circles: 146
• The general tetracyclical coordinates: 150
• Orthogonal coordinates: 153
• The linear transformation: 154
• The metrical transformation: 155
• Inversion: 156
• The linear group: 159
• Duals of tetracyclical coordinates: 161
• A SPECIAL SYSTEM OF COORDINATES 70 The coordinate system: 164
• The straight line and the equilateral hyperbola: 166
• The bilinear equation: 167
• The bilinear transformation: 169
• THREEDIMENSIONAL GEOMETRY CHAPTER XI CIRCLE COORDINATES 74 Elementary circle coordinates: 171
• The quadratic circle complex: 173
• Higher circle coordinates: 177
• POINT AND PLANE COORDINATES 77 Cartesian point coordinates: 180
• Distance: 181
• The straight line: 182
• The plane: 185
• Direction and angle: 188
• Quadriplanar point coordinates: 193
• Straight line and plane: 194
• Plane coordinates: 197
• Onedimensional extents of points: 200
• Locus of an equation in point coordinates: 205
• Onedimensional extents of planes: 210
• Locus of an equation in plane coordinates: 215
• Change of coordinates: 218
• SURFACES OF SECOND ORDER AND OF SECOND CLASS SECTION PAGE 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 coordinates: 227
• 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 COORDINATES 111 Pencils of spheres: 266
• Bundles of spheres: 268
• Complexes of spheres: 269
• Inversion: 270
• Dupins cyelide: 276
• Cyclides: 279
• PENTASPHERICAL COORDINATES 117 Specialized coordinates: 282
• The sphere: 284
• Angle between spheres: 286
• The power of a point with respect to a sphere: 287
• General orthogonal coordinates: 288
• SECTION PAGE 122 The linear transformation: 291
• Relation between pentaspherical and Cartesian coordinates: 293
• Tangent circles and spheres: 295
• Cyclides in pentaspherical coordinates: 297
• GEOMETRY OF FOUR AND HIGHER DIMENSIONS CHAPTER XVII LINE COORDINATES IN THREE DIMENSIONAL SPACE 127 The Pluck...: 301
• Dualistic definition: 303
• Intersecting lines: 304
• General line coordinates: 305
• Pencils and bundles of lines: 306
• Complexes congruences series: 308
• The linear line complex: 310
• Conjugate lines: 314
• Complexes in point coordinates: 316
• Complexes in Cartesian coordinates: 317
• The bilinear equation in point coordinates: 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
• Pliickers complex surfaces: 334
• The 2 2 congruence: 335
• Line congruences in general: 336
• SPHERE COORDINATES 146 Elementary sphere coordinates: 341
• Higher sphere coordinates: 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 COORDINATES SECTION PAGE 155 Definitions: 362
• 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 Sn 1: 407
• Application to line geometry: 410
• Metrical space of n dimensions: 413
• Minimum projection of Sn upon Sn_1: 419
• INDEX: 421

Popular passages

Page 9 - X = x cos a + y sin a Y = —x sin a + y...‎

Page 117 - The sum of the angles of a triangle is less than two right angles by a quantity proportional to the area of the triangle.‎

Page 5 - The length of one side of a triangle must be greater than the difference and less than the sum of the lengths of the other two sides.‎

Page 380 - If p is the length of the perpendicular from the origin to the plane, and a, ft, y, are the direction angles of this perpendicular, then cosa = ._ — . . the same sign ± v^l2 + B* + C2 ± VA* + B* +C*' of the radical being used throughout, and so chosen that p is positive.‎

Page 143 - A circle is the locus of a point at a constant distance from a fixed point. The fixed point is the center of the circle, and the constant distance is the radius.‎

Page 51 - ... 2. Three points F, G, H are taken on the side BC of a triangle ABC: through G any line is drawn cutting AB and AC in L and M respectively; FL and HM intersect in K; prove that K lies on a fixed straight line passing through A.‎

Page 40 - V s(s — a)(s — b)(s — c) in which a, b, and c are the lengths of the sides of the triangle and s = %(a + b + c).‎

Page 153 - To prove that the electric force at a point outside a charged sphere is inversely proportional to the square of the distance of the point from the center of the sphere.‎

Page 226 - To prove that the locus of the middle points of a system of parallel chords of a parabola is a straight line parallel to the axis of the parabola.‎

Page 72 - The point is called the pole of the line, and the line is called the polar of the point with respect to the curve of second class given by equation (1), § 37.‎

Bibliographic information

• Title: Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry; Volume 737 of Dover Books. S; Dover books on advanced mathematics; Dover books on science
• Author: Frederick Shenstone Woods
• Edition: unabridged
• Publisher: Ginn, 1922
• Original from: Harvard University
• Digitized: Aug 16, 2007
• Length: 423 pages