The geometry of the conic sections

Definitions ......Page 19
Formulae connecting Cartesians and Polars ......Page 24
Distance between two points ......Page 25
Area of triangle—Condition of collinearity of three points ......Page 27
Definitions ......Page 31
Relativity of equations of loci ......Page 32
Line through two given points ......Page 35
General linear equation ......Page 37
Standard forms ......Page 39
Length of perpendicular on a given line ......Page 47
Parallel lines. Perpendicular lines ......Page 50
Intersection of lines ......Page 53
Polar equation ......Page 56
Area of triangle ......Page 58
Angle between two lines ......Page 63
General equation representing two lines ......Page 66
Bisectors of angles between two lines ......Page 70
General equation of the circle ......Page 77
Constant rectangle of segments of chords through a point ......Page 79
Imaginary points ......Page 81
Tangent ......Page 84
Circle referred to its centre ......Page 85
Poles and polars ......Page 89
Conjugate points and lines ......Page 92
Equation of chord in terms of its middle point ......Page 93
Trigonometrical notation ......Page 95
Systems of circles—Coaxial circles ......Page 98
Formulae of transition ......Page 109
Invariants ......Page 113
Oblique axes ......Page 116
Definitions ......Page 120
General equation ......Page 121
Standard forms for parabola, ellipse and hyperbola ......Page 122
Criteria of discrimination for General Equation ......Page 137
Tangent. Chord of contact. Poles and polars. Conjugate points and lines ......Page 140
Equation of chord in terms of its middle point ......Page 148
Pair of tangents ......Page 150
Special equations and properties ......Page 154
Normals ......Page 157
Equation of parabola referred to diameter and tangent ......Page 159
The parabola and the general equation ......Page 161
Special equations and properties ......Page 172
Director circle ......Page 175
Eccentric angle ......Page 177
Properties of normals ......Page 180
Conjugate diameters ......Page 187
Special equations ......Page 203
Asymptotes ......Page 205
Conjugate hyperbola ......Page 210
Conjugate diameters ......Page 212
Hyperbola referred to its asymptotes ......Page 216
General polar equation ......Page 221
Focus pole ......Page 222
Chord, tangent and normal ......Page 224
Intersections of two conies ......Page 231
Contact of conies ......Page 232
Conic through five points ......Page 233
Conies through points of intersection of two conies ......Page 235
Pair of tangents ......Page 238
Foci of conies ......Page 239
Equation of axes ......Page 242
Length of axes ......Page 244
Eccentricity ......Page 248
Director circle ......Page 251
A theorem of Newton's ......Page 254
Contact of conies and circles of curvature ......Page 256
Conic referred to two tangents ......Page 263
System of conies through four points ......Page 265
Definitions and propositions respecting similarity ......Page 274
Confocals and their properties ......Page 283
Definitions ......Page 291
Formulae of transition from Cartesians to areals ......Page 292
Area of triangle—Distance between points—Equation of line ......Page 294
The line at infinity ......Page 297
Parallel lines—Length of perpendicular ......Page 298
Special form of equation of line ......Page 301
General equation of second degree ......Page 302
Criteria of discrimination ......Page 305
Tangents &c ......Page 308
Centre. Foci, Axes. Asymptotes ......Page 311
Conies circumscribing the triangle of reference ......Page 315
Conies inscribed in the triangle of reference ......Page 316
Conies for which the triangle of reference is self-polar ......Page 318
Equations of special circles ......Page 319
The circular points at infinity ......Page 322
Radical axis ......Page 324
Orthogonal circles ......Page 326
Definitions ......Page 335
Geometrical interpretation ......Page 337
Transformation to Cartesians ......Page 338
General equation of second degree ......Page 339
Criteria of discrimination ......Page 341
Foci. Axes ......Page 343
Trilinears ......Page 344
Cartesians as a homogeneous system ......Page 345
Polar reciprocals ......Page 347
Conies expressed by a single parameter ......Page 349
The equations of two conies ......Page 350
Double contact ......Page 351
Single contact ......Page 353
Three point contact. Four point contact ......Page 354
Analytical representation of cross ratios ......Page 360
Harmonic conjugates ......Page 362
The representation of four coplanar points ......Page 364
Conies through four points ......Page 366
The representation of four coplanar lines ......Page 367
Conies touching four lines ......Page 369
Constant cross-ratio property of conies ......Page 370
Involution ......Page 372
Homographic ranges ......Page 377
Invariants of a single conic ......Page 387
Invariants of two conies ......Page 392
Invariants and projection ......Page 393
Illustrations ......Page 397
Conditions for single, double, three point contact ......Page 402
Invariants and reciprocation ......Page 406
Reciprocal relation between point and tangential equations ......Page 412
Envelopes ......Page 416
Tangential equation of conies touching the common tangents of two conies ......Page 420
Points of intersection of two conies ......Page 421
Equation of four common tangents ......Page 423
Circular points at infinity ......Page 426
Confocal conies ......Page 428
Foci of conies ......Page 430
Condition for rectangular hyperbola ......Page 432
Perpendicularity of lines ......Page 433
Definition ......Page 440
Geometrical interpretation ......Page 441
The F conic ......Page 442
Conditions for double contact ......Page 445
Polar reciprocal ......Page 446
The Ф conic ......Page 450
Contravariants ......Page 452
Jaoobian of three conies ......Page 454
The cubic covariant of two conies ......Page 457
The cubic contravariant ......Page 458