Index. in this web service Cambridge University Press

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1 A(2), 71 affine classification of conics, 108 group, 71, 471 properties, 73 transformation, 71 transformations, basic properties, 73 affine geometry, 73 congruence, 86, 111 Fundamental Theorem, 86 rectangular hyperbola, 116 Alberti, Leone Battistuta, 130 altitude, 391, 447 Altitude Theorem, 391 angle, 273, 351 of lune, 441 of parallelism, 395 sum, 383, 384, 388, 439 Angle Theorem, 274 angular defect, 406 angular excess, 440 antipodal point, 425 Apollonian circles, 261, 317 family of, 318 Apollonian Circles Theorem, 317, 319 Apollonius of Perga, 1 arbelos, 330 Archimedes, xi area hyperbolic, 401 spherical, 441 argument, 277 associated circle and plane, 459 asymptote, 17, 474 asymptotic triangle, 388 Aubel s Theorem, van, 120 auxiliary circle, 32 axes ellipse, 15 hyperbola, 17 parabola, 13 B 2 4AC Test, 42, 213 barycentric coordinates, 102 collinearity, 104 equation of line, 104 Section Formula, 105 Bayeux tapestry, 128 Beltrami Klein model, 478 Bessel, Friedrich Wilhelm, 345 Bolyai, Janos, 3, 344 boundary point, 346 Brannan, Michael, xiv Brianchon, Charles J., 177 Brianchon s Theorem, 177, 252 Brisbane Exhibition and Conference Centre, 52 Brunelleschi, 129 camera, aerial, 189 canonical form, 363, 365 Cassegrain focus, 32 centre of inversion, 263 of projective reflection, 475 Ceva, Giovanni, 93 Ceva s Theorem, 94, 107 converse, 97, 108 circle, 6 auxiliary, 32 equation, 7 generalized, 285 great, 426 hyperbolic, 373 little, 426 of Apollonius, 317 of inversion, 263 circles intersecting, 9, 321 kissing, 335 orthogonal, 9 Soddy, 335 coaxal circles, 317, 322, 458 Coaxal Circles Theorem, 322 colatitude, 428 collinearity in RP 2, 141, 142, 145 common perpendicular, 388 Common Perpendiculars Theorem,

2 584 compass bearing, constant, 463 complex number, 276 computer graphics, xii, 127 Concentricity Theorem, 326 congruence, 69, 163, 215, 316, 385 affine, 111 Euclidean, 69 G-congruence, 70 conic, 6 classification, 41, 42 focus directrix definition, 11 general equation, 36 matrix representation, 37 polar equation, 18 projective, 202, 205 section, 6, 22, 210 section, degenerate, 6 section, non-degenerate, 6 tangent, 23, 25 conjugate diameters, 78 Conjugate Diameters Theorem, 78, 116 conjugate transformation, 454 cooling tower, 48 Cosine Rule, 449 Coxeter, H. S. M., 409 cross-ratio, 179, 180, 187, 188, 475 on embedding plane, 187 Dandelin, Germinal Pierre, 22 spheres, 22, 53 d-congruent, 385 Desargues, Girard, 1, 2 Desargues Theorem, 135, 173 Descartes, René, 1 diagonal vanishing point, 134 Difference of Focal Distances Property, 20 directed distances, 52 direct transformation, 362, 365, 429 directrix, 12 discriminant, 111 distance, 7, 367 function, 367, 368 hyperbolic, 368, 369, 380, 481 in R 2, 7 spherical, 426, 444 d-line, 346, 347 boundary point, 346 doubling map, 81, 297 doubly asymptotic triangle, 388, 401 d-point, 345 d-quadrilateral, 384 d-triangle, 383 Dual Triangles Theorem, 443 duality, 176, 250 Duality, Principle of, 178, 250 Duccio, 129 Dürer, Albrecht, 130 eccentricity, 12 Eccentricity Formula, 209 École Polytechnique, 2 electrostatics, 2 Elements of Euclid, 1 ellipse, 14 parametric representation, 15 Reflection Property, 29 standard form, 15 embedding plane, 147, 148 standard, 148 envelope, 32 equatorial plane, 428, 433 equidistant curves, 399 equivalence relation, 69 Erlangen Programme, 4, 478 Escher, Maurits Cornelis, 409 Euclid, xi Euclidean, 62 congruence, 68 geometry, 62 properties, 64 transformation, 66 extended complex plane, 283 conjugation function, 286 line, 285 linear function, 287 plane, 284 reciprocal function, 287 External Reflection Property, 31 families of circles, 321 coaxal, 317, 322, 458 Fermat, Pierre de, 1 Five Points Theorem, 230 focal chord, 13 focal distances ellipse, 19 hyperbola, 20 focus, 12, 32 foreshortening, 134 function extended conjugation, 286 extended linear, 287 extended reciprocal, 287 linear, 282 notation, xiii reciprocal, 283 Fundamental Theorem Affine Geometry, 88, 93 Inversive Geometry, 310, 312 Projective Geometry, 162, 172 G D, 350 Gauss, Carl Friedrich, 344, 345, 405 generalized circle, 285 generating lines, 49 generator of cone, 210 geodesic, 377, 409, 412, 426 geometry, 61 affine, 73

3 585 elliptic, 343, 424 Euclidean, 62 hyperbolic, 345, 412 model, 4 non-euclidean, 344 Giotto, 129 Goethe, Johann Wolfgang, xi gradient, 13 great circle, 426 Greenwich meridian, 426 half-plane model, 412 hemisphere, 483 hexagon, 177 hierarchy, 475 Hilbert, David, 485 homogeneous coordinates, 138 homogeneous matrix, 471 horizon, 135 horosphere, 485 hyperbola, 16 asymptotes, 17 parametric representation, 18 rectangular, 18 Reflection Property, 30 standard form, 18 hyperbolic area, 401 centre, 373 circle, 373 distance, 368, 369, 380, 481 geometry, 345, 412 group G D, 350 limit rotation, 363 midpoint, 371 paraboloid, 43, 51 parallel postulate, 344 radius, 373 reflection, 350, 379 rotation, 362 transformation, 350 translation, 363 triangle, 383 hyperboloid of one sheet, 43, 48 ideal Line, 148 ideal Point, 148, 188 Incidence Property of RP 2, 146 indirect transformation, 362, 429 infinity, point at, 284 inside of triangle on S 2, 440 Internal Reflection Property, 31 inverse, 263 inversion, 262, 265 inversive geometry, 295 group, 296 transformation, 295 Inversive Geometry, Fundamental Theorem, 310, 312 isometry, 63, 278, 429 Isosceles Triangle Theorem, 437 Joachimsthal, 216 notation, 218, 227 Section Equation, 219 kaleidoscope, 409 kiss precise, 335 kissing circles, 335 Klein, Christian Felix, 4 Kleinian view, 61 La Hire s Theorem, 225, 244 Lady s and Gentleman s Diary, xi latitude, 428 length on S 2, 426 limit rotation, 363 line on S 2, 426 Line, 141 Conic, 251 equation of, 141, 144 ideal, 148 linear function, 282 linearization map, 478 Little Chef, restaurant, 52 little circle, 426 Lobachevskii, Nikolai Ivanovich, 3, 344 Lobachevskii s Formula, 394, 398 longitude, 428 Lovell radio-telescope, 29 lune, 441 map, planar, 460 Martini, Simone, 129 matrix of rotation, 38, 429 Median Theorem, 93 Menelaus of Alexandria, 93 Menelaus Theorem, 98, 106 Converse, 101, 106 Mercator, 463 meridian, 426, 461 midpoint, 78 hyperbolic, 371 Midpoint Theorem, 78 Möbius, August Ferdinand, 3 band, 3 transformation, 298, 454 model (of a geometry), 4 disc, 345 elliptic, 424 half-plane, 412 hyperbolic, 345 spherical, 425 modulus, 276 Monge, Gaspard, 2, 177 Newton, Isaac, 1 non-euclidean, 344 normal, 26

4 586 oblate spheroid, 460 Origin Lemma, 351 orthogonal, 9, 65, 322, 432 outside of triangle on S 2, 440, 448 P(2), 152 Pappus Theorem, 175, 177, 186 parabola, 12 parametric representation, 13 Reflection Property, 31 standard form, 13 vertex, 13 parallel, definition, 349 parallel postulate elliptic, 344 Euclidean, 343 Hyperbolic, 344 parallel projection, 74 basic properties, 75 composition, 82 parallels, 461 parameter of projective reflection, 475 Parametrization Theorem, 237, 246 Pascal, Blaise, 2 Pascal s Theorem, 238 converse, 240 perpendicular, 388 common, 388 on S 2, 448 perspective, 128 focused, 129 terraced, 128 transformation, 166, 167 vertical, 129 perspectivity, 131 Perspectivity Theorem, 172 Plato, xi Poincaré, Jules Henri, 4 point at infinity, 284 circle, 318 Point, 137 conic, 251 ideal, 148, 188 unit, 146 polar, 26, 223, 224, 226 equation of conic, 18 form, 277 pole, 224 Pole, 425 Poncelet, Jean Victor, 2 Poncelet s porism, 329 porism, 328 power of a point, 52 principal argument, 277 vanishing point, 134 Principle of Duality, 178, 250 Product Theorem, 436 projection, 461 azimuthal, 462 conformal, 462 conical, 463 cylindrical, 462 equal area, 463 gnomic, 462 Mercator, 463 orthogonal, 462 orthographic, 462 parallel, 74 stereographic, 290, 450, 462 projective congruence, 163 figure, 140 geometry, 156 hierarchy, 475 line, 141 plane, 137 point, 137 properties, 156 reflection, 475 transformation, 151 projective conic, 202, 205 degenerate, 205 inside, 207 non-degenerate, 205 outside, 207 standard form, 237, 242 tangent, 207 Projective Geometry, Fundamental Theorem, 162, 172 Ptolemy s Theorem, 330 punctured, definition, 268 Pythagoras Theorem, 392, 396, 445 quadric, 42 classification, 45 degenerate, 42 matrix representation, 44 surface, 42, 43 quadrilateral, 162, 384 radical axis, 321 railroad lines, 134 real projective plane, 137 reciprocal function, 283 rectangular hyperbola, 18 reflection, 28 hyperbolic, 361, 379, 411, 480 Law, 28 Lemma, 379 Reflection Property ellipse, 29 hyperbola, 30 parabola, 31 Riemann sphere, 290, 450 Roque de los Muchachos Observatory, 31 rotation, 38, 278 elementary, 429 hyperbolic, 362 matrix, 429

5 587 RP 2, 137 RP 2 0, 471 ruled surface, 48 generators, 49 scaling, 280, 461 Scorer, Sam, 51 Section Equation, Joachimsthal s, 219 Section Formula, 105, 496 self-dual, definition, 176 self-inverse, 264 shoemaker s knife, 330 simply asymptotic, 388 Sine Formula, 29, 396 Sine Rule, 448 slope, 13 Soddy, Frederick, 335 sphere, Riemann, 290, 450 spherical distance, 426, 444 geometry, 425 isometry, 429 polar coordinates, 428 triangle, 437, 438 trigonometry, 438 Sputnik, xi standard embedding plane, 148 projective conic, 237, 242 Steiner, Jakob, 328 Steiner s porism, 328 stereographic projection, 290, 450, 462 strict line segment, 440 strict triangle, 440 subgeometry, 472 Sum of Focal Distances Property, 19 Symmetry Lemma, 273 tangent, 23, 207, 216, 225, 474 pair, 222, 226 to conic in standard form, 114 tapestry, Bayeux, 128 telescope, 31 tessellation, 408 hyperbolic, 409 Three Points Theorem, 233 Three Tangents and Three Chords Theorem, 244 Three Tangents Theorem, 216 translation, 277 trebly asymptotic, 388, 401, 402 triangle dual, 442 spherical, 438 triangle of reference, 146 Triangle Inequality, 367 Two Apollonian Circles Theorem, 327 ultra-parallel, definition, 349 Unique Fourth Point Theorem, 185 unit Point, 146 van Aubel s Theorem, 120 vanishing line, 135 vanishing point diagonal, 134 principal, 134 vertex of parabola, 13 Vinci, Leonardo da, 130 wallpaper patterns, 408