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| Eisenhart L.P. — A Treatise on the Differential Geometry of Curves and Surfaces |
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| Предметный указатель |
Pseudospherical surface, geodesies 275—277 283 317 318
Pseudospherical surface, involute 318
Pseudospherical surface, lines of curvature 190 203 280 320
Pseudospherical surface, surfaces with the same spherical representation of their lines of curvature as 320 371 437 439 443 444
Pseudospherical surface, transformations of 280—290 318 320 370 456
Pseudospherical surfaces, a family of Lame of 452—456 464
Quadratic form see “Fundamental”
Quadrics, associate surfaces 390 391
Quadrics, asymptotic lines 233
Quadrics, confocal 226 401
Quadrics, fundamental quantities 229
Quadrics, geodesies 234—236 239 240
Quadrics, lines of curvature 233 239 240
Quadrics, normals to 422 (see also “Ellipsoid” “Hyperboloid” “Paraboloid”)
Representation, conformal see “Conformal”
Representation, conformal-conjugate 224
Representation, equivalent 113 188
Representation, Gaussian 141
Representation, geodesic 225 317
Representation, isometric 100 113
Representation, spherical see “Spherical”
Revolution, surfaces of see “Surface”
Ribaucour, (asymptotic lines on surfaces of center) 184
Ribaucour, (cyclic systems associated with a triply orthogonal system) 457
Ribaucour, (cyclic systems of equal circles) 280
Ribaucour, (cyclic systems) 426 428 432
Ribaucour, (deformation of the envelope of the planes of a cyclic system) 429 430
Ribaucour, (limit surfaces) 389
Ribaucour, (middle envelope of an isotropic congruence) 413
Ribaucour, congruence of 420—422 424 425 435 442 443
Ribaucour, triple systems of 452 455 463
Riccati equation 25 26 50 248 429
Rodrigues, equations of 122
Ruled surface of a congruence 393—395 398 401 422 423 “Hyperboloid” “Paraboloid”)
Ruled surface of tangents to a surface 188
Ruled surface, asymptotic lines 248—250
Ruled surface, central plane 244
Ruled surface, central point 243
Ruled surface, conjugate 268
Ruled surface, definition 241
Ruled surface, deformation 343—348 350 367
Ruled surface, director-cone 241
Ruled surface, directrix 241
Ruled surface, doubly 234
Ruled surface, generators 241
Ruled surface, infinitesimal deformation 381
Ruled surface, line of striction 243 244 248 268 348 351 352 369 401 422
Ruled surface, linear element 241 247
Ruled surface, lines of curvature 250 268
Ruled surface, mean curvature 249
Ruled surface, normals to 195 247
Ruled surface, parameter of distribution 245 247 268 348 424 425
Ruled surface, spherical indicatrix of 351
Ruled surface, tangent plane 246 247 268
Ruled surface, total curvature 247
Scheffers (equations of a curve) 28
Scherk, surface of 260
Schwarz, formulas of 264—267 269
Singular line of a surface 71
Singular point of a surface 71
Sinistrorsum 19
Sphere, asymptotic lines 223 422
Sphere, conformal representation 109—111
Sphere, equations 52 77 81
Sphere, equivalent representation 113
Sphere, fundamental quantities 116 171
Sphere, minimal lines 81
Sphere, principal radii 120
Spheres, family of see “Family”
Spherical curve 36 38 47 50 149 314—316 317 319 320 465
Spherical indicatrix of a ruled surface 351
Spherical indicatrix of the binormals to a curve 50 177
Spherical indicatrix of the tangents to a curve 9 13 50 177
Spherical representation of a congruence, definition 393
Spherical representation of a congruence, developables 409 412—414 422 432—435 437 441
Spherical representation of a congruence, principal surfaces 397 408
Spherical representation of a surface, area of closed portion 145
Spherical representation of a surface, asymptotic lines 144 148 191—195 254 340 390 414
Spherical representation of a surface, conjugate system 200—202 257 385
Spherical representation of a surface, definition 141
Spherical representation of a surface, fundamental quantities 141—143 160—165 173
Spherical representation of a surface, lines of curvature 143 148 150 151 188 201 253 279 280 292 296 301 302 308 314 315 320 371 387 437 442—445
Spherical representation of an axis of a moving trihedral 354
Spherical surface of Enneper 317
Spherical surface of revolution 270—272
Spherical surface, definition 270
Spherical surface, deformation 276 323
Spherical surface, geodesies 275—279 318
Spherical surface, involute 300
Spherical surface, lines of curvature 278
Spherical surface, parallels to 179
Spherical surface, surface with the same spherical representation of its lines of curvature as 338 (see also “Surface of constant total curvature”)
Spherical surface, transformation 278—280 297
Spherical surfaces, a family of Lame of 463
Spiral surface, asymptotic lines 151
Spiral surface, definition 151
Spiral surface, deformation 349
Spiral surface, generation 151
Spiral surface, geodesies 219
Spiral surface, lines of curvature 151
Spiral surface, minimal tines 151
Stereographic projection 110 112
Superosculating circle 21
Superosculating lines on a surface 187
| Surface of constant mean curvature, definition 179
Surface of constant mean curvature, deformation 298
Surface of constant mean curvature, lines of curvature 296—298
Surface of constant mean curvature, minimal curves 318
Surface of constant mean curvature, parallels to 179
Surface of constant mean curvature, transformation 297
Surface of constant total curvature, area of geodesic triangle 219
Surface of constant total curvature, asymptotic lines 317
Surface of constant total curvature, definition 179
Surface of constant total curvature, geodesies 224
Surface of constant total curvature, lines of curvature 317
Surface of constant total curvature, spherical representation 372 (see also “Pseudospherical surface and Spherical surface”)
Surface of reference 392
Surface of revolution, asymptotic lines 131
Surface of revolution, definition 107
Surface of revolution, deformation 108 112 147 149 260 276 277 283 326—331 341 349—350 362—364 369 370 372 444
Surface of revolution, equivalent representation 113
Surface of revolution, fundamental quantities 107 147
Surface of revolution, geodesies 205 209 224
Surface of revolution, lines of curvature 126
Surface of revolution, loxodromic curve see “Loxodromic”
Surface of revolution, parallel surfaces 185
Surface of revolution, particular 111 160 320
Surface of translation, associate surface 381 390;
Surface of translation, asymptotic lines 198
Surface of translation, congruence of tangents 405
Surface of translation, definition 197 198
Surface of translation, deformation 349 350
Surface of translation, equations 197
Surface of translation, generators 198 203
Surface of translation, middle surface of a W-congruence 422 424
Surface with plane lines of curvature see “Lines of curvature”
Surface with spherical lines of curvature see “Lines of curvature”
Surface with the same spherical representation of its lines of curvature as a pseudospherical surface see “Pseudospherical surface”
Surface with the same spherical representation of its lines of curvature as a spherical surface see “Spherical surface”
Surface, limit 389
Surfaces of revolution, a family of Lame of 451
Tangent plane to a surface, characteristic of 126
Tangent plane to a surface, definition 56 114
Tangent plane to a surface, developable surface 57
Tangent plane to a surface, distance to 114
Tangent plane to a surface, equation 57
Tangent plane to a surface, is the osculating plane of asymptotic line 128
Tangent plane to a surface, meets the surface 123
Tangent surface of a curve 41—44 57
Tangent surface of a curve, applicable to the plane 101 156
Tangent to a curve 6 7 49 50 59
Tangent to a curve, spherical indicatrix of 9 13 50 177
Tangent to a surface 112
Tangential coordinates 163 194 201
Tetrahedral surface, asymptotic lines 267
Tetrahedral surface, definition 267
Tetrahedral surface, deformation 341
Tetrahedral surfaces, triple system of 465
Tore 124
Torsion of a curve of a plane curve 16
Torsion of a curve of asymptotic line 140
Torsion of a curve, constant 50
Torsion of a curve, definition 16
Torsion of a curve, radius of 16 17 21
Torsion of a curve, sign of 19
Torsion, geodesic 137—140 174 176
Tractrix, equations 35
Tractrix, helicoid whose meridian is a 291
Tractrix, surface of revolution of 274 290
Transformation by reciprocal radii 104 196 203
Transformation of curvilinear coordinates 53—55 73 74
Transformation of rectangular coordinates 72
Transformation, projective see “Projective”
Triply orthogonal system of surfaces of Bianchi 452—454 464 465
Triply orthogonal system of surfaces of cyclides of Dupin 465
Triply orthogonal system of surfaces of isothermic surfaces 465
Triply orthogonal system of surfaces of Ribaucour 452 463
Triply orthogonal system of surfaces of Weingarten 455 456 463 464
Triply orthogonal system of surfaces, associated with a cyclic system 446
Triply orthogonal system of surfaces, definition 447
Triply orthogonal system of surfaces, fundamental quantities 447—451
Triply orthogonal system of surfaces, transformation of 462 463
Triply orthogonal system of surfaces, with one family of molding surfaces 465
Triply orthogonal system of surfaces, with one family of surfaces of revolution 451 452
Umbilical point of a surface of quadrics 230 232 234 236—238 240 267
Umbilical point of a surface, definition 120
v. Mangoldt (geodesies on surfaces of positive curvature) 212
Variation of a function 82 83
Voss, surface of 341 390 415 442 443
W-congruence 417—420 422 424
W-surface of Weingarten 298 424
W-surface, definition 291
W-surface, evolute 294 295 318 319
W-surface, fundamental quantities 291—293
W-surface, lines of curvature 293
W-surface, particular 291 300 318 319
W-surface, ruled 299 319
W-surface, spherical representation 292
Weierstrass, (algebraic minimal surfaces) 261
Weierstrass, (equations of a minimal surface) 260
Weingarten, (geodesic ellipses and hyperbolas) 214
Weingarten, (infinitesimal deformation) 374 387
Weingarten, (lines of curvature on an isothermic surface) 389
Weingarten, (tangential coordinates) 163
Weingarten, (W-surfaces) 291 292 294
Weingarten, method of 353—372
Weingarten, surface of 298 424
Weingarten, triple system of 455 456 463 464
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