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| Авторизация |
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| Eisenhart L.P. — A Treatise on the Differential Geometry of Curves and Surfaces |
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| Предметный указатель |
Focal surface of a pseudospherical congruence 416
Focal surface, developable 412
Focal surface, fundamental quantities 409—411
Focal surface, infinitesimal deformation of 420
Focal surface, intersect 423
Focal surface, met by developables in lines of curvature 414
Focal surface, reduces to a curve 406 412
Form of a curve 18
Frenet — Serret formulas 17
Fundamental equations of a congruence 406 407
Fundamental quadratic form of a congruence 393
Fundamental quadratic form of a surface, first 71
Fundamental quadratic form of a surface, second 115
Fundamental quantities of the first order 71
Fundamental quantities of the second order 115
Fundamental theorem of the theory of curves 24
Fundamental theorem of the theory of surfaces 159
G see “E”
Gauss, (area of geodesic triangle) 209
Gauss, (geodesic circles) 207
Gauss, (geodesic parallels) 206
Gauss, (parametric form of equations) 50
Gauss, (spherical representation) 141
Gauss, (total curvature of a surface) 155
Gauss, Gauss, equations of 154 155 187
Generators of a developable surface 41
Generators of a ruled surface 241t
Generators of a surface of translation 198
Geodesic circles 207
Geodesic contingence, angle of 212
Geodesic curvature 132 134 135 136 140 213 223
Geodesic curvature, center of 132 225 423
Geodesic curvature, curves of constaHt 137 140 187 223 319
Geodesic curvature, invariance of 135
Geodesic curvature, radius of 132 150 151 174 176 209 411
Geodesic ellipses and hyperbolas 213—216 225
Geodesic parallels 207
Geodesic parameters 207
Geodesic polar coordinates 207—209 236 276
Geodesic representation 225 317
Geodesic torsion 137—140 174 176
Geodesic torsion, radius of 138 174 176
Geodesic triangle 209 210
Geodesies on surfaces of Liouville 218 219
Geodesies on surfaces of negative curvature 211
Geodesies, definition 133
Geodesies, equations of 204 205 215—219
Geodesies, plane 140
Goursat, surfaces of 366 372
Guichard, (congruences of Ribaucour) 421
Guichard, (spherical representation of the developables of a congruence) 409
Guichard, congruences of 414 416 417 422 442
H, definition 71
Hamilton, equation of 397
Hazzidakis, transformation of 278 279 338
Helicoid, general 146—148
Helicoid, general, applicable to a hyperboloid 347
Helicoid, general, geodesies 149 151 209
Helicoid, general, is W-surface 300
Helicoid, general, meridian of 146
Helicoid, general, minimal 329 331
Helicoid, general, parameter of 146
Helicoid, general, pseudospherical 291
Helicoid, general, surfaces of center of 186
Helicoid, right 146 148 203 247 260 260 267 330 347 381 422
Helix, circular 2 41 45 203
Helix, cylindrical 20 21 29 30 47 64
Henneberg, surface of 267
Hyperbolic point 125 200
Hyperbolic type of pseudospherical surface 273
Hyperbolic type of surface of Bianchi 371 379
Hyperboloid of revolution 247 348
Hyperboloid, deformation of 347 348
Hyperboloid, equations 228
Hyperboloid, evolute of 234
Hyperboloid, fundamental quantities 228—230
Hyperboloid, lines of striction 268
Indicatrix in which lines of curvature are preserved 387 391
Indicatrix of a right helicoid 381
Indicatrix of Dupin see “Dupin”
Indicatrix of ruled surfaces 381
Indicatrix of the focal surfaces of a W-congruence 420
Indicatrix, generatrices 373 420
Indicatrix, Infinitesimal deformation of a surface 373 385—387
Indicatrix, spherical see “Spherical”
Intrinsic equations of a curve 23 29 30 30
Invariants of a differential equation 380 385 406
Invariants, differential 85—90
Inversion by reciprocal radii Involute, of a curve 43—45 311
Inversion of a surface 180 184 300
Inversion, definition 196
Inversion, preserves a triply orthogonal system 463 (see also “Transformation”)
Inversion, preserves an isothermic system of lines of curvature 391
Inversion, preserves lines of curvature 190
Isometric parameters see “Isothermic parameters”
Isometric representation 100 113
Isothermal-conjugate systems of curves 198—200
Isothermal-conjugate systems of curves on associate surfaces 390
Isothermal-conjugate systems of curves, formed of lines of curvature 147 203 233 278
Isothermal-conjugate systems of curves, spherical representation 202
Isothermal-orthogonal system see “Isothermic orthogonal system”
Isothermic orthogonal systems 93—98 209 252 254
Isothermic orthogonal systems, formed of lines of curvature see “Isothermic surface”
Isothermic parameters 93—97 102
Isothermic surface 108 159 232 253 269 297 387—389 391 425 465
Isotropic congruence 412 413 416 422—424
Isotropic developable 72 171 412 424
Isotropic plane 49
Jacebi (geodesic lines) 217
Joachimsthal (geodesies and lines of curvature on central quadrics) 240
Joachimsthal, surfaces of 308 309 319
Joachimsthal, theorem of 140
Kummer (rectilinear congruences) 392
Lagrange (minimal surfaces) 251
Lame (differential parameters) 85
Lame, equations of 449
Lame, family of 461 463 464
Lelieuvre, formulas of 193 195 417 419 420 422
Lie, (double minimal surfaces) 259
Lie, (lines of curvature of W-surfaces) 293
Lie, (surfaces of translation) 197 198
Lie, transformation of 289 297
Limit point 396 399
Limit surface 389
Line of curvature on an isothermic surface 389
Line of curvature, circular 149 310—314 316 446
Line of curvature, definition 121 122 128
Line of curvature, equation of 121 171 247
Line of curvature, geodesic 140
Line of curvature, geodesic torsion of 139
Line of curvature, normal curvature of 121 131
Line of curvature, osculating plane 148
Line of curvature, parametric 122 151 186
Line of curvature, plane 149 150 201 305—314 319 320 463
Line of curvature, plane in both systems 269 300—304 319 320
Line of curvature, spherical 149 314—317 319 320 465
Line of curvature, spherical representation of 143 148 150
Line of curvature, two surfaces intersecting in 140
Line of striction 243 244 248 268 348 351 352 369 401 422
Line, singular 71
Linear element of a curve 4 5
Linear element of a surface 42 71 171
Linear element of space 447
Linear element of the spherical representation 141 173 393
Linear element, reduced form 353
Lines of length zero see “Minimal lines”
Lines of shortest length 212 220
Liouville, (angle of geodesic contingence) 212
Liouville, (form of Gauss equation) 187
Liouville, surfaces of 214 215 218 232
| Loxodromic curve 78 108 112 120 131 140 209
Mainardi, equations of 156
Malus and Dupin, theorem of 403
Mean curvature 123 126 145
Mean evolute 165 166 372
Mean ruled surfaces of a congruence 422 423 425
Mercator chart 109
Meridian curve on a surface 260
Meridian of a helicoid 146
Meridian of a surface of revolution 107
Meusnier, theorem of 118
Middle envelope of a congruence 413 415
Middle point of a line of a congruence 399
Middle surface of a congruence 399 401 408 413 421—424
Minding (geodesic curvature) 222 223
Minding, method of 344
Minding, problem of 321 323 326
Minimal curves 6 47 49 255 257
Minimal curves on a sphere 81 257 364—366 390
Minimal curves on a surface 81 82 85 91 254—265 318 391
Minimal straight lines 48 49 260
Minimal surface of Enneper 269
Minimal surface of Henneberg 267
Minimal surface of revolution 150
Minimal surface of Scherk 260
Minimal surface, adjoint 254 263 267 377
Minimal surface, algebraic 260—262
Minimal surface, associate 263 267 269 330 381
Minimal surface, asymptotic lines 129 186 195 254 257 269
Minimal surface, definition 129 251
Minimal surface, deformation of 264 327—329 349 381
Minimal surface, determination of 265 266
Minimal surface, double 258—260
Minimal surface, evolute 260 372
Minimal surface, geodesies 267
Minimal surface, helicoidal 149 330 331
Minimal surface, lines of curvature 186 253 257 264 269
Minimal surface, minimal lines 177 186 254—265
Minimal surface, parallel plane sections of 160
Minimal surface, ruled 148
Minimal surface, spherical representation 143 251—254
Molding surface, applicable 319 338
Molding surface, associate to right helicoid 381
Molding surface, definition 302
Molding surface, equations of 307 308
Molding surface, lines of curvature 307 308 320
Molding surface, normal to a congruence of Ribaucour 422
Molding surfaces, a family of Lame of 465
Monge, (equations of a surface) 64
Monge, (molding surfaces) 302
Monge, surfaces of 305—308 319
Moving trihedral for a curve 30—33
Moving trihedral for a curve, applications of 33—36 39 40 64—68
Moving trihedral for a surface 166—170
Moving trihedral for a surface, applications of 171—183 281—288 336—338 352—364 426—442
Moving trihedral for a surface, rotations of 169
Normal congruence of lines see “Congruence”
Normal congruence of lines of curves see “Congruence”
Normal curvature of a surface see “Curvature”
Normal plane to a curve 8 15 65
Normal section of a surface 118 234
Normal to a curve 12
Normal to a surface 57 114 117 120 121 141 195
Normal, principal, definition 12
Normal, principal, parallel to a plane 16 21
Normals, principal, which are binormals of another curve 51
Normals, principal, which are principal normals of another curve 41
Order of contact 8 21
Orthogonal system of curves 75 77 80—82 91 119 129 177 187
Orthogonal system of curves, geodesies 187
Orthogonal system of curves, isothermic see “Isothermic”
Orthogonal system of curves, parametric 75 93 122 134
Orthogonal trajectories of a family of curves 50 79 95 112 147 149 150
Orthogonal trajectories of a family of geodesies 216
Orthogonal trajectories of a family of surfaces 446 451 452 456 457 460 463 464
Orthogonal trajectories of a one-parameter family of planes 35 451
Osculating circle 14 21 65
Osculating plane of a geodesic 133
Osculating plane of an asymptotic line 128
Osculating plane of edge of regression 57
Osculating plane, definition 10
Osculating plane, equation of 11
Osculating plane, meets the curve 19
Osculating plane, orthogonal trajectories of 35
Osculating plane, passes through a fixed point 22
Osculating plane, stationary 18
Osculating planes of two curves parallel 50
Osculating sphere 37 38 47 51 65
Parabolic point on a surface 125
Parabolic type of pseudospherical surfaces 274
Parabolic type of surfaces of Bianchi 370 371 442 443 445
Paraboloid of normals to a ruled surface 247
Paraboloid, a right conoid 56
Paraboloid, a surface of translation 203
Paraboloid, asymptotic lines 191 233
Paraboloid, congruence of tangents 401 (see also “Quadrics”)
Paraboloid, deformation of 348 349 367—369 372
Paraboloid, equations 230 330
Paraboloid, evolute of 234
Paraboloid, fundamental quantities 231
Paraboloid, line of striction 268
Paraboloid, lines of curvature 232 240
Paraboloid, tangent plane 112
Parallel curves 44
Parallel on a surface of revolution 107
Parallel surface of surface of constant curvature 179
Parallel surface of surface of revolution 185
Parallel surface, definition 177
Parallel surface, fundamental quantities 178
Parallel surface, lines of curvature 178
Parallel surfaces, a family of Lame of 446
Parallel, geodesic 86 207
Parameter of distribution 246 247 268 348 424 425
Parameter, definition 1
Parametric curves 54 55
Parametric equations see “Equations”
Plane curve, condition for 2 16
Plane curve, condition for, curvature 15
Plane curve, condition for, equations 28 49
Plane curve, condition for, intrinsic equations 36
Plane curves forming a conjugate system 224
Plane lines of curvature see “Lines of curvature”
Point of a surface, elliptic 125 200
Point of a surface, focal see “Focal”
Point of a surface, hyperbolic 125 200
Point of a surface, limit see “Limit”
Point of a surface, middle see “Middle”
Point of a surface, parabolic 126
Point of a surface, singular 71
Polar developable 64 65 112 209
Polar line of a curve 15 38 46
Principal directions at a point 121
Principal normal to a curve see “Normal”
Principal planes of a congruence 396 397
Principal radii of normal curvature 119 120 291 450
Principal surfaces of a congruence 396—398 408
Projective transformation, preserves asymptotic lines and conjugate systems 202
Projective transformation, preserves osculating planes 49
Pseudosphere 274 290
Pseudospherical congruence 415 416 464
Pseudospherical congruence, normal 184
Pseudospherical surface of Dini 291 318
Pseudospherical surface of Enneper 317 320
Pseudospherical surface of revolution of elliptic type 274
Pseudospherical surface of revolution of hyperbolic type 273
Pseudospherical surface of revolution of parabolic type 274
Pseudospherical surface, asymptotic lines 190 290 414
Pseudospherical surface, definition 270
Pseudospherical surface, deformation 277 323
Pseudospherical surface, evolute 318
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