Eisenhart L.P. — A Treatise on the Differential Geometry of Curves and Surfaces :: Электронная библиотека попечительского совета мехмата МГУ

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Eisenhart L.P. — A Treatise on the Differential Geometry of Curves and Surfaces

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Название: A Treatise on the Differential Geometry of Curves and Surfaces

Автор: Eisenhart L.P.

Аннотация:

Created especially for graduate students by a leading writer on mathematics, this introduction to the geometry of curves and surfaces concentrates on problems that students will find most helpful.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1909

Количество страниц: 474

Добавлена в каталог: 25.03.2006

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Предметный указатель

$\dagger$ Surface, definition      53
$\ddagger$ Surface of center, a curve      186 188 308—314
$\ddagger$ Surface of center, asymptotic lines      183 184
$\ddagger$ Surface of center, definition      179
$\ddagger$ Surface of center, developable      186 305—308
$\ddagger$ Surface of center, fundamental quantities      181 182
$\ddagger$ Surface of center, lines of curvature      183 184
$\ddagger$ Surface of center, met by developables in a conjugate system      180 181
$\ddagger$ Surface of center, total curvature      183
$\mathcal{D}$ , $\mathcal{D}'$ , $\mathcal{D}''$ , definition      386
$\mathcal{E}$ , $\mathcal{F}$ , $\mathcal{G}$ , definition      141
$\mathcal{E}$ , $\mathcal{F}$ , $\mathcal{G}$ , for the moving trihedral      174
$\mathcal{F}$       see “ $\mathcal{E}$ ”
$\mathcal{G}$       see “ $\mathcal{E}$ ”
$\mathcal{H}$ , definition      142
Acceleration      15 50
Angle between curves      74 200
Angle of geodesic contingence      212
Applicable surfaces to the plane      101 156
Applicable surfaces, definition      100
Applicable surfaces, invariance of geodesic curvature      135
Applicable surfaces, invariance of total curvature      156
Applicable surfaces, pairs of, derived from a given pair      349 (see also “Deformation of”)
Applicable surfaces, solution of the problem of determining whether two given surfaces are applicable      321—326
Applicable surfaces, surfaces Area, element of      75 145
Area of a portion of a surface      145 250
Area of a portion of a surface, minimum      222
Associate surfaces of a ruled surface      381
Associate surfaces of an isothermic surface      388
Associate surfaces of pseudospherical surfaces      390
Associate surfaces of quadrics      390 391
Associate surfaces of the right helicoid      381
Associate surfaces of the sphere      381
Associate surfaces, applicable      381
Associate surfaces, characteristic property      425
Associate surfaces, definition      378
Associate surfaces, determination      378—381
Asymptotic directions, definition      128
Asymptotic lines, definition      128
Asymptotic lines, orthogonal      129
Asymptotic lines, parametric      129 189—194
Asymptotic lines, preserved by projective transformation      202
Asymptotic lines, preserved in a deformation      342—347
Asymptotic lines, spherical representation      144 191—193
Asymptotic lines, straight      140 234
Beltrami, (applicable ruled surfaces)      345
Beltrami, (differential parameters)      88 90
Beltrami, (geodesic curvature)      183
Beltrami, (normal congruences)      403
Beltrami, (ruled W-surfaces)      299
Bertrand curves on a ruled surface      250
Bertrand curves, definition      39
Bertrand curves, deformation      348
Bertrand curves, parametric equations      51
Bertrand curves, properties      39—41
Bianchi, (associate surfaces)      378
Bianchi, (cyclic congruences of Ribaucour)      435
Bianchi, (cyclic systems)      441
Bianchi, (surfaces with circular lines of curvature)      311
Bianchi, (surfaces with spherical lines of curvature)      315
Bianchi, (theorem of permutability)      286—288
Bianchi, generalized transformation of      439
Bianchi, surfaces of      370 371 442 443 445
Bianchi, transformation of      280—283 290 318 320 370 456
Bianchi, triply orthogonal systems of      452—464 464 465
Binomials which are the principal normals to another curve      51
Binormal to a curve, definition      12
Binormal to a curve, spherical indicatrix      50
Bonnet, (formula of geodesic curvature)      136
Bonnet, (lines of curvature of Liouville type)      232
Bonnet, (ruled surfaces)      248
Bonnet, (surfaces of constant curvature)      179
Bonnet, (surfaces of constant mean curvature)      298
Bour, (associate isothermic surfaces)      388
Bour, (helicoids)      147
BScklund, transformation of      284—290
Canal surfaces, definition      68
Canal surfaces, surfaces of center      186
Catenoid, adjoint surface of      267
Catenoid, definition      150
Catenoid, surfaces applicable to      318
Cauchy, problem of      265 335
Central plane      244
Central point      243
Cesaro (moving trihedral)      32
Characteristic equation      375
Characteristic function      374 377
Characteristic lines      130 131
Characteristic lines, parametric      203
Characteristics of a family of surfaces      59—61
Characteristics of the tangent planes to a surface      126
Christoffel (associate isothermic surfaces)      388
Christoffel symbols, definition      152 153
Christoffel symbols, relations between, for a surface and its spherical representation      162 193 201
Circle of curvature      14
Circle, osculating      14
Circles on the sphere      301
Circles, orthogonal system of, in the plane      80 97
Circular lines of curvature      149 310 316 423 446
Circular point on a surface      124
Codazzi, equations of      155—157 161 168 170 189 200
Combescure transformation of curves      50
Combescure transformation of triple systems      461—465
Complementary surface      184 185 283 290 370 464
Conformal representation of a plane upon itself      104 112
Conformal representation of a pseudospherical surface upon the plane      317
Conformal representation of a sphere upon itself      110 111
Conformal representation of a sphere upon the plane      109
Conformal representation of a surface and its spherical representation      143
Conformal representation of a surface upon itself      101—103
Conformal representation of two surfaces      98—100 391
Conformal-conjugate representation of two surfaces      224
Congruence of curves      426
Congruence of curves, normal      430
Congruence of straight lines (rectilinear) of Guicliard      414 415 417 422 442
Congruence of straight lines (rectilinear) of Ribaucour      420—422 424 425 435 442 443
Congruence of straight lines (rectilinear), associate normal      401—403 411
Congruence of straight lines (rectilinear), cyclic      431—445
Congruence of straight lines (rectilinear), cyclic of Ribaucour      435 442 443
Congruence of straight lines (rectilinear), definition      392
Congruence of straight lines (rectilinear), derived      403—405 411 412
Congruence of straight lines (rectilinear), developables      398 409 414 421 432 437
Congruence of straight lines (rectilinear), developables of cyclic      437 441
Congruence of straight lines (rectilinear), focal planes      400 401 409 416
Congruence of straight lines (rectilinear), focal points      398 399 425
Congruence of straight lines (rectilinear), focal surfaces      400 406 409—411 412 414 416 420
Congruence of straight lines (rectilinear), isotropic      412 413 416
Congruence of straight lines (rectilinear), limit points      396
Congruence of straight lines (rectilinear), mean ruled surfaces      422 423 425
Congruence of straight lines (rectilinear), middle envelope      413 415
Congruence of straight lines (rectilinear), middle point      399
Congruence of straight lines (rectilinear), middle surface      399 401 408 413 421—424
Congruence of straight lines (rectilinear), normal      393 398 401 402 403 412 422 423 437
Congruence of straight lines (rectilinear), normal cyclic      437
Congruence of straight lines (rectilinear), principal planes      396 397
Congruence of straight lines (rectilinear), principal surfaces      396—398 408
Congruence of straight lines (rectilinear), pseudospherical      184 415 416 464
Congruence of straight lines (rectilinear), ruled surfaces      393 398 401
Congruence of straight lines (rectilinear), spherical representation of cyclic      432—433
Congruence of straight lines (rectilinear), W-      417—420 422 424
Conjugate directions      126 173
Conjugate directions, normal radii in      131
Conjugate system of plane curves      224
Conjugate system, definition      127 223
Conjugate system, parametric      195 203 223 224
Conjugate system, preserved by projective transformation      202
Conjugate system, preserved in a deformation      338—342 348 349
Conjugate system, spherical representation of      200
Conjugate systems in correspondence      130
Conoid, right      56 58 59 68 82 98 112 120 195 347

Coordinates, curvilinear, in space      447
Coordinates, curvilinear, on a surface      55
Coordinates, elliptic      227
Coordinates, symmetric      91—93
Coordinates, tangential      163 194 201
Correspondence with orthogonality of linear elements      374—377 390
Corresponding conjugate systems      130
Cosserat (infinitesimal deformation)      380 385
Cross-ratio of four solutions of a Riccati equation      26
Cross-ratio of points of intersection of four-curved asymptotic lines on a ruled surface      249
Cross-ratio of the points in which four surfaces orthogonal to a cyclic system meet the circles      429
Cubic, twisted      4 8 11 12 15 269
Curvature, first, center of      14
Curvature, first, circle of      14
Curvature, first, constant      22 38 51
Curvature, first, of a curve      9
Curvature, first, radius of      9
Curvature, Gaussian      123
Curvature, Gaussian, geodesic      see “Geodesic”
Curvature, mean, of a surface      123 126 145
Curvature, mean, of a surface, surfaces of constant      see “Surface”
Curvature, normal, of a surface, radius of      118 120 130 131 150
Curvature, normal, of a surface, radius of, center of      118 150
Curvature, normal, of a surface, radius of, principal centers of      122
Curvature, normal, of a surface, radius of, principal radii of      119 120 291 450
Curvature, second, constant      50 (see also “Torsion”)
Curvature, second, of a curve      16
Curvature, total, of a surface      123 126 145 155 156 160 172 186 194 208 211
Curvature, total, radius of      189
Curvature, total, surfaces of constant      see “Surface”
Curve of constant first curvature      22 38 51
Curve of constant torsion      50
Curve, definition      2
Curve, form of a      18
Cyclic congruences      see “Congruences”
Cyclic system      426—446
Cyclic system of equal circles      430 443
Cyclic system, associated with a triple system      457; 458
Cyclic system, definition      426
Cyclic system, planes depend on one parameter      442
Cyclic system, planes envelop a curve      439 440
Cyclic system, planes through a point      440 441
Cyclic system, surfaces orthogonal to      436 437 444 457
Cyclic system, triple system associated with a      446
Cyclides of Dupin      188 312—314 412 422 465
D, , for the moving trihedral      174
D, , , definition      115
Darboux, (asymptotic lines and conjugate systems preserved by projective transformation)      202
Darboux, (asymptotic lines parametric)      191
Darboux, (conjugate lines parametric)      195
Darboux, (general problem of deformation)      332
Darboux, (generation of new surfaces of Weingarten)      298
Darboux, (generation of surfaces with plane lines of curvature in both systems)      304
Darboux, (geodesic parallels)      216 217
Darboux, (lines of curvature preserved by an inversion)      196
Darboux, (moving trihedral)      168 169 170
Darboux, (surfaces applicable to paraboloids)      367
Darboux, (triply orthogonal systems)      468—461
Darboux, derived congruences of      404 405
Darboux, twelve surfaces of      391
Deformation of surfaces      see “Applicable surfaces”
Deformation of surfaces of minimal surfaces      264 269 327—330
Deformation of surfaces of paraboloids      348 368 369
Deformation of surfaces of ruled surfaces      343—348 350 367
Deformation of surfaces of surfaces of constant curvature      321—323
Deformation of surfaces of surfaces of revolution      see “Surfaces of revolution”
Deformation of surfaces of the envelope of the planes of a cyclic system      429 430
Deformation of surfaces, general problem      331—333
Deformation of surfaces, method of Weingarten      353—369
Deformation of surfaces, which changes a curve on the surface into a given curve in space      333—336
Deformation of surfaces, which preserves asymptotic lines      336 342 343
Deformation of surfaces, which preserves conjugate systems      338—342 349 350 443
Deformation of surfaces, which preserves lines of curvature      336—338 341
Developable surface of a congruence      see “Congruence”
Developable surface, applicable to the plane      101 156 219 321 322
Developable surface, definition      61
Developable surface, equation      64
Developable surface, formed by normals to a surface at points of a line of curvature      122
Developable surface, fundamental property      244
Developable surface, geodesies on a      224 268 318 322
Developable surface, particular kinds      69
Developable surface, polar      64 65 112 209
Developable surface, principal radii      149
Developable surface, rectifying      62 64 112 209
Developable surface, total curvature      156 250
Dextrorsum      19”
Differential parameters of the first order      84—88 90 91 120 160 166 186
Differential parameters of the second order      88—91 160 165 166 186
Dini, (ruled W-surfaces)      299
Dini, (spherical representation of asymptotic lines)      192
Dini, (surfaces of Liouville)      214
Dini, surface of      291 318
Director-cone of a ruled surface      141
Director-developable of a surface of Monge      305
Directrix of a ruled surface      241
Dobriner (surfaces with spherical lines of curvature)      315
Dupin (triply orthogonal systems)      449
Dupin, cyclide of      see “Cyclide”
Dupin, indicatrix of      124—126 129 150
Dupin, theorem of Malus and      403
e, f, , g, definition      393
E, F, G, definition      70
E, F, G, for the moving trihedral      174
Edge of regression      43 60 69
Element of area      75 145
Element, Ellipsoid, equations      228
Element, linear      see “Linear element”
Element, normal section      234
Element, polar geodesic system      236—238
Element, surface corresponding with parallelism of tangent plane      269 (see also “Quadrics”)
Element, umbilical geodesies      236 267
Elliptic coordinates      227
Elliptic point of a surface      125 200
Elliptic type of pseudospherical surfaces      274
Elliptic type of surfaces of Bianchi      370 371
Enneper, (equations of a minimal surface)      256
Enneper, (torsion of asymptotic lines)      140
Enneper, minimal surface of      269
Enneper, surfaces of constant curvature of      317 320
Envelope of a one-parameter family of planes      61—63 64 69 442
Envelope of a one-parameter family of spheres      66—69
Envelope of a two-parameter family of planes      162 224 426 439
Envelope of a twoparameter family of spheres      391 444
Envelope of geodesies      221
Envelope, definition      59 60
Equations of a curve      1 2 3 21
Equations of a surface      52 53 54
Equations, parametric      1 2 52 53
Equidistance, curves of      456
Equidistantial system      187 203
Equivalent representation of two surfaces      113 188
Euler, equation of      124 221
Evolute of a curve      43 45—47
Evolute of a surface      180 415
Evolute, center) ; of the quadrics      234
Evolute, mean, of a surface      165 166 372
f      see “E”
f       see “e”
Family of Lame      461 463 464
Family, one-parameter, of curves      78—80
Family, one-parameter, of geodesies      216 221
Family, one-parameter, of planes      61—64 69 442 463
Family, one-parameter, of spheres      66—69 309 319 463
Family, one-parameter, of surfaces      59 446 447 451 452 457—461
Family, two-parameter, of planes      162 224 426 439
Family, two-parameter, of spheres      391 444
Focal conic      226 234 313 314
Focal planes      400 401 409 416
Focal points      398 399 425
Focal surface of a congruence      400

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