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Projective surfaces are strongly related to affine surfaces (that is, ordinary algebraic surfaces). One passes from a projective surface to the corresponding affine surface by setting to one some coordinate or indeterminate of the defining polynomials (usually the last one). Conversely, one passes
598:
may take any values. Also, there are surfaces for which there cannot exist a single parametrization that covers the whole surface. Therefore, one often considers surfaces which are parametrized by several parametric equations, whose images cover the surface. This is formalized by the concept of
1366:{\displaystyle {\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial u}}&{\dfrac {\partial f_{1}}{\partial v}}\\{\dfrac {\partial f_{2}}{\partial u}}&{\dfrac {\partial f_{2}}{\partial v}}\\{\dfrac {\partial f_{3}}{\partial u}}&{\dfrac {\partial f_{3}}{\partial v}}\end{bmatrix}}}
2656:
560:
1129:
1821:
1609:
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provide important visual cues to the orientation and slopes of surfaces, and the use of almost self-similar fractal patterns can help create natural looking visual effects. The modeling of the Earth's rough surfaces via
1625:. It is an irregular point that remains irregular, whichever parametrization is chosen (otherwise, there would exist a unique tangent plane). Such an irregular point, where the tangent plane is undefined, is said
1826:
991:
834:, which is outside the plane of the circle) is an algebraic surface which is not a differentiable surface. If one removes the apex, the remainder of the cone is the union of two differentiable surfaces.
565:
Parametric equations of surfaces are often irregular at some points. For example, all but two points of the unit sphere, are the image, by the above parametrization, of exactly one pair of Euler angles
2927:{\displaystyle {\frac {\partial f}{\partial x}}(x_{0},y_{0},z_{0})(x-x_{0})+{\frac {\partial f}{\partial y}}(x_{0},y_{0},z_{0})(y-y_{0})+{\frac {\partial f}{\partial z}}(x_{0},y_{0},z_{0})(z-z_{0})=0.}
3409:
Most authors consider as an algebraic surface only algebraic varieties of dimension two, but some also consider as surfaces all algebraic sets whose irreducible components have the dimension two.
1674:
1513:
428:
3929:
Here "implicit" does not refer to a property of the surface, which may be defined by other means, but instead to how it is defined. Thus this term is an abbreviation of "surface defined by an
3632:. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through
2975:) is a point of the surface where the implicit equation holds and the three partial derivatives of its defining function are all zero. Therefore, the singular points are the solutions of a
3342:
the defining polynomial (in case of surfaces in a space of dimension three), or by homogenizing all polynomials of the defining ideal (for surfaces in a space of higher dimension).
2297:
366:
3732:
Because the intended result of the process is to produce a landscape, rather than a mathematical function, processes are frequently applied to such landscapes that may affect the
423:
2412:
3636:. On the other hand, extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. This is well illustrated by the non-linear
3574:, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the
986:
3265:
2973:
967:
919:
189:
There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and
2058:{\displaystyle {\begin{aligned}x&={\frac {f_{1}(t,u)}{f_{0}(t,u)}},\\y&={\frac {f_{2}(t,u)}{f_{0}(t,u)}},\\z&={\frac {f_{3}(t,u)}{f_{0}(t,u)}}\,,\end{aligned}}}
2979:
of four equations in three indeterminates. As most such systems have no solution, many surfaces do not have any singular point. A surface with no singular point is called
3223:
3063:
2149:
2998:
if there is no other singular point in a neighborhood of it. Otherwise, the singular points may form a curve. This is in particular the case for self-crossing surfaces.
1636:, that is the points where the surface crosses itself. In other words, these are the points which are obtained for (at least) two different values of the parameters.
1387:
of the range of the parametrization. For surfaces in a space of higher dimension, the condition is the same, except for the number of columns of the
Jacobian matrix.
2645:
2339:
is nonzero. An implicit surface has thus, locally, a parametric representation, except at the points of the surface where the three partial derivatives are zero.
3817:
are also sometimes used as temporary ways to represent an object, with the goal of using the points to create one or more of the three permanent representations.
3376:
polynomials, but these polynomials must satisfy further conditions that may be not immediate to verify. Firstly, the polynomials must not define a variety or an
2414:, the tangent plane and the direction of the normal are well defined, and may be deduced, with the implicit function theorem from the definition given above, in
1508:
3101:
are accepted for defining an algebraic surface. However, the field of coefficients of a polynomial is not well defined, as, for example, a polynomial with
3412:
In the case of surfaces in a space of dimension three, every surface is a complete intersection, and a surface is defined by a single polynomial, which is
1669:
694:
There are several kinds of surfaces that are considered in mathematics. An unambiguous terminology is thus necessary to distinguish them when needed. A
1483:
It may occur that an irregular point becomes regular, if one changes the parametrization. This is the case of the poles in the parametrization of the
3746:, the generation of natural looking surfaces and landscapes was a major turning point in art history, where the distinction between geometric,
2154:
Implicit means that the equation defines implicitly one of the variables as a function of the other variables. This is made more exact by the
3350:
One cannot define the concept of an algebraic surface in a space of dimension higher than three without a general definition of an
100:
72:
202:
53:
3750:
and natural, man made art became blurred. The first use of a fractal-generated landscape in a film was in 1982 for the movie
4105:
1444:
maps the tangent plane to the surface at a point to the tangent plane to the image of the surface at the image of the point.
4078:
3391:
polynomials define an algebraic set of dimension two or higher. If the dimension is two, the algebraic set may have several
4056:
4034:
858:) is a differentiable surface and an algebraic surface. It is also a ruled surface, and, for this reason, is often used in
682:
is the locus of a moving line satisfying some constraints; in modern terminology, a ruled surface is a surface, which is a
79:
3806:
3771:
4127:
17:
4101:
4074:
4052:
4030:
3983:
The infinite degree of transcendence is a technical condition, which allows an accurate definition of the concept of
3961:
3537:
3514:
3355:
3327:
1465:
119:
86:
823:
of a line crossing a circle and parallel to a given direction) is an algebraic surface and a differentiable surface.
3648:, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to
3752:
3705:
resulting from the procedure is not a deterministic, but rather a random surface that exhibits fractal behavior.
1453:
at a point of a surface is the unique line passing through the point and perpendicular to the tangent plane; the
636:
3737:
3713:
3416:
or not, depending on whether non-irreducible algebraic sets of dimension two are considered as surfaces or not.
68:
3152:
57:
3339:
3953:
555:{\displaystyle {\begin{aligned}x&=\cos(u)\cos(v)\\y&=\sin(u)\cos(v)\\z&=\sin(v)\,.\end{aligned}}}
2240:
306:
3802:
3637:
4158:
3406:. If there are several components, then one needs further polynomials for selecting a specific component.
1124:{\displaystyle {\begin{aligned}x&=f_{1}(u,v),\\y&=f_{2}(u,v),\\z&=f_{3}(u,v)\,.\end{aligned}}}
2358:
2303:
934:
3089:. Abstract algebraic surfaces, which are not explicitly embedded in another space, are also considered.
3747:
3722:
3443:
3322:
in four variables. More generally, a projective surface is a subset of a projective space, which is a
3241:
2949:
943:
895:
2155:
3859:
3633:
3075:
2990:
The study of surfaces near their singular points and the classification of the singular points is
643:, which are not contained in any other space. On the other hand, this excludes surfaces that have
3486:
3315:
3181:
3018:
2107:
2098:
1760:
Every point of this surface is regular, as the two first columns of the
Jacobian matrix form the
786:
715:
93:
46:
3641:
3597:
3593:
3413:
3319:
2976:
1461:
1380:
two. Here "almost all" means that the values of the parameters where the rank is two contain a
710:
3485:. This allows the characterization of the properties of surfaces in terms of purely algebraic
3798:
3743:
3541:
3403:
3392:
1441:
866:
845:
841:
is a topological surface, which is neither a differentiable surface nor an algebraic surface.
809:
675:
608:
235:
210:
3521:
3012:
Originally, an algebraic surface was a surface which may be defined by an implicit equation
2428:
666:
is the locus of a point which is at a given distance of a fixed point, called the center; a
4154:
3971:
3790:
3579:
3525:
3162:
2995:
1604:{\displaystyle {\begin{aligned}x&=t\cos(u)\\y&=t\sin(u)\\z&=t\,.\end{aligned}}}
1437:
8:
4093:
3952:, American Mathematical Society Colloquium Publications, vol. 29, Providence, R.I.:
3833:
3601:
3589:
3490:
3381:
3098:
3082:
3081:
The concept has been extended in several directions, by defining surfaces over arbitrary
2332:
922:
820:
765:
761:
696:
659:
383:
of two variables (some further conditions are required to ensure that the image is not a
380:
372:
274:
270:
171:
224:
two; this means that a moving point on a surface may move in two directions (it has two
3849:
3733:
3582:, who showed that curvature was an intrinsic property of a surface, independent of its
3575:
3548:
3501:
3482:
3462:
3425:
3323:
2991:
1783:
879:
740:
683:
655:
644:
632:
388:
278:
225:
198:
163:
141:
869:
is an algebraic surface and the union of two non-intersecting differentiable surfaces.
4187:
4182:
4177:
4123:
4097:
4070:
4048:
4026:
3957:
3930:
3726:
3686:
3663:
3552:
3439:
3380:
of higher dimension, which is typically the case if one of the polynomials is in the
3351:
3007:
2072:
926:
816:
801:
722:). Every differentiable surface is a topological surface, but the converse is false.
612:
298:
290:
238:
217:
194:
1663:
be a function of two real variables. This is a parametric surface, parametrized as
639:). This allows defining surfaces in spaces of dimension higher than three, and even
241:
is defined. For example, the surface of the Earth resembles (ideally) a sphere, and
3879:
3810:
3649:
3629:
3613:
3477:
study of such arrangements of triangles (or, more generally, of higher-dimensional
3466:
3311:
3086:
2084:
1773:
1377:
797:
736:
567:
282:
230:
4119:
Fractals: The
Patterns of Chaos : a New Aesthetic of Art, Science, and Nature
1750:{\displaystyle {\begin{aligned}x&=t\\y&=u\\z&=f(t,u)\,.\end{aligned}}}
4117:
4009:
3967:
3947:
3757:
3717:
3709:
3621:
3567:
3545:
3455:
3295:
3102:
2090:
1761:
1492:
1433:
1147:
930:
889:
827:
769:
726:
667:
648:
628:
3898:
3617:
3494:
3110:
747:). A surface that is not supposed to be included in another space is called an
250:
4171:
3984:
3874:
3474:
3447:
3377:
3235:
1499:
1416:
805:
679:
620:
249:
provide two-dimensional coordinates on it (except at the poles and along the
197:. The exact definition of a surface may depend on the context. Typically, in
183:
3943:
3670:
3910:
A Riemannian surface is a smooth surface equipped with a
Riemannian metric.
3869:
3864:
3838:
3827:
2094:
1488:
1381:
859:
830:(locus of a line crossing a circle, and passing through a fixed point, the
616:
611:, a surface is a manifold of dimension two; this means that a surface is a
400:
3500:
The homeomorphism classes of surfaces have been completely described (see
3814:
3605:
3533:
3451:
3106:
1484:
1449:
1384:
772:
624:
294:
155:
4023:
Advances in multimedia modeling: 13th
International Multimedia Modeling
3690:
1812:
1429:
1143:
838:
581:
577:
286:
266:
3897:
A smooth surface is a surface in which each point has a neighborhood
3843:
3645:
3609:
3563:
3085:, and by considering surfaces in spaces of arbitrary dimension or in
404:
376:
246:
221:
175:
3644:: although Euler developed the one variable equations to understand
3334:
from an affine surface to its associated projective surface (called
933:
of dimension at least three. Usually the function is supposed to be
670:
is the locus of a line passing through a fixed point and crossing a
35:
3675:
3583:
3470:
3435:
3431:
2419:
701:
604:
600:
411:
262:
242:
206:
3760:
refined the techniques of
Mandelbrot to create an alien landscape.
3558:
Surfaces have been extensively studied from various perspectives:
1138:(for example, if the three functions are constant with respect to
132:
3702:
3698:
3694:
3678:
3478:
167:
2347:
A point of the surface where at least one partial derivative of
1815:
in two indeterminates, then the parametric surface, defined by
1164:
3625:
663:
190:
145:
137:
3652:, a concept that can only be defined in terms of an embedding.
725:
A "surface" is often implicitly supposed to be contained in a
3854:
3740:, in the interests of producing a more convincing landscape.
3438:
of dimension two. This means that a topological surface is a
3365:
More precisely, an algebraic surface in a space of dimension
1135:
671:
384:
179:
3778:
2335:
near a point of the surface where the partial derivative in
4088:
Rhonda Roland
Shearer "Rethinking Images and Metaphors" in
1436:, because its definition is independent of the choice of a
4160:
3117:
of the surface has been generalized in the following way.
399:. For example, the unit sphere may be parametrized by the
3809:
are one way of representing objects. The other ways are
3794:
3105:
coefficients may also be considered as a polynomial with
228:). In other words, around almost every point, there is a
4045:
Human symmetry perception and its computational analysis
3708:
Many natural phenomena exhibit some form of statistical
1632:
There is another kind of singular points. There are the
1476:
A point of a parametric surface which is not regular is
1142:), a further condition is required, generally that, for
4109:
3145:
be the smallest field containing the coefficients, and
1399:
where the above
Jacobian matrix has rank two is called
3608:. An important role in their study has been played by
2650:
The tangent plane is defined by its implicit equation
4016:
3244:
3184:
3021:
2952:
2659:
2431:
2361:
2243:
2110:
2097:) of dimension 3 is the set of the common zeros of a
1824:
1672:
1511:
1471:
1327:
1296:
1263:
1232:
1199:
1168:
1158:
989:
946:
898:
426:
309:
297:
is an algebraic surface, as it may be defined by the
2415:
937:, and this will be always the case in this article.
281:
of three variables is a surface, which is called an
3228:If the polynomial has real coefficients, the field
2342:
1491:: it suffices to permute the role of the different
1403:, or, more properly, the parametrization is called
60:. Unsourced material may be challenged and removed.
3551:with various additional structures, most often, a
3259:
3217:
3057:
2967:
2926:
2639:
2406:
2291:
2143:
2057:
1749:
1603:
1390:
1365:
1123:
961:
913:
678:is the locus of a curve rotating around a line. A
554:
360:
27:Mathematical idealization of the surface of a body
3830:, the area of a differential element of a surface
3092:
1464:of surfaces, in the neighborhood of a point, see
201:, a surface may cross itself (and may have other
4169:
3990:
3600:, and sometimes appear in parametric form or as
2075:, but most algebraic surfaces are not rational.
1639:
1432:of the Jacobian matrix. The tangent plane is an
3697:behavior that mimics the appearance of natural
3461:Every topological surface is homeomorphic to a
3345:
1480:. There are several kinds of irregular points.
3314:of dimension three is the set of points whose
2331:. In other words, the implicit surface is the
2219:, then there exists a differentiable function
285:. If the defining three-variate function is a
3782:An open surface with u- and v-flow lines and
3238:, and a point of the surface that belongs to
1457:is a vector which is parallel to the normal.
969:is given by three functions of two variables
603:: in the context of manifolds, typically in
3369:is the set of the common zeros of at least
1428:and having a direction parallel to the two
804:and a differentiable surface. It is also a
719:
4069:by Fereydoon Family and Tamas Vicsek 1991
651:or points where a surface crosses itself.
3507:
3402:polynomials define a surface, which is a
3247:
2955:
2047:
1782:is a surface that may be parametrized by
1739:
1593:
1134:As the image of such a function may be a
1113:
949:
901:
544:
387:). In this case, one says that one has a
120:Learn how and when to remove this message
3777:
3689:or fractal surface is generated using a
3669:
3520:
3113:coefficients. Therefore, the concept of
705:
131:
3764:
744:
269:of its points. This is the case of the
14:
4170:
4115:
3434:, a surface is generally defined as a
3419:
3358:. In fact, an algebraic surface is an
2416:§ Tangent plane and normal vector
2292:{\displaystyle f(x,y,\varphi (x,y))=0}
940:Specifically, a parametric surface in
888:is the image of an open subset of the
658:, a surface is generally defined as a
361:{\displaystyle x^{2}+y^{2}+z^{2}-1=0.}
4153:
3996:
3395:. If there is only one component the
3301:
3097:Polynomials with coefficients in any
2418:. The direction of the normal is the
873:
662:of a point or a line. For example, a
576:). For the remaining two points (the
371:A surface may also be defined as the
3942:
3384:generated by the others. Generally,
3175:which is a solution of the equation
3001:
1424:is the unique plane passing through
58:adding citations to reliable sources
29:
3772:Computer representation of surfaces
2407:{\displaystyle (x_{0},y_{0},z_{0})}
2078:
1767:
1614:The apex of the cone is the origin
735:. A surface that is contained in a
24:
4122:. Simon and Schuster. p. 84.
3738:fractal behavior of such a surface
3656:
3360:algebraic variety of dimension two
2845:
2837:
2758:
2750:
2671:
2663:
2578:
2570:
2513:
2505:
2448:
2440:
1472:Irregular point and singular point
1345:
1330:
1314:
1299:
1281:
1266:
1250:
1235:
1217:
1202:
1186:
1171:
25:
4199:
3949:Foundations of Algebraic Geometry
3538:differential geometry of surfaces
3515:Differential geometry of surfaces
3356:dimension of an algebraic variety
2937:
1466:Differential geometry of surfaces
768:of two variables, defined over a
277:of two variables. The set of the
174:, but, unlike a plane, it may be
4010:"The Fractal Geometry of Nature"
3813:(lines and curves) and solids.
3770:This section is an excerpt from
3681:to create a mountainous terrain.
3662:This section is an excerpt from
3513:This section is an excerpt from
3260:{\displaystyle \mathbb {R} ^{3}}
3169:of the surface is an element of
2968:{\displaystyle \mathbb {R} ^{3}}
2343:Regular points and tangent plane
2186:, and the partial derivative in
1498:On the other hand, consider the
962:{\displaystyle \mathbb {R} ^{3}}
914:{\displaystyle \mathbb {R} ^{2}}
34:
4082:
4060:
3753:Star Trek II: The Wrath of Khan
1391:Tangent plane and normal vector
637:Surface (differential geometry)
395:by these two variables, called
261:Often, a surface is defined by
45:needs additional citations for
4163:, Princeton University Library
4038:
4002:
3977:
3936:
3923:
3904:
3891:
3846:, a two-dimensional equivalent
3693:algorithm designed to produce
3586:embedding in Euclidean space.
3206:
3188:
3153:algebraically closed extension
3093:Surfaces over arbitrary fields
3043:
3025:
2915:
2896:
2893:
2854:
2828:
2809:
2806:
2767:
2741:
2722:
2719:
2680:
2626:
2587:
2561:
2522:
2496:
2457:
2401:
2362:
2280:
2277:
2265:
2247:
2132:
2114:
2041:
2029:
2014:
2002:
1966:
1954:
1939:
1927:
1891:
1879:
1864:
1852:
1786:of two variables. That is, if
1736:
1724:
1573:
1567:
1541:
1535:
1146:values of the parameters, the
1110:
1098:
1068:
1056:
1026:
1014:
689:
541:
535:
512:
506:
497:
491:
468:
462:
453:
447:
256:
170:. It is a generalization of a
13:
1:
4047:by Christopher W. Tyler 2002
3954:American Mathematical Society
3789:In technical applications of
1640:Graph of a bivariate function
720:§ Differentiable surface
4067:Dynamics of Fractal Surfaces
3885:
3803:computer-aided manufacturing
3588:Surfaces naturally arise as
3578:, first studied in depth by
3481:) is the starting object of
3442:such that every point has a
3346:In higher dimensional spaces
3267:(a usual point) is called a
615:such that every point has a
7:
3821:
3218:{\displaystyle f(x,y,z)=0.}
3058:{\displaystyle f(x,y,z)=0,}
2946:of an implicit surface (in
2144:{\displaystyle f(x,y,z)=0.}
2093:(or, more generally, in an
935:continuously differentiable
848:(the graph of the function
754:
166:of the common concept of a
10:
4204:
4147:
4090:The languages of the brain
3769:
3723:fractional Brownian motion
3716:. Moreover, variations in
3661:
3512:
3423:
3271:. A point that belongs to
3078:, with real coefficients.
3005:
2082:
1771:
877:
729:of dimension 3, typically
706:§ Topological surface
647:, such as the vertex of a
265:that are satisfied by the
140:is the surface of a solid
69:"Surface" mathematics
3748:computer generated images
3074:is a polynomial in three
2156:implicit function theorem
2089:An implicit surface in a
2071:A rational surface is an
745:§ Projective surface
178:; this is analogous to a
3916:
3860:Signed distance function
3638:Euler–Lagrange equations
1495:for changing the poles.
714:is a surfaces that is a
3712:that can be modeled by
3316:homogeneous coordinates
2099:differentiable function
2068:is a rational surface.
1502:of parametric equation
1462:differential invariants
716:differentiable manifold
700:is a surface that is a
3901:to some open set in E.
3787:
3725:was first proposed by
3701:. In other words, the
3682:
3642:calculus of variations
3612:(in the spirit of the
3529:
3508:Differentiable surface
3393:irreducible components
3320:homogeneous polynomial
3318:are zeros of a single
3261:
3219:
3059:
2994:. A singular point is
2969:
2928:
2641:
2640:{\displaystyle \left.}
2408:
2293:
2145:
2059:
1751:
1618:, and is obtained for
1605:
1440:. In other words, any
1367:
1125:
963:
915:
791:differentiable surface
711:differentiable surface
704:of dimension two (see
556:
362:
151:
4155:Gauss, Carl Friedrich
4116:Briggs, John (1992).
4025:by Tat-Jen Cham 2007
3799:computer-aided design
3781:
3736:and even the overall
3673:
3542:differential geometry
3524:
3404:complete intersection
3336:projective completion
3262:
3220:
3060:
2970:
2929:
2642:
2422:, that is the vector
2409:
2351:is nonzero is called
2294:
2146:
2060:
1752:
1606:
1442:affine transformation
1368:
1126:
964:
916:
867:two-sheet hyperboloid
846:hyperbolic paraboloid
810:surface of revolution
785:. If the function is
676:surface of revolution
609:differential geometry
557:
363:
211:differential geometry
135:
3791:3D computer graphics
3765:In computer graphics
3580:Carl Friedrich Gauss
3562:, relating to their
3526:Carl Friedrich Gauss
3242:
3182:
3163:transcendence degree
3019:
2950:
2657:
2429:
2359:
2241:
2108:
1822:
1670:
1634:self-crossing points
1509:
1156:
987:
944:
896:
592:, and the longitude
424:
307:
289:, the surface is an
54:improve this article
4094:Albert M. Galaburda
3834:Coordinate surfaces
3420:Topological surface
3120:Given a polynomial
2333:graph of a function
2101:of three variables
1420:at a regular point
923:continuous function
783:topological surface
766:continuous function
697:topological surface
381:continuous function
375:, in some space of
293:. For example, the
279:zeros of a function
275:continuous function
3956:, pp. 1–363,
3850:Polyhedral surface
3788:
3683:
3576:Gaussian curvature
3530:
3502:Surface (topology)
3483:algebraic topology
3463:polyhedral surface
3426:Surface (topology)
3324:projective variety
3308:projective surface
3302:Projective surface
3257:
3215:
3055:
2992:singularity theory
2965:
2924:
2637:
2404:
2355:. At such a point
2289:
2141:
2055:
2053:
1784:rational functions
1747:
1745:
1601:
1599:
1363:
1357:
1353:
1322:
1289:
1258:
1225:
1194:
1121:
1119:
959:
911:
886:parametric surface
880:Parametric surface
874:Parametric surface
741:projective surface
656:classical geometry
633:Surface (topology)
552:
550:
389:parametric surface
358:
226:degrees of freedom
199:algebraic geometry
164:mathematical model
152:
18:Surface (geometry)
3931:implicit equation
3727:Benoit Mandelbrot
3687:fractal landscape
3664:Fractal landscape
3553:Riemannian metric
3440:topological space
3352:algebraic variety
3087:projective spaces
3008:Algebraic surface
3002:Algebraic surface
2852:
2765:
2678:
2585:
2520:
2455:
2073:algebraic surface
2045:
1970:
1895:
1352:
1321:
1288:
1257:
1224:
1193:
927:topological space
837:The surface of a
817:circular cylinder
802:algebraic surface
789:, the graph is a
641:abstract surfaces
613:topological space
379:at least 3, of a
299:implicit equation
291:algebraic surface
239:coordinate system
218:topological space
195:Euclidean 3-space
130:
129:
122:
104:
16:(Redirected from
4195:
4164:
4141:
4140:
4138:
4136:
4113:
4107:
4086:
4080:
4064:
4058:
4042:
4036:
4020:
4014:
4013:
4006:
4000:
3994:
3988:
3981:
3975:
3974:
3940:
3934:
3927:
3911:
3908:
3902:
3895:
3880:Surface integral
3786:-contours shown.
3714:fractal surfaces
3650:minimal surfaces
3630:hyperbolic plane
3614:Erlangen program
3401:
3390:
3375:
3368:
3312:projective space
3296:rational numbers
3294:is the field of
3293:
3282:
3276:
3266:
3264:
3263:
3258:
3256:
3255:
3250:
3233:
3224:
3222:
3221:
3216:
3174:
3160:
3150:
3144:
3138:
3073:
3064:
3062:
3061:
3056:
2974:
2972:
2971:
2966:
2964:
2963:
2958:
2933:
2931:
2930:
2925:
2914:
2913:
2892:
2891:
2879:
2878:
2866:
2865:
2853:
2851:
2843:
2835:
2827:
2826:
2805:
2804:
2792:
2791:
2779:
2778:
2766:
2764:
2756:
2748:
2740:
2739:
2718:
2717:
2705:
2704:
2692:
2691:
2679:
2677:
2669:
2661:
2646:
2644:
2643:
2638:
2633:
2629:
2625:
2624:
2612:
2611:
2599:
2598:
2586:
2584:
2576:
2568:
2560:
2559:
2547:
2546:
2534:
2533:
2521:
2519:
2511:
2503:
2495:
2494:
2482:
2481:
2469:
2468:
2456:
2454:
2446:
2438:
2413:
2411:
2410:
2405:
2400:
2399:
2387:
2386:
2374:
2373:
2350:
2338:
2330:
2298:
2296:
2295:
2290:
2233:
2218:
2193:
2189:
2185:
2150:
2148:
2147:
2142:
2085:Implicit surface
2079:Implicit surface
2064:
2062:
2061:
2056:
2054:
2046:
2044:
2028:
2027:
2017:
2001:
2000:
1990:
1971:
1969:
1953:
1952:
1942:
1926:
1925:
1915:
1896:
1894:
1878:
1877:
1867:
1851:
1850:
1840:
1810:
1803:
1780:rational surface
1774:Rational surface
1768:Rational surface
1756:
1754:
1753:
1748:
1746:
1662:
1624:
1617:
1610:
1608:
1607:
1602:
1600:
1427:
1423:
1410:
1398:
1372:
1370:
1369:
1364:
1362:
1361:
1354:
1351:
1343:
1342:
1341:
1328:
1323:
1320:
1312:
1311:
1310:
1297:
1290:
1287:
1279:
1278:
1277:
1264:
1259:
1256:
1248:
1247:
1246:
1233:
1226:
1223:
1215:
1214:
1213:
1200:
1195:
1192:
1184:
1183:
1182:
1169:
1141:
1130:
1128:
1127:
1122:
1120:
1097:
1096:
1055:
1054:
1013:
1012:
976:
972:
968:
966:
965:
960:
958:
957:
952:
920:
918:
917:
912:
910:
909:
904:
857:
780:
749:abstract surface
737:projective space
734:
597:
591:
575:
574:
561:
559:
558:
553:
551:
416:
409:
367:
365:
364:
359:
345:
344:
332:
331:
319:
318:
283:implicit surface
231:coordinate patch
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
4203:
4202:
4198:
4197:
4196:
4194:
4193:
4192:
4168:
4167:
4150:
4145:
4144:
4134:
4132:
4130:
4114:
4110:
4087:
4083:
4065:
4061:
4043:
4039:
4021:
4017:
4008:
4007:
4003:
3995:
3991:
3982:
3978:
3964:
3941:
3937:
3928:
3924:
3919:
3914:
3909:
3905:
3896:
3892:
3888:
3824:
3819:
3818:
3775:
3767:
3762:
3761:
3758:Loren Carpenter
3718:surface texture
3710:self-similarity
3667:
3659:
3657:Fractal surface
3654:
3653:
3622:Euclidean plane
3618:symmetry groups
3568:Euclidean space
3540:deals with the
3518:
3510:
3495:homology groups
3456:Euclidean plane
3428:
3422:
3396:
3385:
3370:
3366:
3348:
3304:
3289:
3280:
3272:
3251:
3246:
3245:
3243:
3240:
3239:
3229:
3183:
3180:
3179:
3170:
3156:
3146:
3140:
3121:
3095:
3069:
3020:
3017:
3016:
3010:
3004:
2959:
2954:
2953:
2951:
2948:
2947:
2940:
2909:
2905:
2887:
2883:
2874:
2870:
2861:
2857:
2844:
2836:
2834:
2822:
2818:
2800:
2796:
2787:
2783:
2774:
2770:
2757:
2749:
2747:
2735:
2731:
2713:
2709:
2700:
2696:
2687:
2683:
2670:
2662:
2660:
2658:
2655:
2654:
2620:
2616:
2607:
2603:
2594:
2590:
2577:
2569:
2567:
2555:
2551:
2542:
2538:
2529:
2525:
2512:
2504:
2502:
2490:
2486:
2477:
2473:
2464:
2460:
2447:
2439:
2437:
2436:
2432:
2430:
2427:
2426:
2395:
2391:
2382:
2378:
2369:
2365:
2360:
2357:
2356:
2348:
2345:
2336:
2328:
2321:
2314:
2307:
2242:
2239:
2238:
2220:
2216:
2209:
2202:
2195:
2194:is not zero at
2191:
2187:
2183:
2176:
2169:
2159:
2109:
2106:
2105:
2091:Euclidean space
2087:
2081:
2052:
2051:
2023:
2019:
2018:
1996:
1992:
1991:
1989:
1982:
1976:
1975:
1948:
1944:
1943:
1921:
1917:
1916:
1914:
1907:
1901:
1900:
1873:
1869:
1868:
1846:
1842:
1841:
1839:
1832:
1825:
1823:
1820:
1819:
1805:
1792:
1787:
1776:
1770:
1762:identity matrix
1744:
1743:
1714:
1708:
1707:
1697:
1691:
1690:
1680:
1673:
1671:
1668:
1667:
1645:
1642:
1619:
1615:
1598:
1597:
1583:
1577:
1576:
1551:
1545:
1544:
1519:
1512:
1510:
1507:
1506:
1493:coordinate axes
1474:
1425:
1421:
1408:
1396:
1393:
1356:
1355:
1344:
1337:
1333:
1329:
1326:
1324:
1313:
1306:
1302:
1298:
1295:
1292:
1291:
1280:
1273:
1269:
1265:
1262:
1260:
1249:
1242:
1238:
1234:
1231:
1228:
1227:
1216:
1209:
1205:
1201:
1198:
1196:
1185:
1178:
1174:
1170:
1167:
1160:
1159:
1157:
1154:
1153:
1148:Jacobian matrix
1139:
1118:
1117:
1092:
1088:
1081:
1075:
1074:
1050:
1046:
1039:
1033:
1032:
1008:
1004:
997:
990:
988:
985:
984:
974:
970:
953:
948:
947:
945:
942:
941:
931:Euclidean space
905:
900:
899:
897:
894:
893:
890:Euclidean plane
882:
876:
849:
776:
757:
730:
727:Euclidean space
692:
668:conical surface
649:conical surface
629:Euclidean plane
593:
585:
572:
570:
549:
548:
522:
516:
515:
478:
472:
471:
434:
427:
425:
422:
421:
414:
407:
340:
336:
327:
323:
314:
310:
308:
305:
304:
259:
236:two-dimensional
216:A surface is a
182:generalizing a
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
4201:
4191:
4190:
4185:
4180:
4166:
4165:
4149:
4146:
4143:
4142:
4129:978-0671742171
4128:
4108:
4104:pages 351–359
4081:
4059:
4055:pages 173–177
4037:
4015:
4001:
3989:
3976:
3962:
3935:
3921:
3920:
3918:
3915:
3913:
3912:
3903:
3889:
3887:
3884:
3883:
3882:
3877:
3872:
3867:
3862:
3857:
3852:
3847:
3841:
3836:
3831:
3823:
3820:
3776:
3768:
3766:
3763:
3668:
3660:
3658:
3655:
3616:), namely the
3604:associated to
3519:
3511:
3509:
3506:
3489:, such as the
3465:such that all
3424:Main article:
3421:
3418:
3347:
3344:
3303:
3300:
3286:rational point
3284:, or simply a
3279:rational over
3254:
3249:
3226:
3225:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3161:, of infinite
3094:
3091:
3076:indeterminates
3066:
3065:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3006:Main article:
3003:
3000:
2962:
2957:
2944:singular point
2939:
2938:Singular point
2936:
2935:
2934:
2923:
2920:
2917:
2912:
2908:
2904:
2901:
2898:
2895:
2890:
2886:
2882:
2877:
2873:
2869:
2864:
2860:
2856:
2850:
2847:
2842:
2839:
2833:
2830:
2825:
2821:
2817:
2814:
2811:
2808:
2803:
2799:
2795:
2790:
2786:
2782:
2777:
2773:
2769:
2763:
2760:
2755:
2752:
2746:
2743:
2738:
2734:
2730:
2727:
2724:
2721:
2716:
2712:
2708:
2703:
2699:
2695:
2690:
2686:
2682:
2676:
2673:
2668:
2665:
2648:
2647:
2636:
2632:
2628:
2623:
2619:
2615:
2610:
2606:
2602:
2597:
2593:
2589:
2583:
2580:
2575:
2572:
2566:
2563:
2558:
2554:
2550:
2545:
2541:
2537:
2532:
2528:
2524:
2518:
2515:
2510:
2507:
2501:
2498:
2493:
2489:
2485:
2480:
2476:
2472:
2467:
2463:
2459:
2453:
2450:
2445:
2442:
2435:
2403:
2398:
2394:
2390:
2385:
2381:
2377:
2372:
2368:
2364:
2344:
2341:
2326:
2319:
2312:
2300:
2299:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2214:
2207:
2200:
2181:
2174:
2167:
2152:
2151:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2083:Main article:
2080:
2077:
2066:
2065:
2050:
2043:
2040:
2037:
2034:
2031:
2026:
2022:
2016:
2013:
2010:
2007:
2004:
1999:
1995:
1988:
1985:
1983:
1981:
1978:
1977:
1974:
1968:
1965:
1962:
1959:
1956:
1951:
1947:
1941:
1938:
1935:
1932:
1929:
1924:
1920:
1913:
1910:
1908:
1906:
1903:
1902:
1899:
1893:
1890:
1887:
1884:
1881:
1876:
1872:
1866:
1863:
1860:
1857:
1854:
1849:
1845:
1838:
1835:
1833:
1831:
1828:
1827:
1790:
1772:Main article:
1769:
1766:
1758:
1757:
1742:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1715:
1713:
1710:
1709:
1706:
1703:
1700:
1698:
1696:
1693:
1692:
1689:
1686:
1683:
1681:
1679:
1676:
1675:
1641:
1638:
1612:
1611:
1596:
1592:
1589:
1586:
1584:
1582:
1579:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1552:
1550:
1547:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1520:
1518:
1515:
1514:
1473:
1470:
1434:affine concept
1392:
1389:
1374:
1373:
1360:
1350:
1347:
1340:
1336:
1332:
1325:
1319:
1316:
1309:
1305:
1301:
1294:
1293:
1286:
1283:
1276:
1272:
1268:
1261:
1255:
1252:
1245:
1241:
1237:
1230:
1229:
1222:
1219:
1212:
1208:
1204:
1197:
1191:
1188:
1181:
1177:
1173:
1166:
1165:
1163:
1132:
1131:
1116:
1112:
1109:
1106:
1103:
1100:
1095:
1091:
1087:
1084:
1082:
1080:
1077:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1053:
1049:
1045:
1042:
1040:
1038:
1035:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1011:
1007:
1003:
1000:
998:
996:
993:
992:
956:
951:
929:, generally a
908:
903:
878:Main article:
875:
872:
871:
870:
863:
842:
835:
824:
819:(that is, the
813:
794:
787:differentiable
756:
753:
691:
688:
563:
562:
547:
543:
540:
537:
534:
531:
528:
525:
523:
521:
518:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
484:
481:
479:
477:
474:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
435:
433:
430:
429:
403:, also called
369:
368:
357:
354:
351:
348:
343:
339:
335:
330:
326:
322:
317:
313:
258:
255:
251:180th meridian
213:, it may not.
144:, here having
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
4200:
4189:
4186:
4184:
4181:
4179:
4176:
4175:
4173:
4162:
4161:
4156:
4152:
4151:
4131:
4125:
4121:
4120:
4112:
4106:
4103:
4102:0-674-00772-7
4099:
4095:
4091:
4085:
4079:
4076:
4075:981-02-0720-4
4072:
4068:
4063:
4057:
4054:
4053:0-8058-4395-7
4050:
4046:
4041:
4035:
4032:
4031:3-540-69428-5
4028:
4024:
4019:
4011:
4005:
3998:
3993:
3986:
3985:generic point
3980:
3973:
3969:
3965:
3963:9780821874622
3959:
3955:
3951:
3950:
3945:
3939:
3932:
3926:
3922:
3907:
3900:
3899:diffeomorphic
3894:
3890:
3881:
3878:
3876:
3875:Surface patch
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3851:
3848:
3845:
3842:
3840:
3837:
3835:
3832:
3829:
3826:
3825:
3816:
3812:
3808:
3804:
3800:
3796:
3792:
3785:
3780:
3773:
3759:
3755:
3754:
3749:
3745:
3744:R. R. Shearer
3742:According to
3741:
3739:
3735:
3730:
3728:
3724:
3719:
3715:
3711:
3706:
3704:
3700:
3696:
3692:
3688:
3680:
3677:
3672:
3665:
3651:
3647:
3643:
3639:
3635:
3631:
3627:
3623:
3619:
3615:
3611:
3607:
3603:
3599:
3596:of a pair of
3595:
3591:
3587:
3585:
3581:
3577:
3573:
3572:intrinsically
3569:
3565:
3561:
3560:extrinsically
3556:
3554:
3550:
3547:
3543:
3539:
3535:
3527:
3523:
3516:
3505:
3503:
3498:
3496:
3492:
3488:
3484:
3480:
3476:
3475:combinatorial
3472:
3468:
3464:
3459:
3457:
3453:
3449:
3445:
3441:
3437:
3433:
3427:
3417:
3415:
3410:
3407:
3405:
3399:
3394:
3388:
3383:
3379:
3378:algebraic set
3373:
3363:
3361:
3357:
3353:
3343:
3341:
3337:
3331:
3329:
3325:
3321:
3317:
3313:
3309:
3299:
3297:
3292:
3287:
3283:
3275:
3270:
3252:
3237:
3236:complex field
3232:
3212:
3209:
3203:
3200:
3197:
3194:
3191:
3185:
3178:
3177:
3176:
3173:
3168:
3164:
3159:
3154:
3149:
3143:
3136:
3132:
3128:
3124:
3118:
3116:
3112:
3108:
3104:
3100:
3090:
3088:
3084:
3079:
3077:
3072:
3052:
3049:
3046:
3040:
3037:
3034:
3031:
3028:
3022:
3015:
3014:
3013:
3009:
2999:
2997:
2993:
2988:
2986:
2982:
2978:
2960:
2945:
2921:
2918:
2910:
2906:
2902:
2899:
2888:
2884:
2880:
2875:
2871:
2867:
2862:
2858:
2848:
2840:
2831:
2823:
2819:
2815:
2812:
2801:
2797:
2793:
2788:
2784:
2780:
2775:
2771:
2761:
2753:
2744:
2736:
2732:
2728:
2725:
2714:
2710:
2706:
2701:
2697:
2693:
2688:
2684:
2674:
2666:
2653:
2652:
2651:
2634:
2630:
2621:
2617:
2613:
2608:
2604:
2600:
2595:
2591:
2581:
2573:
2564:
2556:
2552:
2548:
2543:
2539:
2535:
2530:
2526:
2516:
2508:
2499:
2491:
2487:
2483:
2478:
2474:
2470:
2465:
2461:
2451:
2443:
2433:
2425:
2424:
2423:
2421:
2417:
2396:
2392:
2388:
2383:
2379:
2375:
2370:
2366:
2354:
2340:
2334:
2325:
2318:
2311:
2305:
2304:neighbourhood
2286:
2283:
2274:
2271:
2268:
2262:
2259:
2256:
2253:
2250:
2244:
2237:
2236:
2235:
2231:
2227:
2223:
2213:
2206:
2199:
2180:
2173:
2166:
2162:
2157:
2138:
2135:
2129:
2126:
2123:
2120:
2117:
2111:
2104:
2103:
2102:
2100:
2096:
2092:
2086:
2076:
2074:
2069:
2048:
2038:
2035:
2032:
2024:
2020:
2011:
2008:
2005:
1997:
1993:
1986:
1984:
1979:
1972:
1963:
1960:
1957:
1949:
1945:
1936:
1933:
1930:
1922:
1918:
1911:
1909:
1904:
1897:
1888:
1885:
1882:
1874:
1870:
1861:
1858:
1855:
1847:
1843:
1836:
1834:
1829:
1818:
1817:
1816:
1814:
1808:
1801:
1797:
1793:
1785:
1781:
1775:
1765:
1764:of rank two.
1763:
1740:
1733:
1730:
1727:
1721:
1718:
1716:
1711:
1704:
1701:
1699:
1694:
1687:
1684:
1682:
1677:
1666:
1665:
1664:
1660:
1656:
1652:
1648:
1637:
1635:
1630:
1628:
1622:
1594:
1590:
1587:
1585:
1580:
1570:
1564:
1561:
1558:
1555:
1553:
1548:
1538:
1532:
1529:
1526:
1523:
1521:
1516:
1505:
1504:
1503:
1501:
1500:circular cone
1496:
1494:
1490:
1486:
1481:
1479:
1469:
1467:
1463:
1458:
1456:
1455:normal vector
1452:
1451:
1445:
1443:
1439:
1435:
1431:
1419:
1418:
1417:tangent plane
1412:
1406:
1402:
1388:
1386:
1383:
1379:
1358:
1348:
1338:
1334:
1317:
1307:
1303:
1284:
1274:
1270:
1253:
1243:
1239:
1220:
1210:
1206:
1189:
1179:
1175:
1161:
1152:
1151:
1150:
1149:
1145:
1137:
1114:
1107:
1104:
1101:
1093:
1089:
1085:
1083:
1078:
1071:
1065:
1062:
1059:
1051:
1047:
1043:
1041:
1036:
1029:
1023:
1020:
1017:
1009:
1005:
1001:
999:
994:
983:
982:
981:
980:
954:
938:
936:
932:
928:
924:
906:
891:
887:
881:
868:
864:
861:
856:
852:
847:
843:
840:
836:
833:
829:
828:circular cone
825:
822:
818:
814:
811:
807:
806:ruled surface
803:
799:
795:
792:
788:
784:
779:
774:
771:
767:
763:
759:
758:
752:
750:
746:
742:
738:
733:
728:
723:
721:
717:
713:
712:
707:
703:
699:
698:
687:
685:
681:
680:ruled surface
677:
673:
669:
665:
661:
657:
652:
650:
646:
645:singularities
642:
638:
634:
630:
626:
622:
618:
614:
610:
606:
602:
596:
589:
583:
579:
569:
545:
538:
532:
529:
526:
524:
519:
509:
503:
500:
494:
488:
485:
482:
480:
475:
465:
459:
456:
450:
444:
441:
438:
436:
431:
420:
419:
418:
413:
406:
402:
398:
394:
390:
386:
382:
378:
374:
355:
352:
349:
346:
341:
337:
333:
328:
324:
320:
315:
311:
303:
302:
301:
300:
296:
292:
288:
284:
280:
276:
272:
268:
264:
254:
252:
248:
244:
240:
237:
233:
232:
227:
223:
219:
214:
212:
208:
205:), while, in
204:
203:singularities
200:
196:
192:
187:
185:
184:straight line
181:
177:
173:
169:
165:
161:
157:
150:
147:
143:
139:
134:
124:
121:
113:
110:February 2022
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
4159:
4133:. Retrieved
4118:
4111:
4089:
4084:
4066:
4062:
4044:
4040:
4022:
4018:
4004:
3992:
3979:
3948:
3938:
3925:
3906:
3893:
3870:Surface area
3865:Solid figure
3839:Hypersurface
3828:Area element
3815:Point clouds
3783:
3751:
3734:stationarity
3731:
3707:
3684:
3606:space curves
3571:
3559:
3557:
3531:
3499:
3460:
3448:homeomorphic
3444:neighborhood
3429:
3411:
3408:
3397:
3386:
3371:
3364:
3359:
3349:
3340:homogenizing
3335:
3332:
3307:
3305:
3290:
3285:
3278:
3273:
3268:
3230:
3227:
3171:
3166:
3157:
3147:
3141:
3134:
3130:
3126:
3122:
3119:
3114:
3096:
3080:
3070:
3067:
3011:
2989:
2985:non-singular
2984:
2980:
2943:
2941:
2649:
2352:
2346:
2323:
2316:
2309:
2301:
2229:
2225:
2221:
2211:
2204:
2197:
2178:
2171:
2164:
2160:
2153:
2095:affine space
2088:
2070:
2067:
1809:= 0, 1, 2, 3
1806:
1799:
1795:
1788:
1779:
1777:
1759:
1658:
1654:
1650:
1646:
1643:
1633:
1631:
1626:
1620:
1613:
1497:
1489:Euler angles
1482:
1477:
1475:
1459:
1454:
1448:
1446:
1415:
1413:
1404:
1400:
1394:
1375:
1133:
978:
939:
885:
883:
860:architecture
854:
850:
831:
790:
782:
777:
748:
739:is called a
731:
724:
709:
695:
693:
653:
640:
621:homeomorphic
617:neighborhood
594:
587:
564:
401:Euler angles
396:
393:parametrized
392:
370:
260:
229:
215:
188:
159:
153:
148:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
3944:Weil, André
3634:connections
3534:mathematics
3452:open subset
3414:irreducible
3354:and of the
1813:polynomials
1485:unit sphere
1450:normal line
1430:row vectors
1385:open subset
892:(typically
800:is both an
773:open subset
690:Terminology
625:open subset
584:), one has
582:south poles
391:, which is
295:unit sphere
267:coordinates
257:Definitions
234:on which a
156:mathematics
4172:Categories
3997:Gauss 1902
3797:) such as
3691:stochastic
3676:triangular
3610:Lie groups
3487:invariants
3277:is called
3269:real point
2234:such that
1460:For other
1144:almost all
979:parameters
839:polyhedron
686:of lines.
397:parameters
287:polynomial
80:newspapers
3886:Footnotes
3844:Perimeter
3811:wireframe
3646:geodesics
3598:variables
3594:functions
3584:isometric
3564:embedding
3479:simplexes
3471:triangles
3328:dimension
3165:. Then a
2903:−
2846:∂
2838:∂
2816:−
2759:∂
2751:∂
2729:−
2672:∂
2664:∂
2579:∂
2571:∂
2514:∂
2506:∂
2449:∂
2441:∂
2263:φ
1804:are, for
1616:(0, 0, 0)
1565:
1533:
1478:irregular
1346:∂
1331:∂
1315:∂
1300:∂
1282:∂
1267:∂
1251:∂
1236:∂
1218:∂
1203:∂
1187:∂
1172:∂
977:, called
770:connected
619:which is
533:
504:
489:
460:
445:
405:longitude
377:dimension
347:−
263:equations
247:longitude
222:dimension
4188:Surfaces
4183:Topology
4178:Geometry
4157:(1902),
4077:page 45
3946:(1946),
3822:See also
3807:surfaces
3679:fractals
3628:and the
3549:surfaces
3446:that is
3436:manifold
3432:topology
3103:rational
2996:isolated
2420:gradient
1627:singular
1395:A point
755:Examples
702:manifold
605:topology
601:manifold
412:latitude
243:latitude
207:topology
4148:Sources
4135:15 June
3972:0023093
3703:surface
3699:terrain
3695:fractal
3674:Use of
3640:in the
3620:of the
3528:in 1828
3234:is the
3111:complex
2981:regular
2353:regular
1405:regular
1401:regular
925:, in a
921:) by a
627:of the
193:in the
191:spheres
168:surface
160:surface
94:scholar
4126:
4100:
4073:
4051:
4029:
3970:
3960:
3626:sphere
3624:, the
3590:graphs
3546:smooth
3536:, the
3473:. The
3467:facets
3450:to an
3151:be an
3139:, let
3083:fields
3068:where
2977:system
1438:metric
808:and a
664:sphere
623:to an
568:modulo
176:curved
146:radius
138:sphere
96:
89:
82:
75:
67:
4096:2002
4033:page
3917:Notes
3855:Shape
3491:genus
3454:of a
3382:ideal
3338:) by
3330:two.
3310:in a
3288:, if
3167:point
3115:point
3099:field
2302:in a
2184:) = 0
2158:: if
1382:dense
1136:curve
821:locus
798:plane
781:is a
764:of a
762:graph
743:(see
718:(see
708:). A
684:union
672:curve
660:locus
631:(see
578:north
385:curve
373:image
273:of a
271:graph
180:curve
172:plane
162:is a
101:JSTOR
87:books
4137:2014
4124:ISBN
4098:ISBN
4071:ISBN
4049:ISBN
4027:ISBN
3958:ISBN
3801:and
3602:loci
3570:and
3493:and
3469:are
3107:real
1644:Let
1447:The
1414:The
1378:rank
1376:has
973:and
832:apex
760:The
674:; a
635:and
607:and
586:cos
580:and
417:by
410:and
245:and
209:and
158:, a
142:ball
73:news
4092:by
3795:CAx
3756:.
3592:of
3566:in
3544:of
3532:In
3504:).
3430:In
3400:– 2
3389:– 2
3374:– 2
3326:of
3155:of
3109:or
2983:or
2306:of
2190:of
1623:= 0
1562:sin
1530:cos
1487:by
1407:at
775:of
654:In
590:= 0
530:sin
501:cos
486:sin
457:cos
442:cos
253:).
220:of
186:.
154:In
56:by
4174::
3968:MR
3966:,
3933:".
3805:,
3729:.
3685:A
3555:.
3497:.
3458:.
3362:.
3306:A
3298:.
3213:0.
3133:,
3129:,
2987:.
2942:A
2922:0.
2322:,
2315:,
2228:,
2210:,
2203:,
2177:,
2170:,
2139:0.
1811:,
1798:,
1778:A
1657:,
1649:=
1629:.
1468:.
1411:.
884:A
865:A
855:xy
853:=
844:A
826:A
815:A
796:A
751:.
356:0.
136:A
4139:.
4012:.
3999:.
3987:.
3793:(
3784:Z
3774:.
3666:.
3517:.
3398:n
3387:n
3372:n
3367:n
3291:k
3281:k
3274:k
3253:3
3248:R
3231:K
3210:=
3207:)
3204:z
3201:,
3198:y
3195:,
3192:x
3189:(
3186:f
3172:K
3158:k
3148:K
3142:k
3137:)
3135:z
3131:y
3127:x
3125:(
3123:f
3071:f
3053:,
3050:0
3047:=
3044:)
3041:z
3038:,
3035:y
3032:,
3029:x
3026:(
3023:f
2961:3
2956:R
2919:=
2916:)
2911:0
2907:z
2900:z
2897:(
2894:)
2889:0
2885:z
2881:,
2876:0
2872:y
2868:,
2863:0
2859:x
2855:(
2849:z
2841:f
2832:+
2829:)
2824:0
2820:y
2813:y
2810:(
2807:)
2802:0
2798:z
2794:,
2789:0
2785:y
2781:,
2776:0
2772:x
2768:(
2762:y
2754:f
2745:+
2742:)
2737:0
2733:x
2726:x
2723:(
2720:)
2715:0
2711:z
2707:,
2702:0
2698:y
2694:,
2689:0
2685:x
2681:(
2675:x
2667:f
2635:.
2631:]
2627:)
2622:0
2618:z
2614:,
2609:0
2605:y
2601:,
2596:0
2592:x
2588:(
2582:z
2574:f
2565:,
2562:)
2557:0
2553:z
2549:,
2544:0
2540:y
2536:,
2531:0
2527:x
2523:(
2517:y
2509:f
2500:,
2497:)
2492:0
2488:z
2484:,
2479:0
2475:y
2471:,
2466:0
2462:x
2458:(
2452:x
2444:f
2434:[
2402:)
2397:0
2393:z
2389:,
2384:0
2380:y
2376:,
2371:0
2367:x
2363:(
2349:f
2337:z
2329:)
2327:0
2324:z
2320:0
2317:y
2313:0
2310:x
2308:(
2287:0
2284:=
2281:)
2278:)
2275:y
2272:,
2269:x
2266:(
2260:,
2257:y
2254:,
2251:x
2248:(
2245:f
2232:)
2230:y
2226:x
2224:(
2222:φ
2217:)
2215:0
2212:z
2208:0
2205:y
2201:0
2198:x
2196:(
2192:f
2188:z
2182:0
2179:z
2175:0
2172:y
2168:0
2165:x
2163:(
2161:f
2136:=
2133:)
2130:z
2127:,
2124:y
2121:,
2118:x
2115:(
2112:f
2049:,
2042:)
2039:u
2036:,
2033:t
2030:(
2025:0
2021:f
2015:)
2012:u
2009:,
2006:t
2003:(
1998:3
1994:f
1987:=
1980:z
1973:,
1967:)
1964:u
1961:,
1958:t
1955:(
1950:0
1946:f
1940:)
1937:u
1934:,
1931:t
1928:(
1923:2
1919:f
1912:=
1905:y
1898:,
1892:)
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