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2691:{\displaystyle {\begin{alignedat}{5}W\mathbb {n} {\text{ is perpendicular to }}M\mathbb {t} \quad \,&{\text{ if and only if }}\quad 0=(W\mathbb {n} )\cdot (M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=(W\mathbb {n} )^{\mathrm {T} }(M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=\left(\mathbb {n} ^{\mathrm {T} }W^{\mathrm {T} }\right)(M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=\mathbb {n} ^{\mathrm {T} }\left(W^{\mathrm {T} }M\right)\mathbb {t} \\\end{alignedat}}}
589:
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1902:{\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial x}}\times {\frac {\partial \mathbf {r} }{\partial y}}=\left(1,0,{\tfrac {\partial f}{\partial x}}\right)\times \left(0,1,{\tfrac {\partial f}{\partial y}}\right)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right);}
3849:
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Therefore, one should use the inverse transpose of the linear transformation when transforming surface normals. The inverse transpose is equal to the original matrix if the matrix is orthonormal, that is, purely rotational with no scaling or shearing.
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801:
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4038:{\displaystyle \mathbb {n} =\nabla F\left(x_{1},x_{2},\ldots ,x_{n}\right)=\left({\tfrac {\partial F}{\partial x_{1}}},{\tfrac {\partial F}{\partial x_{2}}},\ldots ,{\tfrac {\partial F}{\partial x_{n}}}\right)\,.}
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That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. Alternatively, if the hyperplane is defined as the solution set of a single linear equation
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hypersurfaces, and the normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point.
3180:{\displaystyle \mathbf {r} \left(t_{1},\ldots ,t_{n-1}\right)=\mathbf {p} _{0}+t_{1}\mathbf {p} _{1}+\cdots +t_{n-1}\mathbf {p} _{n-1},}
3508:
1416:
915:
2106:{\displaystyle \mathbf {n} =\nabla F(x,y,z)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right).}
2213:
When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals.
1027:
1251:{\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}.}
5272: – directional vector associated with a vertex, intended as a replacement to the true geometric normal of the surface
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2145:, but it does not have a unique direction, since its opposite is also a unit normal. For a surface which is the
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If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a
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42:
A normal to a surface at a point is the same as a normal to the tangent plane to the surface at the same point.
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Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the
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are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector
261:(notice the singular, as only one normal will be defined) to determine a surface's orientation toward a
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4288:{\displaystyle f_{1}\left(x_{1},\ldots ,x_{n}\right),\ldots ,f_{k}\left(x_{1},\ldots ,x_{n}\right).}
3387:{\displaystyle P={\begin{bmatrix}\mathbf {p} _{1}&\cdots &\mathbf {p} _{n-1}\end{bmatrix}},}
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are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both
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2121:. In general, it is possible to define a normal almost everywhere for a surface that is
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5252: – vector bundle, complementary to the tangent bundle, associated to an embedding
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The definition of a normal to a surface in three-dimensional space can be extended to
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is by definition a normal to a tangent plane, given by the cross product of the
5202:
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796:{\displaystyle \mathbf {r} (s,t)=\mathbf {r} _{0}+s\mathbf {p} +t\mathbf {q} ,}
619:
481:
404:{\displaystyle \mathbf {N} =R{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}}
63:
4148:
is the set of the common zeros of a finite set of differentiable functions in
573:{\displaystyle \mathbf {T} ={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} s}}}
38:
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964:
A curved surface showing the unit normal vectors (blue arrows) to the surface
910:
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2117:, it has no well-defined normal at that point: for example, the vertex of a
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is not zero. At these points a normal vector is given by the gradient:
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1589:
an upward-pointing normal can be found either from the parametrization
597:
113:, which may be used for indicating sides (e.g., interior or exterior).
588:
5353:
455:
102:
5181:
5155:, the shapes of 3D objects are estimated from surface normals using
4597:
be the variety defined in the 3-dimensional space by the equations
4380:
3843:
1411:
604:
47:
5261: – Physical quantity that changes sign with improper rotation
4383:
in the neighborhood of a point where the
Jacobian matrix has rank
2133:
342:
30:
600:
2149:
of a set in three dimensions, one can distinguish between two
3562:{\displaystyle \mathbb {n} =\left(a_{1},\ldots ,a_{n}\right)}
946:{\displaystyle \mathbf {n} =\mathbf {p} \times \mathbf {q} .}
165:
4485:
In other words, a variety is defined as the intersection of
1470:
the gradient at any point is perpendicular to the level set
5162:
The normal vector may be obtained as the gradient of the
2141:
The normal to a (hyper)surface is usually scaled to have
1126:{\displaystyle \mathbf {r} (s,t)=(x(s,t),y(s,t),z(s,t)),}
955:
726:
For a plane whose equation is given in parametric form
2938:
2210:
matrix, as translation is irrelevant to the calculation
269:, or the orientation of each of the surface's corners (
5343:
5217:
are respectively the angle between the normal and the
4832:
Thus the normal affine space is the plane of equation
3998:
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3333:
2068:
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1864:
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246:
Normal vectors are of special interest in the case of
5148:
to change the apparent lighting of rendered elements.
5144:
containing surface normal information may be used in
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Pages displaying wikidata descriptions as a fallback
5254:
Pages displaying wikidata descriptions as a fallback
5241: – In mathematics, vector space of linear forms
607:), a surface normal can be calculated as the vector
27:
Line or vector perpendicular to a curve or a surface
4585:to the points where the variety is not a manifold.
2113:Since a surface does not have a tangent plane at a
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219:is the set of vectors which are orthogonal to the
211:
82:at a given point is the line perpendicular to the
5378:
3498:{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=c,}
2216:Specifically, given a 3×3 transformation matrix
583:
346:Normal direction (in red) to a curve (in black).
296:on the surface where the normal vector contains
4433:is the vector space generated by the values at
3779:{\displaystyle F(x_{1},x_{2},\ldots ,x_{n})=0,}
2336:perpendicular to the transformed tangent plane
1661:{\displaystyle \mathbf {r} (x,y)=(x,y,f(x,y)),}
172:, etc. The concept of normality generalizes to
105:of the object. Multiplying a normal vector by
5170:
1461:{\displaystyle \mathbf {n} =\nabla F(x,y,z).}
4556:and generated by the normal vector space at
4066:
4058:
611:of two (non-parallel) edges of the polygon.
5225:) and the angle between the normal and the
4085:Varieties defined by implicit equations in
4051:is the one-dimensional subspace with basis
3702:{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}
3423:{\displaystyle P\mathbf {n} =\mathbf {0} .}
2817:will satisfy the above equation, giving a
2810:{\displaystyle W=(M^{-1})^{\mathrm {T} },}
2165:, the normal is usually determined by the
5295:"Radiometry, BRDF and Photometric Stereo"
4607:
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997:
855:{\displaystyle \mathbf {p} ,\mathbf {q} }
325:
5180:
3644:defined implicitly as the set of points
2132:
959:
587:
341:
101:is a normal vector whose length is the
37:
29:
5111:Surface normals are useful in defining
3034:given by its parametric representation
2924:{\displaystyle \mathbf {t} ^{\prime },}
2179:
14:
5379:
2892:{\displaystyle \mathbf {n} ^{\prime }}
2329:{\displaystyle \mathbf {n} ^{\prime }}
2184:in this section we only use the upper
2137:A vector field of normals to a surface
1909:or more simply from its implicit form
956:Normal to general surfaces in 3D space
5344:
5122:Surface normals are commonly used in
4099:defined by implicit equations in the
187:of arbitrary dimension embedded in a
152:is also used as an adjective: a line
5003:the rows of the Jacobian matrix are
4753:the rows of the Jacobian matrix are
2756:{\displaystyle W^{\mathrm {T} }M=I,}
2169:or its analog in higher dimensions.
1979:{\displaystyle F(x,y,z)=z-f(x,y)=0,}
716:{\displaystyle \mathbf {n} =(a,b,c)}
183:The concept has been generalized to
74:to a given object. For example, the
34:A polygon and its two normal vectors
5322:from the original on April 27, 2009
5292:
2285:perpendicular to the tangent plane
284:of a normal at a point of interest
24:
5373:from either a triangle or polygon.
4581:These definitions may be extended
4009:
4001:
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560:
548:
484:, in terms of the curve position
391:
379:
310:to a curve or to a surface is the
25:
5413:
5337:
4640:This variety is the union of the
4633:{\displaystyle x\,y=0,\quad z=0.}
4075:{\displaystyle \{\mathbf {n} \}.}
3633:{\displaystyle \mathbb {R} ^{n}.}
3216:is a point on the hyperplane and
1545:given as the graph of a function
968:If a (possibly non-flat) surface
273:) to mimic a curved surface with
138:is a vector perpendicular to the
5265:Tangential and normal components
4141:{\displaystyle \mathbb {R} ^{n}}
4062:
3413:
3405:
3357:
3338:
3297:
3238:{\displaystyle \mathbf {p} _{i}}
3225:
3209:{\displaystyle \mathbf {p} _{0}}
3196:
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3027:{\displaystyle \mathbb {R} ^{n}}
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1538:{\displaystyle \mathbb {R} ^{3}}
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1010:{\displaystyle \mathbb {R} ^{3}}
936:
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825:{\displaystyle \mathbf {r} _{0}}
812:
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553:
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466:
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4453:of the gradient vectors of the
3282:{\displaystyle i=1,\ldots ,n-1}
2633:
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2512:
2464:
2455:
2444: is perpendicular to
1410:on the surface is given by the
5316:The Physics Classroom Tutorial
5304:
5286:
5185:Diagram of specular reflection
5066:
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5010:
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2954:
2863:{\displaystyle M\mathbb {t} ,}
2793:
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2620:
2609:
2553:
2542:
2531:
2519:
2499:
2488:
2482:
2471:
2392:{\displaystyle \mathbf {Wn} .}
2357:{\displaystyle \mathbf {Mt} ,}
2128:
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2007:
1964:
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692:
443:{\displaystyle R=\kappa ^{-1}}
292:) can be defined at the point
13:
1:
5364:explanation of normal vectors
5279:
2835:{\displaystyle W\mathbb {n} }
2417:{\displaystyle \mathbf {W} .}
2234:{\displaystyle \mathbf {M} ,}
902:{\displaystyle \mathbf {q} ,}
670:{\displaystyle ax+by+cz+d=0,}
592:Plane equation in normal form
584:Normal to planes and polygons
5371:calculating a surface normal
5197:is the outward-pointing ray
3304:{\displaystyle \mathbf {n} }
2715:{\displaystyle \mathbf {W} }
2300:{\displaystyle \mathbf {t} }
2278:{\displaystyle \mathbf {n} }
2256:{\displaystyle \mathbf {W} }
2241:we can determine the matrix
1176:variables, then a normal to
877:{\displaystyle \mathbf {p} }
832:is a point on the plane and
499:{\displaystyle \mathbf {r} }
473:{\displaystyle \mathbf {T} }
257:The normal is often used in
7:
5232:
4345:-th row is the gradient of
3834:then the hypersurface is a
3832:continuously differentiable
1365:{\displaystyle F(x,y,z)=0,}
86:to the curve at the point.
10:
5418:
5300:. Northwestern University.
5174:
5171:Normal in geometric optics
4588:
2630: if and only if
2563: if and only if
2509: if and only if
2461: if and only if
2183:
909:which can be found as the
618:given by the general form
350:The normal direction to a
335:
329:
4934:is the plane of equation
4377:implicit function theorem
4318:{\displaystyle k\times n}
2263:that transforms a vector
2203:{\displaystyle 3\times 3}
1582:{\displaystyle z=f(x,y),}
1372:then a normal at a point
5164:signed distance function
5075:{\displaystyle (0,0,0).}
4883:{\displaystyle b\neq 0,}
4825:{\displaystyle (0,a,0).}
4746:{\displaystyle a\neq 0,}
4717:{\displaystyle (a,0,0),}
3842:of the points where the
2364:by the following logic:
185:differentiable manifolds
5312:"The Law of Reflection"
5245:Ellipsoid normal vector
5105:
5034:{\displaystyle (0,0,1)}
4996:{\displaystyle (0,0,0)}
4927:{\displaystyle (0,b,0)}
4784:{\displaystyle (0,0,1)}
3836:differentiable manifold
3709:satisfying an equation
1403:{\displaystyle (x,y,z)}
1315:{\displaystyle (x,y,z)}
1023:curvilinear coordinates
290:foot of a perpendicular
199:of a manifold at point
118:three-dimensional space
5186:
5096:
5076:
5035:
4997:
4957:
4928:
4884:
4855:
4826:
4785:
4747:
4718:
4674:
4654:
4634:
4573:
4550:
4532:of the variety is the
4526:
4499:
4477:
4476:{\displaystyle f_{i}.}
4447:
4423:
4400:
4369:
4368:{\displaystyle f_{i}.}
4339:
4319:
4299:of the variety is the
4289:
4162:
4142:
4113:
4076:
4039:
3824:
3800:
3780:
3703:
3640:A hypersurface may be
3634:
3598:
3563:
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3181:
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2155:inward-pointing normal
2138:
2107:
1980:
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500:
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444:
405:
347:
332:Frenet–Serret formulas
326:Normal to space curves
240:
213:
43:
35:
5369:Clear pseudocode for
5366:from Microsoft's MSDN
5205:at a given point. In
5201:to the surface of an
5184:
5134:), often adjusted by
5097:
5077:
5036:
4998:
4958:
4929:
4885:
4856:
4827:
4786:
4748:
4719:
4675:
4655:
4635:
4574:
4551:
4527:
4510:normal (affine) space
4500:
4478:
4448:
4424:
4401:
4370:
4340:
4320:
4290:
4163:
4143:
4114:
4077:
4040:
3825:
3801:
3781:
3704:
3635:
3599:
3597:{\displaystyle (n-1)}
3564:
3500:
3425:
3389:
3306:
3284:
3240:
3211:
3182:
3029:
2999:
2974:
2972:{\displaystyle (n-1)}
2926:
2894:
2865:
2837:
2812:
2758:
2717:
2693:
2419:
2394:
2359:
2331:
2302:
2280:
2258:
2236:
2205:
2159:outer-pointing normal
2136:
2108:
1981:
1904:
1663:
1584:
1540:
1511:
1489:
1463:
1405:
1367:
1317:
1284:as the set of points
1275:
1253:
1168:
1148:
1128:
1012:
983:
963:
948:
904:
879:
857:
827:
798:
718:
672:
591:
575:
521:
501:
475:
445:
406:
345:
336:Further information:
241:
214:
41:
33:
5397:3D computer graphics
5132:Lambert's cosine law
5124:3D computer graphics
5086:
5045:
5007:
4969:
4956:{\displaystyle y=b.}
4938:
4900:
4865:
4854:{\displaystyle x=a.}
4836:
4795:
4757:
4728:
4687:
4664:
4644:
4601:
4560:
4540:
4516:
4489:
4457:
4437:
4410:
4387:
4349:
4329:
4303:
4172:
4152:
4123:
4103:
4096:differential variety
4055:
3850:
3814:
3790:
3713:
3648:
3612:
3576:
3509:
3434:
3398:
3319:
3293:
3249:
3220:
3191:
3038:
3009:
2988:
2951:
2903:
2874:
2846:
2821:
2767:
2726:
2704:
2427:
2403:
2375:
2340:
2311:
2289:
2267:
2245:
2220:
2188:
2180:Transforming normals
2147:topological boundary
2123:Lipschitz continuous
1990:
1913:
1672:
1593:
1549:
1520:
1500:
1475:
1417:
1376:
1326:
1288:
1264:
1187:
1157:
1137:
1028:
992:
972:
916:
888:
866:
836:
807:
730:
681:
625:
533:
510:
488:
462:
418:
361:
259:3D computer graphics
227:
203:
91:vector of length one
5215:angle of reflection
5207:reflection of light
5177:Specular reflection
5146:digital compositing
4431:normal vector space
4379:, the variety is a
4119:-dimensional space
2151:normal orientations
1182:partial derivatives
452:radius of curvature
193:normal vector space
5346:Weisstein, Eric W.
5223:plane of incidence
5211:angle of incidence
5187:
5157:photometric stereo
5130:calculations (see
5092:
5072:
5031:
4993:
4953:
4924:
4880:
4851:
4822:
4781:
4743:
4714:
4670:
4650:
4630:
4572:{\displaystyle P.}
4569:
4546:
4522:
4495:
4473:
4443:
4422:{\displaystyle P,}
4419:
4399:{\displaystyle k.}
4396:
4365:
4335:
4315:
4285:
4158:
4138:
4109:
4089:-dimensional space
4072:
4035:
4024:
3986:
3954:
3820:
3796:
3776:
3699:
3630:
3594:
3559:
3495:
3420:
3384:
3375:
3301:
3279:
3235:
3206:
3177:
3024:
3004:-dimensional space
2994:
2969:
2943:-dimensional space
2921:
2889:
2860:
2832:
2807:
2753:
2712:
2688:
2686:
2414:
2389:
2354:
2326:
2297:
2275:
2253:
2231:
2200:
2139:
2103:
2087:
2059:
1976:
1899:
1883:
1855:
1817:
1770:
1658:
1579:
1535:
1506:
1487:{\displaystyle S.}
1484:
1458:
1400:
1362:
1312:
1270:
1248:
1163:
1143:
1123:
1007:
978:
966:
943:
899:
874:
852:
822:
793:
713:
667:
594:
570:
516:
496:
470:
440:
401:
348:
312:Euclidean distance
288:(analogous to the
239:{\displaystyle P.}
236:
209:
142:of the surface at
95:unit normal vector
44:
36:
5113:surface integrals
5095:{\displaystyle z}
4673:{\displaystyle y}
4653:{\displaystyle x}
4549:{\displaystyle P}
4525:{\displaystyle P}
4498:{\displaystyle k}
4446:{\displaystyle P}
4338:{\displaystyle i}
4161:{\displaystyle n}
4112:{\displaystyle n}
4023:
3985:
3953:
3823:{\displaystyle F}
3799:{\displaystyle F}
2997:{\displaystyle n}
2939:Hypersurfaces in
2899:perpendicular to
2842:perpendicular to
2631:
2564:
2510:
2462:
2445:
2086:
2058:
1882:
1854:
1816:
1769:
1728:
1703:
1509:{\displaystyle S}
1273:{\displaystyle S}
1243:
1218:
1166:{\displaystyle t}
1146:{\displaystyle s}
981:{\displaystyle S}
568:
519:{\displaystyle s}
399:
212:{\displaystyle P}
16:(Redirected from
5409:
5359:
5358:
5331:
5330:
5328:
5327:
5308:
5302:
5301:
5299:
5290:
5275:
5255:
5195:
5194:
5101:
5099:
5098:
5093:
5081:
5079:
5078:
5073:
5040:
5038:
5037:
5032:
5002:
5000:
4999:
4994:
4962:
4960:
4959:
4954:
4933:
4931:
4930:
4925:
4889:
4887:
4886:
4881:
4860:
4858:
4857:
4852:
4831:
4829:
4828:
4823:
4790:
4788:
4787:
4782:
4752:
4750:
4749:
4744:
4723:
4721:
4720:
4715:
4679:
4677:
4676:
4671:
4659:
4657:
4656:
4651:
4639:
4637:
4636:
4631:
4578:
4576:
4575:
4570:
4555:
4553:
4552:
4547:
4536:passing through
4531:
4529:
4528:
4523:
4504:
4502:
4501:
4496:
4482:
4480:
4479:
4474:
4469:
4468:
4452:
4450:
4449:
4444:
4428:
4426:
4425:
4420:
4406:At such a point
4405:
4403:
4402:
4397:
4374:
4372:
4371:
4366:
4361:
4360:
4344:
4342:
4341:
4336:
4324:
4322:
4321:
4316:
4294:
4292:
4291:
4286:
4281:
4277:
4276:
4275:
4257:
4256:
4242:
4241:
4223:
4219:
4218:
4217:
4199:
4198:
4184:
4183:
4167:
4165:
4164:
4159:
4147:
4145:
4144:
4139:
4137:
4136:
4131:
4118:
4116:
4115:
4110:
4081:
4079:
4078:
4073:
4065:
4044:
4042:
4041:
4036:
4030:
4026:
4025:
4022:
4021:
4020:
4007:
3999:
3987:
3984:
3983:
3982:
3969:
3961:
3955:
3952:
3951:
3950:
3937:
3929:
3918:
3914:
3913:
3912:
3894:
3893:
3881:
3880:
3857:
3829:
3827:
3826:
3821:
3805:
3803:
3802:
3797:
3785:
3783:
3782:
3777:
3763:
3762:
3744:
3743:
3731:
3730:
3708:
3706:
3705:
3700:
3695:
3694:
3676:
3675:
3663:
3662:
3639:
3637:
3636:
3631:
3626:
3625:
3620:
3603:
3601:
3600:
3595:
3568:
3566:
3565:
3560:
3558:
3554:
3553:
3552:
3534:
3533:
3516:
3505:then the vector
3504:
3502:
3501:
3496:
3485:
3484:
3475:
3474:
3456:
3455:
3446:
3445:
3429:
3427:
3426:
3421:
3416:
3408:
3393:
3391:
3390:
3385:
3380:
3379:
3372:
3371:
3360:
3347:
3346:
3341:
3310:
3308:
3307:
3302:
3300:
3288:
3286:
3285:
3280:
3244:
3242:
3241:
3236:
3234:
3233:
3228:
3215:
3213:
3212:
3207:
3205:
3204:
3199:
3186:
3184:
3183:
3178:
3173:
3172:
3161:
3155:
3154:
3130:
3129:
3124:
3118:
3117:
3105:
3104:
3099:
3090:
3086:
3085:
3084:
3060:
3059:
3045:
3033:
3031:
3030:
3025:
3023:
3022:
3017:
3003:
3001:
3000:
2995:
2978:
2976:
2975:
2970:
2930:
2928:
2927:
2922:
2917:
2916:
2911:
2898:
2896:
2895:
2890:
2888:
2887:
2882:
2869:
2867:
2866:
2861:
2856:
2841:
2839:
2838:
2833:
2831:
2816:
2814:
2813:
2808:
2803:
2802:
2801:
2791:
2790:
2762:
2760:
2759:
2754:
2740:
2739:
2738:
2721:
2719:
2718:
2713:
2711:
2697:
2695:
2694:
2689:
2687:
2683:
2678:
2674:
2670:
2669:
2668:
2653:
2652:
2651:
2645:
2632:
2629:
2626:
2619:
2608:
2604:
2603:
2602:
2601:
2591:
2590:
2589:
2583:
2565:
2562:
2559:
2552:
2541:
2540:
2539:
2529:
2511:
2508:
2505:
2498:
2481:
2463:
2460:
2454:
2446:
2443:
2441:
2423:
2421:
2420:
2415:
2410:
2398:
2396:
2395:
2390:
2385:
2363:
2361:
2360:
2355:
2350:
2335:
2333:
2332:
2327:
2325:
2324:
2319:
2306:
2304:
2303:
2298:
2296:
2284:
2282:
2281:
2276:
2274:
2262:
2260:
2259:
2254:
2252:
2240:
2238:
2237:
2232:
2227:
2209:
2207:
2206:
2201:
2163:oriented surface
2112:
2110:
2109:
2104:
2099:
2095:
2088:
2085:
2077:
2069:
2060:
2057:
2049:
2041:
1997:
1985:
1983:
1982:
1977:
1908:
1906:
1905:
1900:
1895:
1891:
1884:
1881:
1873:
1865:
1856:
1853:
1845:
1837:
1823:
1819:
1818:
1815:
1807:
1799:
1776:
1772:
1771:
1768:
1760:
1752:
1729:
1727:
1719:
1718:
1709:
1704:
1702:
1694:
1693:
1684:
1679:
1667:
1665:
1664:
1659:
1600:
1588:
1586:
1585:
1580:
1544:
1542:
1541:
1536:
1534:
1533:
1528:
1515:
1513:
1512:
1507:
1493:
1491:
1490:
1485:
1467:
1465:
1464:
1459:
1424:
1409:
1407:
1406:
1401:
1371:
1369:
1368:
1363:
1321:
1319:
1318:
1313:
1279:
1277:
1276:
1271:
1257:
1255:
1254:
1249:
1244:
1242:
1234:
1233:
1224:
1219:
1217:
1209:
1208:
1199:
1194:
1172:
1170:
1169:
1164:
1152:
1150:
1149:
1144:
1132:
1130:
1129:
1124:
1035:
1016:
1014:
1013:
1008:
1006:
1005:
1000:
987:
985:
984:
979:
952:
950:
949:
944:
939:
931:
923:
908:
906:
905:
900:
895:
883:
881:
880:
875:
873:
861:
859:
858:
853:
851:
843:
831:
829:
828:
823:
821:
820:
815:
802:
800:
799:
794:
789:
778:
767:
766:
761:
737:
722:
720:
719:
714:
688:
676:
674:
673:
668:
579:
577:
576:
571:
569:
567:
563:
557:
556:
551:
545:
540:
525:
523:
522:
517:
505:
503:
502:
497:
495:
479:
477:
476:
471:
469:
449:
447:
446:
441:
439:
438:
410:
408:
407:
402:
400:
398:
394:
388:
387:
382:
376:
368:
338:Curvature vector
245:
243:
242:
237:
218:
216:
215:
210:
147:
137:
108:
99:curvature vector
21:
5417:
5416:
5412:
5411:
5410:
5408:
5407:
5406:
5392:Vector calculus
5377:
5376:
5349:"Normal Vector"
5340:
5335:
5334:
5325:
5323:
5310:
5309:
5305:
5297:
5291:
5287:
5282:
5273:
5253:
5235:
5192:
5191:
5179:
5173:
5153:computer vision
5108:
5087:
5084:
5083:
5046:
5043:
5042:
5008:
5005:
5004:
4970:
4967:
4966:
4939:
4936:
4935:
4901:
4898:
4897:
4866:
4863:
4862:
4837:
4834:
4833:
4796:
4793:
4792:
4758:
4755:
4754:
4729:
4726:
4725:
4688:
4685:
4684:
4665:
4662:
4661:
4645:
4642:
4641:
4602:
4599:
4598:
4591:
4561:
4558:
4557:
4541:
4538:
4537:
4534:affine subspace
4517:
4514:
4513:
4490:
4487:
4486:
4464:
4460:
4458:
4455:
4454:
4438:
4435:
4434:
4411:
4408:
4407:
4388:
4385:
4384:
4356:
4352:
4350:
4347:
4346:
4330:
4327:
4326:
4304:
4301:
4300:
4297:Jacobian matrix
4271:
4267:
4252:
4248:
4247:
4243:
4237:
4233:
4213:
4209:
4194:
4190:
4189:
4185:
4179:
4175:
4173:
4170:
4169:
4153:
4150:
4149:
4132:
4127:
4126:
4124:
4121:
4120:
4104:
4101:
4100:
4091:
4061:
4056:
4053:
4052:
4016:
4012:
4008:
4000:
3997:
3978:
3974:
3970:
3962:
3959:
3946:
3942:
3938:
3930:
3927:
3926:
3922:
3908:
3904:
3889:
3885:
3876:
3872:
3871:
3867:
3853:
3851:
3848:
3847:
3815:
3812:
3811:
3808:scalar function
3791:
3788:
3787:
3758:
3754:
3739:
3735:
3726:
3722:
3714:
3711:
3710:
3690:
3686:
3671:
3667:
3658:
3654:
3649:
3646:
3645:
3621:
3616:
3615:
3613:
3610:
3609:
3577:
3574:
3573:
3548:
3544:
3529:
3525:
3524:
3520:
3512:
3510:
3507:
3506:
3480:
3476:
3470:
3466:
3451:
3447:
3441:
3437:
3435:
3432:
3431:
3412:
3404:
3399:
3396:
3395:
3374:
3373:
3361:
3356:
3355:
3353:
3348:
3342:
3337:
3336:
3329:
3328:
3320:
3317:
3316:
3315:of the matrix
3296:
3294:
3291:
3290:
3250:
3247:
3246:
3229:
3224:
3223:
3221:
3218:
3217:
3200:
3195:
3194:
3192:
3189:
3188:
3162:
3157:
3156:
3144:
3140:
3125:
3120:
3119:
3113:
3109:
3100:
3095:
3094:
3074:
3070:
3055:
3051:
3050:
3046:
3041:
3039:
3036:
3035:
3018:
3013:
3012:
3010:
3007:
3006:
2989:
2986:
2985:
2952:
2949:
2948:
2945:
2912:
2907:
2906:
2904:
2901:
2900:
2883:
2878:
2877:
2875:
2872:
2871:
2852:
2847:
2844:
2843:
2827:
2822:
2819:
2818:
2797:
2796:
2792:
2783:
2779:
2768:
2765:
2764:
2734:
2733:
2729:
2727:
2724:
2723:
2707:
2705:
2702:
2701:
2685:
2684:
2679:
2664:
2663:
2659:
2658:
2654:
2647:
2646:
2641:
2640:
2628:
2624:
2623:
2615:
2597:
2596:
2592:
2585:
2584:
2579:
2578:
2577:
2573:
2561:
2557:
2556:
2548:
2535:
2534:
2530:
2525:
2507:
2503:
2502:
2494:
2477:
2459:
2457:
2450:
2442:
2437:
2430:
2428:
2425:
2424:
2406:
2404:
2401:
2400:
2378:
2376:
2373:
2372:
2343:
2341:
2338:
2337:
2320:
2315:
2314:
2312:
2309:
2308:
2292:
2290:
2287:
2286:
2270:
2268:
2265:
2264:
2248:
2246:
2243:
2242:
2223:
2221:
2218:
2217:
2211:
2189:
2186:
2185:
2182:
2167:right-hand rule
2131:
2078:
2070:
2067:
2050:
2042:
2039:
2035:
2031:
1993:
1991:
1988:
1987:
1914:
1911:
1910:
1874:
1866:
1863:
1846:
1838:
1835:
1831:
1827:
1808:
1800:
1797:
1784:
1780:
1761:
1753:
1750:
1737:
1733:
1720:
1714:
1710:
1708:
1695:
1689:
1685:
1683:
1675:
1673:
1670:
1669:
1596:
1594:
1591:
1590:
1550:
1547:
1546:
1529:
1524:
1523:
1521:
1518:
1517:
1501:
1498:
1497:
1476:
1473:
1472:
1420:
1418:
1415:
1414:
1377:
1374:
1373:
1327:
1324:
1323:
1289:
1286:
1285:
1265:
1262:
1261:
1235:
1229:
1225:
1223:
1210:
1204:
1200:
1198:
1190:
1188:
1185:
1184:
1158:
1155:
1154:
1138:
1135:
1134:
1031:
1029:
1026:
1025:
1021:by a system of
1001:
996:
995:
993:
990:
989:
973:
970:
969:
958:
935:
927:
919:
917:
914:
913:
891:
889:
886:
885:
869:
867:
864:
863:
847:
839:
837:
834:
833:
816:
811:
810:
808:
805:
804:
785:
774:
762:
757:
756:
733:
731:
728:
727:
684:
682:
679:
678:
626:
623:
622:
586:
559:
558:
552:
547:
546:
544:
536:
534:
531:
530:
511:
508:
507:
506:and arc-length
491:
489:
486:
485:
465:
463:
460:
459:
431:
427:
419:
416:
415:
390:
389:
383:
378:
377:
375:
364:
362:
359:
358:
340:
334:
328:
303:normal distance
252:smooth surfaces
228:
225:
224:
204:
201:
200:
189:Euclidean space
164:component of a
143:
133:
111:opposite vector
109:results in the
106:
28:
23:
22:
15:
12:
11:
5:
5415:
5405:
5404:
5399:
5394:
5389:
5375:
5374:
5367:
5360:
5339:
5338:External links
5336:
5333:
5332:
5303:
5284:
5283:
5281:
5278:
5277:
5276:
5267:
5262:
5256:
5247:
5242:
5234:
5231:
5203:optical medium
5175:Main article:
5172:
5169:
5168:
5167:
5160:
5149:
5139:
5136:normal mapping
5120:
5107:
5104:
5091:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5050:
5030:
5027:
5024:
5021:
5018:
5015:
5012:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4952:
4949:
4946:
4943:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4879:
4876:
4873:
4870:
4861:Similarly, if
4850:
4847:
4844:
4841:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4780:
4777:
4774:
4771:
4768:
4765:
4762:
4742:
4739:
4736:
4733:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4669:
4660:-axis and the
4649:
4629:
4626:
4623:
4619:
4616:
4613:
4610:
4606:
4590:
4587:
4584:
4568:
4565:
4545:
4521:
4494:
4472:
4467:
4463:
4442:
4418:
4415:
4395:
4392:
4364:
4359:
4355:
4334:
4314:
4311:
4308:
4284:
4280:
4274:
4270:
4266:
4263:
4260:
4255:
4251:
4246:
4240:
4236:
4232:
4229:
4226:
4222:
4216:
4212:
4208:
4205:
4202:
4197:
4193:
4188:
4182:
4178:
4157:
4135:
4130:
4108:
4090:
4083:
4071:
4068:
4064:
4060:
4034:
4029:
4019:
4015:
4011:
4006:
4003:
3996:
3993:
3990:
3981:
3977:
3973:
3968:
3965:
3958:
3949:
3945:
3941:
3936:
3933:
3925:
3921:
3917:
3911:
3907:
3903:
3900:
3897:
3892:
3888:
3884:
3879:
3875:
3870:
3866:
3863:
3860:
3856:
3819:
3795:
3775:
3772:
3769:
3766:
3761:
3757:
3753:
3750:
3747:
3742:
3738:
3734:
3729:
3725:
3721:
3718:
3698:
3693:
3689:
3685:
3682:
3679:
3674:
3670:
3666:
3661:
3657:
3653:
3629:
3624:
3619:
3593:
3590:
3587:
3584:
3581:
3557:
3551:
3547:
3543:
3540:
3537:
3532:
3528:
3523:
3519:
3515:
3494:
3491:
3488:
3483:
3479:
3473:
3469:
3465:
3462:
3459:
3454:
3450:
3444:
3440:
3419:
3415:
3411:
3407:
3403:
3383:
3378:
3370:
3367:
3364:
3359:
3354:
3352:
3349:
3345:
3340:
3335:
3334:
3332:
3327:
3324:
3299:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3232:
3227:
3203:
3198:
3176:
3171:
3168:
3165:
3160:
3153:
3150:
3147:
3143:
3139:
3136:
3133:
3128:
3123:
3116:
3112:
3108:
3103:
3098:
3093:
3089:
3083:
3080:
3077:
3073:
3069:
3066:
3063:
3058:
3054:
3049:
3044:
3021:
3016:
2993:
2968:
2965:
2962:
2959:
2956:
2944:
2937:
2920:
2915:
2910:
2886:
2881:
2859:
2855:
2851:
2830:
2826:
2806:
2800:
2795:
2789:
2786:
2782:
2778:
2775:
2772:
2752:
2749:
2746:
2743:
2737:
2732:
2710:
2682:
2677:
2673:
2667:
2662:
2657:
2650:
2644:
2639:
2636:
2627:
2625:
2622:
2618:
2614:
2611:
2607:
2600:
2595:
2588:
2582:
2576:
2572:
2569:
2560:
2558:
2555:
2551:
2547:
2544:
2538:
2533:
2528:
2524:
2521:
2518:
2515:
2506:
2504:
2501:
2497:
2493:
2490:
2487:
2484:
2480:
2476:
2473:
2470:
2467:
2458:
2453:
2449:
2440:
2436:
2433:
2432:
2413:
2409:
2388:
2384:
2381:
2353:
2349:
2346:
2323:
2318:
2307:into a vector
2295:
2273:
2251:
2230:
2226:
2199:
2196:
2193:
2181:
2178:
2130:
2127:
2115:singular point
2102:
2098:
2094:
2091:
2084:
2081:
2076:
2073:
2066:
2063:
2056:
2053:
2048:
2045:
2038:
2034:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1996:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1898:
1894:
1890:
1887:
1880:
1877:
1872:
1869:
1862:
1859:
1852:
1849:
1844:
1841:
1834:
1830:
1826:
1822:
1814:
1811:
1806:
1803:
1796:
1793:
1790:
1787:
1783:
1779:
1775:
1767:
1764:
1759:
1756:
1749:
1746:
1743:
1740:
1736:
1732:
1726:
1723:
1717:
1713:
1707:
1701:
1698:
1692:
1688:
1682:
1678:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1599:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1532:
1527:
1505:
1496:For a surface
1483:
1480:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1423:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1269:
1247:
1241:
1238:
1232:
1228:
1222:
1216:
1213:
1207:
1203:
1197:
1193:
1162:
1142:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1034:
1004:
999:
977:
957:
954:
942:
938:
934:
930:
926:
922:
898:
894:
872:
850:
846:
842:
819:
814:
792:
788:
784:
781:
777:
773:
770:
765:
760:
755:
752:
749:
746:
743:
740:
736:
712:
709:
706:
703:
700:
697:
694:
691:
687:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
620:plane equation
585:
582:
581:
580:
566:
562:
555:
550:
543:
539:
515:
494:
482:tangent vector
468:
437:
434:
430:
426:
423:
412:
411:
397:
393:
386:
381:
374:
371:
367:
330:Main article:
327:
324:
235:
232:
208:
122:surface normal
26:
9:
6:
4:
3:
2:
5414:
5403:
5402:Orthogonality
5400:
5398:
5395:
5393:
5390:
5388:
5385:
5384:
5382:
5372:
5368:
5365:
5361:
5356:
5355:
5350:
5347:
5342:
5341:
5321:
5317:
5313:
5307:
5296:
5289:
5285:
5271:
5270:Vertex normal
5268:
5266:
5263:
5260:
5257:
5251:
5250:Normal bundle
5248:
5246:
5243:
5240:
5237:
5236:
5230:
5228:
5227:reflected ray
5224:
5220:
5216:
5212:
5208:
5204:
5200:
5199:perpendicular
5196:
5183:
5178:
5165:
5161:
5158:
5154:
5150:
5147:
5143:
5142:Render layers
5140:
5137:
5133:
5129:
5125:
5121:
5118:
5117:vector fields
5114:
5110:
5109:
5103:
5089:
5069:
5063:
5060:
5057:
5054:
5051:
5025:
5022:
5019:
5016:
5013:
4987:
4984:
4981:
4978:
4975:
4965:At the point
4963:
4950:
4947:
4944:
4941:
4918:
4915:
4912:
4909:
4906:
4895:
4894:
4877:
4874:
4871:
4868:
4848:
4845:
4842:
4839:
4819:
4813:
4810:
4807:
4804:
4801:
4775:
4772:
4769:
4766:
4763:
4740:
4737:
4734:
4731:
4711:
4705:
4702:
4699:
4696:
4693:
4681:
4667:
4647:
4627:
4624:
4621:
4617:
4614:
4611:
4608:
4604:
4596:
4586:
4582:
4579:
4566:
4563:
4543:
4535:
4519:
4511:
4506:
4492:
4483:
4470:
4465:
4461:
4440:
4432:
4416:
4413:
4393:
4390:
4382:
4378:
4362:
4357:
4353:
4332:
4325:matrix whose
4312:
4309:
4306:
4298:
4282:
4278:
4272:
4268:
4264:
4261:
4258:
4253:
4249:
4244:
4238:
4234:
4230:
4227:
4224:
4220:
4214:
4210:
4206:
4203:
4200:
4195:
4191:
4186:
4180:
4176:
4155:
4133:
4106:
4098:
4097:
4088:
4082:
4069:
4050:
4045:
4032:
4027:
4017:
4013:
4004:
3994:
3991:
3988:
3979:
3975:
3966:
3956:
3947:
3943:
3934:
3923:
3919:
3915:
3909:
3905:
3901:
3898:
3895:
3890:
3886:
3882:
3877:
3873:
3868:
3864:
3858:
3845:
3841:
3840:neighbourhood
3837:
3833:
3817:
3809:
3793:
3773:
3770:
3767:
3759:
3755:
3751:
3748:
3745:
3740:
3736:
3732:
3727:
3723:
3716:
3691:
3687:
3683:
3680:
3677:
3672:
3668:
3664:
3659:
3655:
3643:
3627:
3622:
3607:
3606:hypersurfaces
3604:-dimensional
3588:
3585:
3582:
3570:
3569:is a normal.
3555:
3549:
3545:
3541:
3538:
3535:
3530:
3526:
3521:
3517:
3492:
3489:
3486:
3481:
3477:
3471:
3467:
3463:
3460:
3457:
3452:
3448:
3442:
3438:
3417:
3409:
3401:
3381:
3376:
3368:
3365:
3362:
3350:
3343:
3330:
3325:
3322:
3314:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3230:
3201:
3174:
3169:
3166:
3163:
3151:
3148:
3145:
3141:
3137:
3134:
3131:
3126:
3114:
3110:
3106:
3101:
3091:
3087:
3081:
3078:
3075:
3071:
3067:
3064:
3061:
3056:
3052:
3047:
3019:
3005:
2991:
2982:
2979:-dimensional
2963:
2960:
2957:
2942:
2936:
2932:
2931:as required.
2918:
2857:
2849:
2824:
2804:
2787:
2784:
2780:
2773:
2770:
2750:
2747:
2744:
2741:
2730:
2698:
2675:
2671:
2660:
2655:
2637:
2634:
2612:
2605:
2593:
2574:
2570:
2567:
2545:
2522:
2516:
2513:
2491:
2485:
2474:
2468:
2465:
2447:
2434:
2411:
2399:We must find
2386:
2370:
2365:
2351:
2228:
2214:
2197:
2194:
2191:
2177:
2175:
2170:
2168:
2164:
2160:
2156:
2152:
2148:
2144:
2135:
2126:
2124:
2120:
2116:
2100:
2096:
2092:
2089:
2082:
2074:
2064:
2061:
2054:
2046:
2036:
2032:
2028:
2022:
2019:
2016:
2013:
2010:
2004:
1998:
1973:
1970:
1967:
1961:
1958:
1955:
1949:
1946:
1943:
1940:
1934:
1931:
1928:
1925:
1922:
1916:
1896:
1892:
1888:
1885:
1878:
1870:
1860:
1857:
1850:
1842:
1832:
1828:
1824:
1820:
1812:
1804:
1794:
1791:
1788:
1785:
1781:
1777:
1773:
1765:
1757:
1747:
1744:
1741:
1738:
1734:
1730:
1724:
1705:
1699:
1680:
1655:
1646:
1643:
1640:
1634:
1631:
1628:
1625:
1622:
1616:
1610:
1607:
1604:
1576:
1570:
1567:
1564:
1558:
1555:
1552:
1530:
1503:
1494:
1481:
1478:
1471:
1455:
1449:
1446:
1443:
1440:
1437:
1431:
1425:
1413:
1394:
1391:
1388:
1385:
1382:
1359:
1356:
1353:
1347:
1344:
1341:
1338:
1335:
1329:
1306:
1303:
1300:
1297:
1294:
1283:
1267:
1260:If a surface
1258:
1245:
1239:
1220:
1214:
1195:
1183:
1179:
1175:
1160:
1140:
1120:
1111:
1108:
1105:
1099:
1096:
1090:
1087:
1084:
1078:
1075:
1069:
1066:
1063:
1057:
1051:
1045:
1042:
1039:
1024:
1020:
1019:parameterized
1002:
975:
962:
953:
940:
932:
924:
912:
911:cross product
896:
844:
817:
790:
782:
779:
771:
768:
763:
753:
747:
744:
741:
724:
723:is a normal.
707:
704:
701:
698:
695:
689:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
621:
617:
612:
610:
609:cross product
606:
602:
599:
590:
564:
541:
529:
528:
527:
513:
483:
457:
453:
435:
432:
428:
424:
421:
395:
372:
369:
357:
356:
355:
353:
344:
339:
333:
323:
321:
318:and its foot
317:
313:
309:
305:
304:
299:
295:
291:
287:
283:
278:
276:
275:Phong shading
272:
268:
264:
260:
255:
253:
249:
248:smooth curves
233:
230:
222:
221:tangent space
206:
198:
194:
190:
186:
181:
179:
175:
174:orthogonality
171:
170:normal vector
167:
163:
159:
155:
151:
146:
141:
140:tangent plane
136:
131:
127:
123:
119:
114:
112:
104:
100:
96:
92:
87:
85:
81:
77:
73:
72:perpendicular
69:
65:
61:
57:
53:
49:
40:
32:
19:
18:Normal vector
5352:
5324:. Retrieved
5315:
5306:
5288:
5259:Pseudovector
5219:incident ray
5190:
5188:
4964:
4893:normal plane
4891:
4682:
4594:
4592:
4580:
4509:
4507:
4484:
4430:
4094:
4092:
4086:
4048:
4046:
3571:
2946:
2940:
2933:
2699:
2368:
2366:
2215:
2212:
2174:pseudovector
2171:
2158:
2154:
2150:
2140:
1495:
1322:satisfying
1259:
1177:
988:in 3D space
967:
725:
613:
595:
454:(reciprocal
413:
349:
319:
315:
307:
301:
297:
293:
285:
281:
279:
267:flat shading
263:light source
256:
197:normal space
196:
192:
182:
178:right angles
169:
161:
153:
149:
144:
134:
125:
124:, or simply
121:
115:
94:
93:is called a
88:
84:tangent line
75:
51:
45:
4683:At a point
4512:at a point
4049:normal line
3806:is a given
2722:such that
2143:unit length
2129:Orientation
677:the vector
603:(such as a
352:space curve
306:of a point
148:. The word
80:plane curve
76:normal line
5381:Categories
5326:2008-03-31
5280:References
5239:Dual space
5193:normal ray
4168:variables
3313:null space
2981:hyperplane
1282:implicitly
70:) that is
5354:MathWorld
5293:Ying Wu.
4872:≠
4735:≠
4310:×
4262:…
4228:…
4204:…
4010:∂
4002:∂
3992:…
3972:∂
3964:∂
3940:∂
3932:∂
3899:…
3862:∇
3749:…
3681:…
3586:−
3539:…
3461:⋯
3366:−
3351:⋯
3274:−
3265:…
3167:−
3149:−
3135:⋯
3079:−
3065:…
2961:−
2914:′
2885:′
2785:−
2700:Choosing
2486:⋅
2322:′
2195:×
2161:. For an
2080:∂
2072:∂
2065:−
2052:∂
2044:∂
2037:−
2002:∇
1947:−
1876:∂
1868:∂
1861:−
1848:∂
1840:∂
1833:−
1810:∂
1802:∂
1778:×
1763:∂
1755:∂
1722:∂
1712:∂
1706:×
1697:∂
1687:∂
1429:∇
1280:is given
1237:∂
1227:∂
1221:×
1212:∂
1202:∂
933:×
456:curvature
433:−
429:κ
132:at point
103:curvature
89:A normal
5387:Surfaces
5320:Archived
5233:See also
5221:(on the
5213:and the
5128:lighting
4583:verbatim
4381:manifold
3844:gradient
3394:meaning
2369:n′
1412:gradient
605:triangle
314:between
271:vertices
58:(e.g. a
48:geometry
5102:-axis.
4680:-axis.
4589:Example
4375:By the
3838:in the
3642:locally
3311:in the
2947:For an
1986:giving
1668:giving
601:polygon
480:is the
450:is the
130:surface
128:, to a
5209:, the
4724:where
3786:where
3187:where
2870:or an
2367:Write
2153:, the
1468:since
803:where
614:For a
598:convex
596:For a
414:where
300:. The
191:. The
168:, the
162:normal
160:, the
154:normal
150:normal
126:normal
68:vector
56:object
54:is an
52:normal
5298:(PDF)
3810:. If
1133:with
616:plane
166:force
158:plane
156:to a
78:to a
66:, or
5189:The
5126:for
5106:Uses
5041:and
4890:the
4791:and
4593:Let
4508:The
4429:the
4295:The
4047:The
3245:for
2157:and
2119:cone
1174:real
1153:and
884:and
354:is:
282:foot
280:The
265:for
250:and
120:, a
97:. A
60:line
50:, a
5362:An
5151:In
5115:of
4896:at
3830:is
3608:in
2983:in
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2371:as
1516:in
1017:is
458:);
223:at
195:or
180:).
116:In
64:ray
46:In
5383::
5351:.
5318:.
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4628:0.
4093:A
2176:.
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322:.
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4991:)
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4951:.
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4945:=
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4907:0
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4802:0
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2020:,
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2014:,
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1935:z
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1103:(
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739:(
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708:c
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696:a
693:(
690:=
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653:+
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647:c
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638:b
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561:d
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542:=
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493:r
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436:1
425:=
422:R
396:s
392:d
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380:d
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370:=
366:N
320:P
316:Q
308:Q
298:Q
294:P
286:Q
234:.
231:P
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176:(
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