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2680:{\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}}}
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1891:{\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);}
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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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4027:{\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.
3169:{\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},}
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2095:{\displaystyle \mathbf {n} =\nabla F(x,y,z)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right).}
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When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals.
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1240:{\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}.}
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5261: – directional vector associated with a vertex, intended as a replacement to the true geometric normal of the surface
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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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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
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4277:{\displaystyle f_{1}\left(x_{1},\ldots ,x_{n}\right),\ldots ,f_{k}\left(x_{1},\ldots ,x_{n}\right).}
3376:{\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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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
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785:{\displaystyle \mathbf {r} (s,t)=\mathbf {r} _{0}+s\mathbf {p} +t\mathbf {q} ,}
393:{\displaystyle \mathbf {N} =R{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}}
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is the set of the common zeros of a finite set of differentiable functions in
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562:{\displaystyle \mathbf {T} ={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} s}}}
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A curved surface showing the unit normal vectors (blue arrows) to the surface
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is not zero. At these points a normal vector is given by the gradient:
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an upward-pointing normal can be found either from the parametrization
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102:, which may be used for indicating sides (e.g., interior or exterior).
5144:, the shapes of 3D objects are estimated from surface normals using
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be the variety defined in the 3-dimensional space by the equations
5250: – Physical quantity that changes sign with improper rotation
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in the neighborhood of a point where the
Jacobian matrix has rank
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of a set in three dimensions, one can distinguish between two
3551:{\displaystyle \mathbb {n} =\left(a_{1},\ldots ,a_{n}\right)}
935:{\displaystyle \mathbf {n} =\mathbf {p} \times \mathbf {q} .}
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In other words, a variety is defined as the intersection of
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the gradient at any point is perpendicular to the level set
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The normal vector may be obtained as the gradient of the
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The normal to a (hyper)surface is usually scaled to have
1115:{\displaystyle \mathbf {r} (s,t)=(x(s,t),y(s,t),z(s,t)),}
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For a plane whose equation is given in parametric form
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matrix, as translation is irrelevant to the calculation
258:, or the orientation of each of the surface's corners (
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are respectively the angle between the normal and the
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Thus the normal affine space is the plane of equation
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Normal vectors are of special interest in the case of
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to change the apparent lighting of rendered elements.
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containing surface normal information may be used in
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Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
5230: – In mathematics, vector space of linear forms
596:), a surface normal can be calculated as the vector
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Line or vector perpendicular to a curve or a surface
4574:to the points where the variety is not a manifold.
2102:Since a surface does not have a tangent plane at a
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71:at a given point is the line perpendicular to the
3487:{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=c,}
2205:Specifically, given a 3×3 transformation matrix
335:Normal direction (in red) to a curve (in black).
285:on the surface where the normal vector contains
4422:is the vector space generated by the values at
3768:{\displaystyle F(x_{1},x_{2},\ldots ,x_{n})=0,}
2325:perpendicular to the transformed tangent plane
1650:{\displaystyle \mathbf {r} (x,y)=(x,y,f(x,y)),}
161:, etc. The concept of normality generalizes to
94:of the object. Multiplying a normal vector by
1450:{\displaystyle \mathbf {n} =\nabla F(x,y,z).}
8:
4545:and generated by the normal vector space at
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600:of two (non-parallel) edges of the polygon.
5214:) and the angle between the normal and the
4074:Varieties defined by implicit equations in
4040:is the one-dimensional subspace with basis
3691:{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}
3412:{\displaystyle P\mathbf {n} =\mathbf {0} .}
2806:will satisfy the above equation, giving a
2799:{\displaystyle W=(M^{-1})^{\mathrm {T} },}
2154:, the normal is usually determined by the
5284:"Radiometry, BRDF and Photometric Stereo"
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844:{\displaystyle \mathbf {p} ,\mathbf {q} }
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3633:defined implicitly as the set of points
90:is a normal vector whose length is the
5274:
5100:Surface normals are useful in defining
3023:given by its parametric representation
2913:{\displaystyle \mathbf {t} ^{\prime },}
2881:{\displaystyle \mathbf {n} ^{\prime }}
2318:{\displaystyle \mathbf {n} ^{\prime }}
2173:in this section we only use the upper
2126:A vector field of normals to a surface
1898:or more simply from its implicit form
945:Normal to general surfaces in 3D space
5111:Surface normals are commonly used in
4088:defined by implicit equations in the
176:of arbitrary dimension embedded in a
141:is also used as an adjective: a line
7:
4992:the rows of the Jacobian matrix are
4742:the rows of the Jacobian matrix are
2745:{\displaystyle W^{\mathrm {T} }M=I,}
2158:or its analog in higher dimensions.
1968:{\displaystyle F(x,y,z)=z-f(x,y)=0,}
705:{\displaystyle \mathbf {n} =(a,b,c)}
172:The concept has been generalized to
63:to a given object. For example, the
23:A polygon and its two normal vectors
5311:from the original on April 27, 2009
2274:perpendicular to the tangent plane
273:of a normal at a point of interest
5362:from either a triangle or polygon.
4570:These definitions may be extended
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473:, in terms of the curve position
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299:to a curve or to a surface is the
14:
4629:This variety is the union of the
4622:{\displaystyle x\,y=0,\quad z=0.}
4064:{\displaystyle \{\mathbf {n} \}.}
3622:{\displaystyle \mathbb {R} ^{n}.}
3205:is a point on the hyperplane and
1534:given as the graph of a function
957:If a (possibly non-flat) surface
262:) to mimic a curved surface with
127:is a vector perpendicular to the
5254:Tangential and normal components
4130:{\displaystyle \mathbb {R} ^{n}}
4051:
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3227:{\displaystyle \mathbf {p} _{i}}
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3198:{\displaystyle \mathbf {p} _{0}}
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999:{\displaystyle \mathbb {R} ^{3}}
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814:{\displaystyle \mathbf {r} _{0}}
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4442:of the gradient vectors of the
3271:{\displaystyle i=1,\ldots ,n-1}
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2433: is perpendicular to
1399:on the surface is given by the
5305:The Physics Classroom Tutorial
5174:Diagram of specular reflection
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2852:{\displaystyle M\mathbb {t} ,}
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2381:{\displaystyle \mathbf {Wn} .}
2346:{\displaystyle \mathbf {Mt} ,}
2014:
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432:{\displaystyle R=\kappa ^{-1}}
281:) can be defined at the point
1:
5353:explanation of normal vectors
2824:{\displaystyle W\mathbb {n} }
2406:{\displaystyle \mathbf {W} .}
2223:{\displaystyle \mathbf {M} ,}
891:{\displaystyle \mathbf {q} ,}
659:{\displaystyle ax+by+cz+d=0,}
581:Plane equation in normal form
573:Normal to planes and polygons
5360:calculating a surface normal
5186:is the outward-pointing ray
3293:{\displaystyle \mathbf {n} }
2704:{\displaystyle \mathbf {W} }
2289:{\displaystyle \mathbf {t} }
2267:{\displaystyle \mathbf {n} }
2245:{\displaystyle \mathbf {W} }
2230:we can determine the matrix
1165:variables, then a normal to
866:{\displaystyle \mathbf {p} }
821:is a point on the plane and
488:{\displaystyle \mathbf {r} }
462:{\displaystyle \mathbf {T} }
246:The normal is often used in
4334:-th row is the gradient of
3823:then the hypersurface is a
3821:continuously differentiable
1354:{\displaystyle F(x,y,z)=0,}
75:to the curve at the point.
5409:
5289:. Northwestern University.
5163:
5160:Normal in geometric optics
2619: if and only if
2552: if and only if
2498: if and only if
2450: if and only if
2172:
898:which can be found as the
607:given by the general form
339:The normal direction to a
324:
318:
4923:is the plane of equation
4366:implicit function theorem
4307:{\displaystyle k\times n}
2252:that transforms a vector
2192:{\displaystyle 3\times 3}
1571:{\displaystyle z=f(x,y),}
1361:then a normal at a point
5153:signed distance function
5064:{\displaystyle (0,0,0).}
4872:{\displaystyle b\neq 0,}
4814:{\displaystyle (0,a,0).}
4735:{\displaystyle a\neq 0,}
4706:{\displaystyle (a,0,0),}
3831:of the points where the
2353:by the following logic:
174:differentiable manifolds
5301:"The Law of Reflection"
5234:Ellipsoid normal vector
5023:{\displaystyle (0,0,1)}
4985:{\displaystyle (0,0,0)}
4916:{\displaystyle (0,b,0)}
4773:{\displaystyle (0,0,1)}
3825:differentiable manifold
3698:satisfying an equation
1392:{\displaystyle (x,y,z)}
1304:{\displaystyle (x,y,z)}
1012:curvilinear coordinates
279:foot of a perpendicular
188:of a manifold at point
107:three-dimensional space
5175:
5085:
5065:
5024:
4986:
4946:
4917:
4873:
4844:
4815:
4774:
4736:
4707:
4663:
4643:
4623:
4562:
4539:
4521:of the variety is the
4515:
4488:
4466:
4465:{\displaystyle f_{i}.}
4436:
4412:
4389:
4358:
4357:{\displaystyle f_{i}.}
4328:
4308:
4288:of the variety is the
4278:
4151:
4131:
4102:
4065:
4028:
3813:
3789:
3769:
3692:
3629:A hypersurface may be
3623:
3587:
3552:
3488:
3413:
3377:
3294:
3272:
3228:
3199:
3170:
3017:
2987:
2962:
2914:
2882:
2853:
2825:
2800:
2746:
2705:
2681:
2407:
2382:
2347:
2319:
2290:
2268:
2246:
2224:
2193:
2144:inward-pointing normal
2127:
2096:
1969:
1892:
1651:
1572:
1528:
1499:
1477:
1451:
1393:
1355:
1305:
1263:
1241:
1156:
1136:
1116:
1000:
971:
954:
936:
892:
867:
845:
815:
786:
706:
660:
582:
563:
509:
489:
463:
433:
394:
336:
321:Frenet–Serret formulas
315:Normal to space curves
229:
202:
32:
24:
5358:Clear pseudocode for
5355:from Microsoft's MSDN
5194:at a given point. In
5190:to the surface of an
5173:
5123:), often adjusted by
5086:
5066:
5025:
4987:
4947:
4918:
4874:
4845:
4816:
4775:
4737:
4708:
4664:
4644:
4624:
4563:
4540:
4516:
4499:normal (affine) space
4489:
4467:
4437:
4413:
4390:
4359:
4329:
4309:
4279:
4152:
4132:
4103:
4066:
4029:
3814:
3790:
3770:
3693:
3624:
3588:
3586:{\displaystyle (n-1)}
3553:
3489:
3414:
3378:
3295:
3273:
3229:
3200:
3171:
3018:
2988:
2963:
2961:{\displaystyle (n-1)}
2915:
2883:
2854:
2826:
2801:
2747:
2706:
2682:
2408:
2383:
2348:
2320:
2291:
2269:
2247:
2225:
2194:
2148:outer-pointing normal
2125:
2097:
1970:
1893:
1652:
1573:
1529:
1500:
1478:
1452:
1394:
1356:
1306:
1273:as the set of points
1264:
1242:
1157:
1137:
1117:
1001:
972:
952:
937:
893:
868:
846:
816:
787:
707:
661:
580:
564:
510:
490:
464:
434:
395:
334:
325:Further information:
230:
203:
30:
22:
5386:3D computer graphics
5121:Lambert's cosine law
5113:3D computer graphics
5075:
5034:
4996:
4958:
4945:{\displaystyle y=b.}
4927:
4889:
4854:
4843:{\displaystyle x=a.}
4825:
4784:
4746:
4717:
4676:
4653:
4633:
4590:
4549:
4529:
4505:
4478:
4446:
4426:
4399:
4376:
4338:
4318:
4292:
4161:
4141:
4112:
4092:
4085:differential variety
4044:
3839:
3803:
3779:
3702:
3637:
3601:
3565:
3498:
3423:
3387:
3308:
3282:
3238:
3209:
3180:
3027:
2998:
2977:
2940:
2892:
2863:
2835:
2810:
2756:
2715:
2693:
2416:
2392:
2364:
2329:
2300:
2278:
2256:
2234:
2209:
2177:
2169:Transforming normals
2136:topological boundary
2112:Lipschitz continuous
1979:
1902:
1661:
1582:
1538:
1509:
1489:
1464:
1406:
1365:
1315:
1277:
1253:
1176:
1146:
1126:
1017:
981:
961:
905:
877:
855:
825:
796:
719:
670:
614:
522:
499:
477:
451:
407:
350:
248:3D computer graphics
216:
192:
80:vector of length one
5204:angle of reflection
5196:reflection of light
5166:Specular reflection
5135:digital compositing
4420:normal vector space
4368:, the variety is a
4108:-dimensional space
2140:normal orientations
1171:partial derivatives
441:radius of curvature
182:normal vector space
5335:Weisstein, Eric W.
5212:plane of incidence
5200:angle of incidence
5176:
5146:photometric stereo
5119:calculations (see
5081:
5061:
5020:
4982:
4942:
4913:
4869:
4840:
4811:
4770:
4732:
4703:
4659:
4639:
4619:
4561:{\displaystyle P.}
4558:
4535:
4511:
4484:
4462:
4432:
4411:{\displaystyle P,}
4408:
4388:{\displaystyle k.}
4385:
4354:
4324:
4304:
4274:
4147:
4127:
4098:
4078:-dimensional space
4061:
4024:
4013:
3975:
3943:
3809:
3785:
3765:
3688:
3619:
3583:
3548:
3484:
3409:
3373:
3364:
3290:
3268:
3224:
3195:
3166:
3013:
2993:-dimensional space
2983:
2958:
2932:-dimensional space
2910:
2878:
2849:
2821:
2796:
2742:
2701:
2677:
2675:
2403:
2378:
2343:
2315:
2286:
2264:
2242:
2220:
2189:
2128:
2092:
2076:
2048:
1965:
1888:
1872:
1844:
1806:
1759:
1647:
1568:
1524:
1495:
1476:{\displaystyle S.}
1473:
1447:
1389:
1351:
1301:
1259:
1237:
1152:
1132:
1112:
996:
967:
955:
932:
888:
863:
841:
811:
782:
702:
656:
583:
559:
505:
485:
459:
429:
390:
337:
301:Euclidean distance
277:(analogous to the
228:{\displaystyle P.}
225:
198:
131:of the surface at
84:unit normal vector
33:
25:
5102:surface integrals
5084:{\displaystyle z}
4662:{\displaystyle y}
4642:{\displaystyle x}
4538:{\displaystyle P}
4514:{\displaystyle P}
4487:{\displaystyle k}
4435:{\displaystyle P}
4327:{\displaystyle i}
4150:{\displaystyle n}
4101:{\displaystyle n}
4012:
3974:
3942:
3812:{\displaystyle F}
3788:{\displaystyle F}
2986:{\displaystyle n}
2928:Hypersurfaces in
2888:perpendicular to
2831:perpendicular to
2620:
2553:
2499:
2451:
2434:
2075:
2047:
1871:
1843:
1805:
1758:
1717:
1692:
1498:{\displaystyle S}
1262:{\displaystyle S}
1232:
1207:
1155:{\displaystyle t}
1135:{\displaystyle s}
970:{\displaystyle S}
557:
508:{\displaystyle s}
388:
201:{\displaystyle P}
5398:
5348:
5347:
5320:
5319:
5317:
5316:
5297:
5291:
5290:
5288:
5279:
5264:
5244:
5184:
5183:
5090:
5088:
5087:
5082:
5070:
5068:
5067:
5062:
5029:
5027:
5026:
5021:
4991:
4989:
4988:
4983:
4951:
4949:
4948:
4943:
4922:
4920:
4919:
4914:
4878:
4876:
4875:
4870:
4849:
4847:
4846:
4841:
4820:
4818:
4817:
4812:
4779:
4777:
4776:
4771:
4741:
4739:
4738:
4733:
4712:
4710:
4709:
4704:
4668:
4666:
4665:
4660:
4648:
4646:
4645:
4640:
4628:
4626:
4625:
4620:
4567:
4565:
4564:
4559:
4544:
4542:
4541:
4536:
4525:passing through
4520:
4518:
4517:
4512:
4493:
4491:
4490:
4485:
4471:
4469:
4468:
4463:
4458:
4457:
4441:
4439:
4438:
4433:
4417:
4415:
4414:
4409:
4395:At such a point
4394:
4392:
4391:
4386:
4363:
4361:
4360:
4355:
4350:
4349:
4333:
4331:
4330:
4325:
4313:
4311:
4310:
4305:
4283:
4281:
4280:
4275:
4270:
4266:
4265:
4264:
4246:
4245:
4231:
4230:
4212:
4208:
4207:
4206:
4188:
4187:
4173:
4172:
4156:
4154:
4153:
4148:
4136:
4134:
4133:
4128:
4126:
4125:
4120:
4107:
4105:
4104:
4099:
4070:
4068:
4067:
4062:
4054:
4033:
4031:
4030:
4025:
4019:
4015:
4014:
4011:
4010:
4009:
3996:
3988:
3976:
3973:
3972:
3971:
3958:
3950:
3944:
3941:
3940:
3939:
3926:
3918:
3907:
3903:
3902:
3901:
3883:
3882:
3870:
3869:
3846:
3818:
3816:
3815:
3810:
3794:
3792:
3791:
3786:
3774:
3772:
3771:
3766:
3752:
3751:
3733:
3732:
3720:
3719:
3697:
3695:
3694:
3689:
3684:
3683:
3665:
3664:
3652:
3651:
3628:
3626:
3625:
3620:
3615:
3614:
3609:
3592:
3590:
3589:
3584:
3557:
3555:
3554:
3549:
3547:
3543:
3542:
3541:
3523:
3522:
3505:
3494:then the vector
3493:
3491:
3490:
3485:
3474:
3473:
3464:
3463:
3445:
3444:
3435:
3434:
3418:
3416:
3415:
3410:
3405:
3397:
3382:
3380:
3379:
3374:
3369:
3368:
3361:
3360:
3349:
3336:
3335:
3330:
3299:
3297:
3296:
3291:
3289:
3277:
3275:
3274:
3269:
3233:
3231:
3230:
3225:
3223:
3222:
3217:
3204:
3202:
3201:
3196:
3194:
3193:
3188:
3175:
3173:
3172:
3167:
3162:
3161:
3150:
3144:
3143:
3119:
3118:
3113:
3107:
3106:
3094:
3093:
3088:
3079:
3075:
3074:
3073:
3049:
3048:
3034:
3022:
3020:
3019:
3014:
3012:
3011:
3006:
2992:
2990:
2989:
2984:
2967:
2965:
2964:
2959:
2919:
2917:
2916:
2911:
2906:
2905:
2900:
2887:
2885:
2884:
2879:
2877:
2876:
2871:
2858:
2856:
2855:
2850:
2845:
2830:
2828:
2827:
2822:
2820:
2805:
2803:
2802:
2797:
2792:
2791:
2790:
2780:
2779:
2751:
2749:
2748:
2743:
2729:
2728:
2727:
2710:
2708:
2707:
2702:
2700:
2686:
2684:
2683:
2678:
2676:
2672:
2667:
2663:
2659:
2658:
2657:
2642:
2641:
2640:
2634:
2621:
2618:
2615:
2608:
2597:
2593:
2592:
2591:
2590:
2580:
2579:
2578:
2572:
2554:
2551:
2548:
2541:
2530:
2529:
2528:
2518:
2500:
2497:
2494:
2487:
2470:
2452:
2449:
2443:
2435:
2432:
2430:
2412:
2410:
2409:
2404:
2399:
2387:
2385:
2384:
2379:
2374:
2352:
2350:
2349:
2344:
2339:
2324:
2322:
2321:
2316:
2314:
2313:
2308:
2295:
2293:
2292:
2287:
2285:
2273:
2271:
2270:
2265:
2263:
2251:
2249:
2248:
2243:
2241:
2229:
2227:
2226:
2221:
2216:
2198:
2196:
2195:
2190:
2152:oriented surface
2101:
2099:
2098:
2093:
2088:
2084:
2077:
2074:
2066:
2058:
2049:
2046:
2038:
2030:
1986:
1974:
1972:
1971:
1966:
1897:
1895:
1894:
1889:
1884:
1880:
1873:
1870:
1862:
1854:
1845:
1842:
1834:
1826:
1812:
1808:
1807:
1804:
1796:
1788:
1765:
1761:
1760:
1757:
1749:
1741:
1718:
1716:
1708:
1707:
1698:
1693:
1691:
1683:
1682:
1673:
1668:
1656:
1654:
1653:
1648:
1589:
1577:
1575:
1574:
1569:
1533:
1531:
1530:
1525:
1523:
1522:
1517:
1504:
1502:
1501:
1496:
1482:
1480:
1479:
1474:
1456:
1454:
1453:
1448:
1413:
1398:
1396:
1395:
1390:
1360:
1358:
1357:
1352:
1310:
1308:
1307:
1302:
1268:
1266:
1265:
1260:
1246:
1244:
1243:
1238:
1233:
1231:
1223:
1222:
1213:
1208:
1206:
1198:
1197:
1188:
1183:
1161:
1159:
1158:
1153:
1141:
1139:
1138:
1133:
1121:
1119:
1118:
1113:
1024:
1005:
1003:
1002:
997:
995:
994:
989:
976:
974:
973:
968:
941:
939:
938:
933:
928:
920:
912:
897:
895:
894:
889:
884:
872:
870:
869:
864:
862:
850:
848:
847:
842:
840:
832:
820:
818:
817:
812:
810:
809:
804:
791:
789:
788:
783:
778:
767:
756:
755:
750:
726:
711:
709:
708:
703:
677:
665:
663:
662:
657:
568:
566:
565:
560:
558:
556:
552:
546:
545:
540:
534:
529:
514:
512:
511:
506:
494:
492:
491:
486:
484:
468:
466:
465:
460:
458:
438:
436:
435:
430:
428:
427:
399:
397:
396:
391:
389:
387:
383:
377:
376:
371:
365:
357:
327:Curvature vector
234:
232:
231:
226:
207:
205:
204:
199:
136:
126:
97:
88:curvature vector
5408:
5407:
5401:
5400:
5399:
5397:
5396:
5395:
5381:Vector calculus
5366:
5365:
5338:"Normal Vector"
5333:
5332:
5329:
5324:
5323:
5314:
5312:
5299:
5298:
5294:
5286:
5281:
5280:
5276:
5271:
5262:
5242:
5224:
5181:
5180:
5168:
5162:
5142:computer vision
5097:
5073:
5072:
5032:
5031:
4994:
4993:
4956:
4955:
4925:
4924:
4887:
4886:
4852:
4851:
4823:
4822:
4782:
4781:
4744:
4743:
4715:
4714:
4674:
4673:
4651:
4650:
4631:
4630:
4588:
4587:
4580:
4547:
4546:
4527:
4526:
4523:affine subspace
4503:
4502:
4476:
4475:
4449:
4444:
4443:
4424:
4423:
4397:
4396:
4374:
4373:
4341:
4336:
4335:
4316:
4315:
4290:
4289:
4286:Jacobian matrix
4256:
4237:
4236:
4232:
4222:
4198:
4179:
4178:
4174:
4164:
4159:
4158:
4139:
4138:
4115:
4110:
4109:
4090:
4089:
4080:
4042:
4041:
4001:
3997:
3989:
3963:
3959:
3951:
3931:
3927:
3919:
3915:
3911:
3893:
3874:
3861:
3860:
3856:
3837:
3836:
3801:
3800:
3797:scalar function
3777:
3776:
3743:
3724:
3711:
3700:
3699:
3675:
3656:
3643:
3635:
3634:
3604:
3599:
3598:
3563:
3562:
3533:
3514:
3513:
3509:
3496:
3495:
3465:
3455:
3436:
3426:
3421:
3420:
3385:
3384:
3363:
3362:
3344:
3342:
3337:
3325:
3318:
3306:
3305:
3304:of the matrix
3280:
3279:
3236:
3235:
3212:
3207:
3206:
3183:
3178:
3177:
3145:
3129:
3108:
3098:
3083:
3059:
3040:
3039:
3035:
3025:
3024:
3001:
2996:
2995:
2975:
2974:
2938:
2937:
2934:
2895:
2890:
2889:
2866:
2861:
2860:
2833:
2832:
2808:
2807:
2781:
2768:
2754:
2753:
2718:
2713:
2712:
2691:
2690:
2674:
2673:
2648:
2647:
2643:
2629:
2613:
2612:
2581:
2567:
2566:
2562:
2546:
2545:
2519:
2492:
2491:
2446:
2414:
2413:
2390:
2389:
2362:
2361:
2327:
2326:
2303:
2298:
2297:
2276:
2275:
2254:
2253:
2232:
2231:
2207:
2206:
2200:
2175:
2174:
2171:
2156:right-hand rule
2120:
2067:
2059:
2039:
2031:
2024:
2020:
1977:
1976:
1900:
1899:
1863:
1855:
1835:
1827:
1820:
1816:
1797:
1789:
1773:
1769:
1750:
1742:
1726:
1722:
1709:
1699:
1684:
1674:
1659:
1658:
1580:
1579:
1536:
1535:
1512:
1507:
1506:
1487:
1486:
1462:
1461:
1404:
1403:
1363:
1362:
1313:
1312:
1275:
1274:
1251:
1250:
1224:
1214:
1199:
1189:
1174:
1173:
1144:
1143:
1124:
1123:
1015:
1014:
1010:by a system of
984:
979:
978:
959:
958:
947:
903:
902:
875:
874:
853:
852:
823:
822:
799:
794:
793:
745:
717:
716:
668:
667:
612:
611:
575:
547:
535:
520:
519:
497:
496:
495:and arc-length
475:
474:
449:
448:
416:
405:
404:
378:
366:
348:
347:
329:
323:
317:
292:normal distance
241:smooth surfaces
214:
213:
190:
189:
178:Euclidean space
153:component of a
132:
122:
100:opposite vector
98:results in the
95:
17:
12:
11:
5:
5406:
5405:
5402:
5394:
5393:
5388:
5383:
5378:
5368:
5367:
5364:
5363:
5356:
5349:
5328:
5327:External links
5325:
5322:
5321:
5292:
5273:
5272:
5270:
5267:
5266:
5265:
5256:
5251:
5245:
5236:
5231:
5223:
5220:
5192:optical medium
5164:Main article:
5161:
5158:
5157:
5156:
5149:
5138:
5128:
5125:normal mapping
5109:
5096:
5093:
5080:
5060:
5057:
5054:
5051:
5048:
5045:
5042:
5039:
5019:
5016:
5013:
5010:
5007:
5004:
5001:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4941:
4938:
4935:
4932:
4912:
4909:
4906:
4903:
4900:
4897:
4894:
4868:
4865:
4862:
4859:
4850:Similarly, if
4839:
4836:
4833:
4830:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4769:
4766:
4763:
4760:
4757:
4754:
4751:
4731:
4728:
4725:
4722:
4702:
4699:
4696:
4693:
4690:
4687:
4684:
4681:
4658:
4649:-axis and the
4638:
4618:
4615:
4612:
4608:
4605:
4602:
4599:
4595:
4579:
4576:
4573:
4557:
4554:
4534:
4510:
4483:
4461:
4456:
4452:
4431:
4407:
4404:
4384:
4381:
4353:
4348:
4344:
4323:
4303:
4300:
4297:
4273:
4269:
4263:
4259:
4255:
4252:
4249:
4244:
4240:
4235:
4229:
4225:
4221:
4218:
4215:
4211:
4205:
4201:
4197:
4194:
4191:
4186:
4182:
4177:
4171:
4167:
4146:
4124:
4119:
4097:
4079:
4072:
4060:
4057:
4053:
4049:
4023:
4018:
4008:
4004:
4000:
3995:
3992:
3985:
3982:
3979:
3970:
3966:
3962:
3957:
3954:
3947:
3938:
3934:
3930:
3925:
3922:
3914:
3910:
3906:
3900:
3896:
3892:
3889:
3886:
3881:
3877:
3873:
3868:
3864:
3859:
3855:
3852:
3849:
3845:
3808:
3784:
3764:
3761:
3758:
3755:
3750:
3746:
3742:
3739:
3736:
3731:
3727:
3723:
3718:
3714:
3710:
3707:
3687:
3682:
3678:
3674:
3671:
3668:
3663:
3659:
3655:
3650:
3646:
3642:
3618:
3613:
3608:
3582:
3579:
3576:
3573:
3570:
3546:
3540:
3536:
3532:
3529:
3526:
3521:
3517:
3512:
3508:
3504:
3483:
3480:
3477:
3472:
3468:
3462:
3458:
3454:
3451:
3448:
3443:
3439:
3433:
3429:
3408:
3404:
3400:
3396:
3392:
3372:
3367:
3359:
3356:
3353:
3348:
3343:
3341:
3338:
3334:
3329:
3324:
3323:
3321:
3316:
3313:
3288:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3221:
3216:
3192:
3187:
3165:
3160:
3157:
3154:
3149:
3142:
3139:
3136:
3132:
3128:
3125:
3122:
3117:
3112:
3105:
3101:
3097:
3092:
3087:
3082:
3078:
3072:
3069:
3066:
3062:
3058:
3055:
3052:
3047:
3043:
3038:
3033:
3010:
3005:
2982:
2957:
2954:
2951:
2948:
2945:
2933:
2926:
2909:
2904:
2899:
2875:
2870:
2848:
2844:
2840:
2819:
2815:
2795:
2789:
2784:
2778:
2775:
2771:
2767:
2764:
2761:
2741:
2738:
2735:
2732:
2726:
2721:
2699:
2671:
2666:
2662:
2656:
2651:
2646:
2639:
2633:
2628:
2625:
2616:
2614:
2611:
2607:
2603:
2600:
2596:
2589:
2584:
2577:
2571:
2565:
2561:
2558:
2549:
2547:
2544:
2540:
2536:
2533:
2527:
2522:
2517:
2513:
2510:
2507:
2504:
2495:
2493:
2490:
2486:
2482:
2479:
2476:
2473:
2469:
2465:
2462:
2459:
2456:
2447:
2442:
2438:
2429:
2425:
2422:
2421:
2402:
2398:
2377:
2373:
2370:
2342:
2338:
2335:
2312:
2307:
2296:into a vector
2284:
2262:
2240:
2219:
2215:
2188:
2185:
2182:
2170:
2167:
2119:
2116:
2104:singular point
2091:
2087:
2083:
2080:
2073:
2070:
2065:
2062:
2055:
2052:
2045:
2042:
2037:
2034:
2027:
2023:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1985:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1887:
1883:
1879:
1876:
1869:
1866:
1861:
1858:
1851:
1848:
1841:
1838:
1833:
1830:
1823:
1819:
1815:
1811:
1803:
1800:
1795:
1792:
1785:
1782:
1779:
1776:
1772:
1768:
1764:
1756:
1753:
1748:
1745:
1738:
1735:
1732:
1729:
1725:
1721:
1715:
1712:
1706:
1702:
1696:
1690:
1687:
1681:
1677:
1671:
1667:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1588:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1521:
1516:
1494:
1485:For a surface
1472:
1469:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1412:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1258:
1236:
1230:
1227:
1221:
1217:
1211:
1205:
1202:
1196:
1192:
1186:
1182:
1151:
1131:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1023:
993:
988:
966:
946:
943:
931:
927:
923:
919:
915:
911:
887:
883:
861:
839:
835:
831:
808:
803:
781:
777:
773:
770:
766:
762:
759:
754:
749:
744:
741:
738:
735:
732:
729:
725:
701:
698:
695:
692:
689:
686:
683:
680:
676:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
609:plane equation
574:
571:
570:
569:
555:
551:
544:
539:
532:
528:
504:
483:
471:tangent vector
457:
426:
423:
419:
415:
412:
401:
400:
386:
382:
375:
370:
363:
360:
356:
319:Main article:
316:
313:
224:
221:
197:
111:surface normal
15:
13:
10:
9:
6:
4:
3:
2:
5404:
5403:
5392:
5391:Orthogonality
5389:
5387:
5384:
5382:
5379:
5377:
5374:
5373:
5371:
5361:
5357:
5354:
5350:
5345:
5344:
5339:
5336:
5331:
5330:
5326:
5310:
5306:
5302:
5296:
5293:
5285:
5278:
5275:
5268:
5260:
5259:Vertex normal
5257:
5255:
5252:
5249:
5246:
5240:
5239:Normal bundle
5237:
5235:
5232:
5229:
5226:
5225:
5221:
5219:
5217:
5216:reflected ray
5213:
5209:
5205:
5201:
5197:
5193:
5189:
5188:perpendicular
5185:
5172:
5167:
5159:
5154:
5150:
5147:
5143:
5139:
5136:
5132:
5131:Render layers
5129:
5126:
5122:
5118:
5114:
5110:
5107:
5106:vector fields
5103:
5099:
5098:
5094:
5092:
5078:
5058:
5052:
5049:
5046:
5043:
5040:
5014:
5011:
5008:
5005:
5002:
4976:
4973:
4970:
4967:
4964:
4954:At the point
4952:
4939:
4936:
4933:
4930:
4907:
4904:
4901:
4898:
4895:
4884:
4883:
4866:
4863:
4860:
4857:
4837:
4834:
4831:
4828:
4808:
4802:
4799:
4796:
4793:
4790:
4764:
4761:
4758:
4755:
4752:
4729:
4726:
4723:
4720:
4700:
4694:
4691:
4688:
4685:
4682:
4670:
4656:
4636:
4616:
4613:
4610:
4606:
4603:
4600:
4597:
4593:
4585:
4577:
4575:
4571:
4568:
4555:
4552:
4532:
4524:
4508:
4500:
4495:
4481:
4472:
4459:
4454:
4450:
4429:
4421:
4405:
4402:
4382:
4379:
4371:
4367:
4351:
4346:
4342:
4321:
4314:matrix whose
4301:
4298:
4295:
4287:
4271:
4267:
4261:
4257:
4253:
4250:
4247:
4242:
4238:
4233:
4227:
4223:
4219:
4216:
4213:
4209:
4203:
4199:
4195:
4192:
4189:
4184:
4180:
4175:
4169:
4165:
4144:
4122:
4095:
4087:
4086:
4077:
4073:
4071:
4058:
4039:
4034:
4021:
4016:
4006:
4002:
3993:
3983:
3980:
3977:
3968:
3964:
3955:
3945:
3936:
3932:
3923:
3912:
3908:
3904:
3898:
3894:
3890:
3887:
3884:
3879:
3875:
3871:
3866:
3862:
3857:
3853:
3847:
3834:
3830:
3829:neighbourhood
3826:
3822:
3806:
3798:
3782:
3762:
3759:
3756:
3748:
3744:
3740:
3737:
3734:
3729:
3725:
3721:
3716:
3712:
3705:
3680:
3676:
3672:
3669:
3666:
3661:
3657:
3653:
3648:
3644:
3632:
3616:
3611:
3596:
3595:hypersurfaces
3593:-dimensional
3577:
3574:
3571:
3559:
3558:is a normal.
3544:
3538:
3534:
3530:
3527:
3524:
3519:
3515:
3510:
3506:
3481:
3478:
3475:
3470:
3466:
3460:
3456:
3452:
3449:
3446:
3441:
3437:
3431:
3427:
3406:
3398:
3390:
3370:
3365:
3357:
3354:
3351:
3339:
3332:
3319:
3314:
3311:
3303:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3219:
3190:
3163:
3158:
3155:
3152:
3140:
3137:
3134:
3130:
3126:
3123:
3120:
3115:
3103:
3099:
3095:
3090:
3080:
3076:
3070:
3067:
3064:
3060:
3056:
3053:
3050:
3045:
3041:
3036:
3008:
2994:
2980:
2971:
2968:-dimensional
2952:
2949:
2946:
2931:
2927:
2925:
2921:
2920:as required.
2907:
2846:
2838:
2813:
2793:
2776:
2773:
2769:
2762:
2759:
2739:
2736:
2733:
2730:
2719:
2687:
2664:
2660:
2649:
2644:
2626:
2623:
2601:
2594:
2582:
2563:
2559:
2556:
2534:
2511:
2505:
2502:
2480:
2474:
2463:
2457:
2454:
2436:
2423:
2400:
2388:We must find
2375:
2359:
2354:
2340:
2217:
2203:
2186:
2183:
2180:
2168:
2166:
2164:
2159:
2157:
2153:
2149:
2145:
2141:
2137:
2133:
2124:
2117:
2115:
2113:
2109:
2105:
2089:
2085:
2081:
2078:
2071:
2063:
2053:
2050:
2043:
2035:
2025:
2021:
2017:
2011:
2008:
2005:
2002:
1999:
1993:
1987:
1962:
1959:
1956:
1950:
1947:
1944:
1938:
1935:
1932:
1929:
1923:
1920:
1917:
1914:
1911:
1905:
1885:
1881:
1877:
1874:
1867:
1859:
1849:
1846:
1839:
1831:
1821:
1817:
1813:
1809:
1801:
1793:
1783:
1780:
1777:
1774:
1770:
1766:
1762:
1754:
1746:
1736:
1733:
1730:
1727:
1723:
1719:
1713:
1694:
1688:
1669:
1644:
1635:
1632:
1629:
1623:
1620:
1617:
1614:
1611:
1605:
1599:
1596:
1593:
1565:
1559:
1556:
1553:
1547:
1544:
1541:
1519:
1492:
1483:
1470:
1467:
1460:
1444:
1438:
1435:
1432:
1429:
1426:
1420:
1414:
1402:
1383:
1380:
1377:
1374:
1371:
1348:
1345:
1342:
1336:
1333:
1330:
1327:
1324:
1318:
1295:
1292:
1289:
1286:
1283:
1272:
1256:
1249:If a surface
1247:
1234:
1228:
1209:
1203:
1184:
1172:
1168:
1164:
1149:
1129:
1109:
1100:
1097:
1094:
1088:
1085:
1079:
1076:
1073:
1067:
1064:
1058:
1055:
1052:
1046:
1040:
1034:
1031:
1028:
1013:
1009:
1008:parameterized
991:
964:
951:
944:
942:
929:
921:
913:
901:
900:cross product
885:
833:
806:
779:
771:
768:
760:
757:
752:
742:
736:
733:
730:
713:
712:is a normal.
696:
693:
690:
687:
684:
678:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
617:
610:
606:
601:
599:
598:cross product
595:
591:
588:
579:
572:
553:
530:
518:
517:
516:
502:
472:
446:
442:
424:
421:
417:
413:
410:
384:
361:
358:
346:
345:
344:
342:
333:
328:
322:
314:
312:
310:
307:and its foot
306:
302:
298:
294:
293:
288:
284:
280:
276:
272:
267:
265:
264:Phong shading
261:
257:
253:
249:
244:
242:
238:
237:smooth curves
222:
219:
211:
210:tangent space
195:
187:
183:
179:
175:
170:
168:
164:
163:orthogonality
160:
159:normal vector
156:
152:
148:
144:
140:
135:
130:
129:tangent plane
125:
120:
116:
112:
108:
103:
101:
93:
89:
85:
81:
76:
74:
70:
66:
62:
61:perpendicular
58:
54:
50:
46:
42:
38:
29:
21:
5341:
5313:. Retrieved
5304:
5295:
5277:
5248:Pseudovector
5208:incident ray
5179:
5177:
4953:
4882:normal plane
4880:
4671:
4583:
4581:
4569:
4498:
4496:
4473:
4419:
4083:
4081:
4075:
4037:
4035:
3560:
2935:
2929:
2922:
2688:
2357:
2355:
2204:
2201:
2163:pseudovector
2160:
2147:
2143:
2139:
2129:
1484:
1311:satisfying
1248:
1166:
977:in 3D space
956:
714:
602:
584:
443:(reciprocal
402:
338:
308:
304:
296:
290:
286:
282:
274:
270:
268:
256:flat shading
252:light source
245:
186:normal space
185:
181:
171:
167:right angles
158:
150:
142:
138:
133:
123:
114:
113:, or simply
110:
104:
83:
82:is called a
77:
73:tangent line
64:
40:
34:
4672:At a point
4501:at a point
4038:normal line
3795:is a given
2711:such that
2132:unit length
2118:Orientation
666:the vector
592:(such as a
341:space curve
295:of a point
137:. The word
69:plane curve
65:normal line
5370:Categories
5315:2008-03-31
5269:References
5228:Dual space
5182:normal ray
4157:variables
3302:null space
2970:hyperplane
1271:implicitly
59:) that is
5343:MathWorld
5282:Ying Wu.
4861:≠
4724:≠
4299:×
4251:…
4217:…
4193:…
3999:∂
3991:∂
3981:…
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3953:∂
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3921:∂
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3670:…
3575:−
3528:…
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3355:−
3340:⋯
3263:−
3254:…
3156:−
3138:−
3124:⋯
3068:−
3054:…
2950:−
2903:′
2874:′
2774:−
2689:Choosing
2475:⋅
2311:′
2184:×
2150:. For an
2069:∂
2061:∂
2054:−
2041:∂
2033:∂
2026:−
1991:∇
1936:−
1865:∂
1857:∂
1850:−
1837:∂
1829:∂
1822:−
1799:∂
1791:∂
1767:×
1752:∂
1744:∂
1711:∂
1701:∂
1695:×
1686:∂
1676:∂
1418:∇
1269:is given
1226:∂
1216:∂
1210:×
1201:∂
1191:∂
922:×
445:curvature
422:−
418:κ
121:at point
92:curvature
78:A normal
5376:Surfaces
5309:Archived
5222:See also
5210:(on the
5202:and the
5117:lighting
4572:verbatim
4370:manifold
3833:gradient
3383:meaning
2358:n′
1401:gradient
594:triangle
303:between
260:vertices
47:(e.g. a
37:geometry
5091:-axis.
4669:-axis.
4578:Example
4364:By the
3827:in the
3631:locally
3300:in the
2936:For an
1975:giving
1657:giving
590:polygon
469:is the
439:is the
119:surface
117:, to a
5198:, the
4713:where
3775:where
3176:where
2859:or an
2356:Write
2142:, the
1457:since
792:where
603:For a
587:convex
585:For a
403:where
289:. The
180:. The
157:, the
151:normal
149:, the
143:normal
139:normal
115:normal
57:vector
45:object
43:is an
41:normal
5287:(PDF)
3799:. If
1122:with
605:plane
155:force
147:plane
145:to a
67:to a
55:, or
5178:The
5115:for
5095:Uses
5030:and
4879:the
4780:and
4582:Let
4497:The
4418:the
4284:The
4036:The
3234:for
2146:and
2108:cone
1163:real
1142:and
873:and
343:is:
271:foot
269:The
254:for
239:and
109:, a
86:. A
49:line
39:, a
5351:An
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1006:is
447:);
212:at
184:or
169:).
105:In
53:ray
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