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Normal (geometry)
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<h2 id="normal-to-space-curves">Normal to space curves</h2> <p class="note">Main article: <a href="/facts/Frenet%E2%80%93Serret_formulas/gwE2vhLI">Frenet–Serret formulas</a></p> <p class="note">Further information: <a href="/facts/Curvature_vector/UbbkBrs0">Curvature vector</a></p> <p>The normal direction to a <a href="/facts/Space_curve/xBQ0pTBW">space curve</a> is: </p> N = R d T d s {\displaystyle \mathbf {N} =R{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}} <p>where R = κ − 1 {\displaystyle R=\kappa ^{-1}} is the <a href="/facts/Radius_of_curvature/FYeD7eyl">radius of curvature</a> (reciprocal <a href="/facts/Curvature/RFCAEsj8">curvature</a>); T {\displaystyle \mathbf {T} } is the <a href="/facts/Tangent_vector/UEq5yffm">tangent vector</a>, in terms of the curve position r {\displaystyle \mathbf {r} } and arc-length s {\displaystyle s} : </p> T = d r d s {\displaystyle \mathbf {T} ={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} s}}} <h2 id="normal-to-planes-and-polygons">Normal to planes and polygons</h2> <p>For a <a href="/facts/Convex_set/vdAuJRJl">convex</a> <a href="/facts/Polygon/yAq7RjPz">polygon</a> (such as a <a href="/facts/Triangle/cRXUwsMn">triangle</a>), a surface normal can be calculated as the vector <a href="/facts/Cross_product/uRBJzkcW">cross product</a> of two (non-parallel) edges of the polygon. </p><p>For a <a href="/facts/Plane_(mathematics)/tAUtRFcn">plane</a> given by the general form <a href="/facts/Plane_equation/tAUtRFcn">plane equation</a> a x + b y + c z + d = 0 , {\displaystyle ax+by+cz+d=0,} the vector n = ( a , b , c ) {\displaystyle \mathbf {n} =(a,b,c)} is a normal. </p><p>For a plane whose equation is given in parametric form r ( s , t ) = r 0 + s p + t q , {\displaystyle \mathbf {r} (s,t)=\mathbf {r} _{0}+s\mathbf {p} +t\mathbf {q} ,} where r 0 {\displaystyle \mathbf {r} _{0}} is a point on the plane and p , q {\displaystyle \mathbf {p} ,\mathbf {q} } are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both p {\displaystyle \mathbf {p} } and q , {\displaystyle \mathbf {q} ,} which can be found as the <a href="/facts/Cross_product/uRBJzkcW">cross product</a> n = p × q . {\displaystyle \mathbf {n} =\mathbf {p} \times \mathbf {q} .} </p> <h2 id="normal-to-general-surfaces-in-3d-space">Normal to general surfaces in 3D space</h2> <p>If a (possibly non-flat) surface S {\displaystyle S} in 3D space R 3 {\displaystyle \mathbb {R} ^{3}} is <a href="/facts/Coordinate_system/Mtd0q4SP">parameterized</a> by a system of <a href="/facts/Curvilinear_coordinates/RQD7Cajd">curvilinear coordinates</a> r ( s , t ) = ( x ( s , t ) , y ( s , t ) , z ( s , t ) ) , {\displaystyle \mathbf {r} (s,t)=(x(s,t),y(s,t),z(s,t)),} with s {\displaystyle s} and t {\displaystyle t} <a href="/facts/Real_number/R02gw5Pb">real</a> variables, then a normal to <i>S</i> is by definition a normal to a tangent plane, given by the cross product of the <a href="/facts/Partial_derivative/JWHsBkAu">partial derivatives</a> n = ∂ r ∂ s × ∂ r ∂ t . {\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}.} </p><p>If a surface S {\displaystyle S} is given <a href="/facts/Implicit_function/UJmQ2Cyk">implicitly</a> as the set of points ( x , y , z ) {\displaystyle (x,y,z)} satisfying F ( x , y , z ) = 0 , {\displaystyle F(x,y,z)=0,} then a normal at a point ( x , y , z ) {\displaystyle (x,y,z)} on the surface is given by the <a href="/facts/Gradient/TsR7uukj">gradient</a> n = ∇ F ( x , y , z ) . {\displaystyle \mathbf {n} =\nabla F(x,y,z).} since <a href="/facts/Level_set/JUmNbdNh">the gradient at any point is perpendicular to the level set</a> S . {\displaystyle S.} </p><p>For a surface S {\displaystyle S} in R 3 {\displaystyle \mathbb {R} ^{3}} given as the graph of a function z = f ( x , y ) , {\displaystyle z=f(x,y),} an upward-pointing normal can be found either from the parametrization r ( x , y ) = ( x , y , f ( x , y ) ) , {\displaystyle \mathbf {r} (x,y)=(x,y,f(x,y)),} giving n = ∂ r ∂ x × ∂ r ∂ y = ( 1 , 0 , ∂ f ∂ x ) × ( 0 , 1 , ∂ f ∂ y ) = ( − ∂ f ∂ x , − ∂ f ∂ y , 1 ) ; {\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);} or more simply from its implicit form F ( x , y , z ) = z − f ( x , y ) = 0 , {\displaystyle F(x,y,z)=z-f(x,y)=0,} giving n = ∇ F ( x , y , z ) = ( − ∂ f ∂ x , − ∂ f ∂ y , 1 ) . {\displaystyle \mathbf {n} =\nabla F(x,y,z)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right).} Since a surface does not have a tangent plane at a <a href="/facts/Singularity_(mathematics)/Y5sMQY4I">singular point</a>, it has no well-defined normal at that point: for example, the vertex of a <a href="/facts/Cone_(geometry)/2c9GIzYD">cone</a>. In general, it is possible to define a normal almost everywhere for a surface that is <a href="/facts/Lipschitz_continuous/Hw20EPEU">Lipschitz continuous</a>. </p> <h3>Orientation</h3> <p>The normal to a (hyper)surface is usually scaled to have <a href="/facts/Unit_vector/GyaXneDn">unit length</a>, but it does not have a unique direction, since its opposite is also a unit normal. For a surface which is the <a href="/facts/Boundary_(topology)/WgdH18kp">topological boundary</a> of a set in three dimensions, one can distinguish between two normal orientations, the inward-pointing normal and outer-pointing normal. For an <a href="/facts/Orientability/1SEQM2Pt">oriented surface</a>, the normal is usually determined by the <a href="/facts/Right-hand_rule/LJe0Qqh4">right-hand rule</a> or its analog in higher dimensions. </p><p>If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a <a href="/facts/Pseudovector/c3b5GgWD">pseudovector</a>. </p> <h3>Transforming normals</h3> <p class="note">in this section we only use the upper 3 × 3 {\displaystyle 3\times 3} matrix, as translation is irrelevant to the calculation</p> <p>When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals. </p><p>Specifically, given a 3×3 transformation matrix M , {\displaystyle \mathbf {M} ,} we can determine the matrix W {\displaystyle \mathbf {W} } that transforms a vector n {\displaystyle \mathbf {n} } perpendicular to the tangent plane t {\displaystyle \mathbf {t} } into a vector n ′ {\displaystyle \mathbf {n} ^{\prime }} perpendicular to the transformed tangent plane M t , {\displaystyle \mathbf {Mt} ,} by the following logic: </p><p>Write n′ as W n . {\displaystyle \mathbf {Wn} .} We must find W . {\displaystyle \mathbf {W} .} W n is perpendicular to M t if and only if 0 = ( W n ) ⋅ ( M t ) if and only if 0 = ( W n ) T ( M t ) if and only if 0 = ( n T W T ) ( M t ) if and only if 0 = n T ( W T M ) t {\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}}} </p><p>Choosing W {\displaystyle \mathbf {W} } such that W T M = I , {\displaystyle W^{\mathrm {T} }M=I,} or W = ( M − 1 ) T , {\displaystyle W=(M^{-1})^{\mathrm {T} },} will satisfy the above equation, giving a W n {\displaystyle W\mathbb {n} } perpendicular to M t , {\displaystyle M\mathbb {t} ,} or an n ′ {\displaystyle \mathbf {n} ^{\prime }} perpendicular to t ′ , {\displaystyle \mathbf {t} ^{\prime },} as required. </p><p>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. </p> <h2 id="hypersurfaces-in-n-dimensional-space">Hypersurfaces in <i>n</i>-dimensional space</h2> <p>For an ( n − 1 ) {\displaystyle (n-1)} -dimensional <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a> in <a href="/facts/N-dimensional_space/0ktmUyac"> n {\displaystyle n} -dimensional space</a> R n {\displaystyle \mathbb {R} ^{n}} given by its parametric representation r ( t 1 , … , t n − 1 ) = p 0 + t 1 v 1 + ⋯ + t n − 1 v n − 1 , {\displaystyle \mathbf {r} \left(t_{1},\ldots ,t_{n-1}\right)=\mathbf {p} _{0}+t_{1}\mathbf {v} _{1}+\cdots +t_{n-1}\mathbf {v} _{n-1},} where p 0 {\displaystyle \mathbf {p} _{0}} is a point on the hyperplane and v i {\displaystyle \mathbf {v} _{i}} for i = 1 , … , n − 1 {\displaystyle i=1,\ldots ,n-1} are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector n {\displaystyle \mathbf {n} } in the <a href="/facts/Null_space/adhGUYa0">null space</a> of the matrix V = [ v 1 ⋯ v n − 1 ] , {\displaystyle V={\begin{bmatrix}\mathbf {v} _{1}&\cdots &\mathbf {v} _{n-1}\end{bmatrix}},} meaning V n = 0 {\displaystyle V\mathbf {n} =\mathbf {0} } . 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 a 1 x 1 + ⋯ + a n x n = c {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=c} , then the vector n = ( a 1 , … , a n ) {\displaystyle \mathbf {n} =\left(a_{1},\ldots ,a_{n}\right)} is a normal. </p><p>The definition of a normal to a surface in three-dimensional space can be extended to ( n − 1 ) {\displaystyle (n-1)} -dimensional <a href="/facts/Hypersurface/cTvtLVsV">hypersurfaces</a> in R n {\displaystyle \mathbb {R} ^{n}} . A hypersurface may be <a href="/facts/Local_property/sdBEwIy2">locally</a> defined implicitly as the set of points ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} satisfying an equation F ( x 1 , x 2 , … , x n ) = 0 {\displaystyle F(x_{1},x_{2},\ldots ,x_{n})=0} , where F {\displaystyle F} is a given <a href="/facts/Scalar_field/8MtPTGob">scalar function</a>. If F {\displaystyle F} is <a href="/facts/Continuously_differentiable/jQqTgLk1">continuously differentiable</a> then the hypersurface is a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a> in the <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> of the points where the <a href="/facts/Gradient/TsR7uukj">gradient</a> is not zero. At these points a normal vector is given by the gradient: n = ∇ F ( x 1 , x 2 , … , x n ) = ( ∂ F ∂ x 1 , ∂ F ∂ x 2 , … , ∂ F ∂ x n ) . {\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)\,.} </p><p>The normal line is the one-dimensional subspace with basis { n } . {\displaystyle \{\mathbf {n} \}.} </p><p>A vector that is normal to the space spanned by the linearly independent vectors v1, ..., v<i>r</i>−1 and falls within the r-dimensional space spanned by the linearly independent vectors v1, ..., v<i>r</i> is given by the r-th column of the matrix Λ = <i>V</i>(<i>V</i>T<i>V</i>)−1, where the matrix <i>V</i> = (v1, ..., v<i>r</i>) is the juxtaposition of the r column vectors. (Proof: <i>V</i>TΛ = <i>I</i> so each of v1, ..., v<i>r</i>−1 is perpendicular to the last column of Λ.) This formula works even when r is less than the dimension of the Euclidean space n. The formula simplifies to Λ = (<i>V</i>T)−1 when <i>r</i> = <i>n</i>. </p> <h2 id="varieties-defined-by-implicit-equations-in-n-dimensional-space">Varieties defined by implicit equations in <i>n</i>-dimensional space</h2> <p>A <a href="/facts/Analytic_variety/qFKYv9NA">differential variety</a> defined by implicit equations in the n {\displaystyle n} -dimensional space R n {\displaystyle \mathbb {R} ^{n}} is the set of the common zeros of a finite set of differentiable functions in n {\displaystyle n} variables f 1 ( x 1 , … , x n ) , … , f k ( x 1 , … , x n ) . {\displaystyle f_{1}\left(x_{1},\ldots ,x_{n}\right),\ldots ,f_{k}\left(x_{1},\ldots ,x_{n}\right).} The <a href="/facts/Jacobian_matrix/0tNv6o8j">Jacobian matrix</a> of the variety is the k × n {\displaystyle k\times n} matrix whose i {\displaystyle i} -th row is the gradient of f i . {\displaystyle f_{i}.} By the <a href="/facts/Implicit_function_theorem/r4h0RpWw">implicit function theorem</a>, the variety is a <a href="/facts/Manifold/rXPERJtu">manifold</a> in the neighborhood of a point where the Jacobian matrix has rank k . {\displaystyle k.} At such a point P , {\displaystyle P,} the normal vector space is the vector space generated by the values at P {\displaystyle P} of the gradient vectors of the f i . {\displaystyle f_{i}.} </p><p>In other words, a variety is defined as the intersection of k {\displaystyle k} hypersurfaces, and the normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point. </p><p>The normal (affine) space at a point P {\displaystyle P} of the variety is the <a href="/facts/Affine_subspace/tlvI3oX1">affine subspace</a> passing through P {\displaystyle P} and generated by the normal vector space at P . {\displaystyle P.} </p><p>These definitions may be extended verbatim to the points where the variety is not a manifold. </p> <h3>Example</h3> <p>Let <i>V</i> be the variety defined in the 3-dimensional space by the equations x y = 0 , z = 0. {\displaystyle x\,y=0,\quad z=0.} This variety is the union of the x {\displaystyle x} -axis and the y {\displaystyle y} -axis. </p><p>At a point ( a , 0 , 0 ) , {\displaystyle (a,0,0),} where a ≠ 0 , {\displaystyle a\neq 0,} the rows of the Jacobian matrix are ( 0 , 0 , 1 ) {\displaystyle (0,0,1)} and ( 0 , a , 0 ) . {\displaystyle (0,a,0).} Thus the normal affine space is the plane of equation x = a . {\displaystyle x=a.} Similarly, if b ≠ 0 , {\displaystyle b\neq 0,} the <i><a href="/facts/Normal_plane_(geometry)/fflIjA8j">normal plane</a></i> at ( 0 , b , 0 ) {\displaystyle (0,b,0)} is the plane of equation y = b . {\displaystyle y=b.} </p><p>At the point ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} the rows of the Jacobian matrix are ( 0 , 0 , 1 ) {\displaystyle (0,0,1)} and ( 0 , 0 , 0 ) . {\displaystyle (0,0,0).} Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the z {\displaystyle z} -axis. </p> <h2 id="uses">Uses</h2> <ul><li>Surface normals are useful in defining <a href="/facts/Surface_integral/oPk5IRI1">surface integrals</a> of <a href="/facts/Vector_field/ccFNuPPp">vector fields</a>.</li> <li>Surface normals are commonly used in <a href="/facts/3D_computer_graphics/zlWW6eaF">3D computer graphics</a> for <a href="/facts/Computer_graphics_lighting/kHshvyKY">lighting</a> calculations (see <a href="/facts/Lambert%27s_cosine_law/phlTwVvS">Lambert's cosine law</a>), often adjusted by <a href="/facts/Normal_mapping/7YmzxkPp">normal mapping</a>.</li> <li><a href="/facts/Render_layers/mebvnyll">Render layers</a> containing surface normal information may be used in <a href="/facts/Digital_compositing/HcfAWpXB">digital compositing</a> to change the apparent lighting of rendered elements.</li> <li>In <a href="/facts/Computer_vision/Tl2Yyk66">computer vision</a>, the shapes of 3D objects are estimated from surface normals using <a href="/facts/Photometric_stereo/puLyprWu">photometric stereo</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li> <li>The normal vector may be obtained as the gradient of the <a href="/facts/Signed_distance_function/lqgd3TAv">signed distance function</a>.</li></ul> <h2 id="normal-in-geometric-optics">Normal in geometric optics</h2> <p class="note">Main article: <a href="/facts/Specular_reflection/dwVIqLVb">Specular reflection</a></p> <p>The normal ray is the outward-pointing ray <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a> to the surface of an <a href="/facts/Optical_medium/gm6LqshC">optical medium</a> at a given point.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> In <a href="/facts/Reflection_of_light/BhcjDs79">reflection of light</a>, the <a href="/facts/Angle_of_incidence_(optics)/DTaCjhTU">angle of incidence</a> and the <a href="/facts/Angle_of_reflection/BhcjDs79">angle of reflection</a> are respectively the angle between the normal and the <a href="/facts/Incident_ray/PgUEC7Sm">incident ray</a> (on the <a href="/facts/Plane_of_incidence/d2tJAscI">plane of incidence</a>) and the angle between the normal and the <a href="/facts/Reflected_ray/PgUEC7Sm">reflected ray</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Dual_space/4Uw4Knz1">Dual space</a> – In mathematics, vector space of linear forms</li> <li><a href="/facts/Ellipsoid_normal_vector/TJ2R6WwW">Ellipsoid normal vector</a></li> <li><a href="/facts/Normal_bundle/E44swhHQ">Normal bundle</a> – vector bundle, complementary to the tangent bundle, associated to an embeddingPages displaying wikidata descriptions as a fallback</li> <li><a href="/facts/Pseudovector/c3b5GgWD">Pseudovector</a> – Physical quantity that changes sign with improper rotation</li> <li><a href="/facts/Tangential_and_normal_components/1gswuKMY">Tangential and normal components</a></li> <li><a href="/facts/Vertex_normal/l8b5aISk">Vertex normal</a> – directional vector associated with a vertex, intended as a replacement to the true geometric normal of the surfacePages displaying wikidata descriptions as a fallback</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/NormalVector.html">"Normal Vector"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li>An <a href="http://msdn.microsoft.com/en-us/library/bb324491(VS.85).aspx">explanation of normal vectors</a> from Microsoft's MSDN</li> <li>Clear pseudocode for <a href="http://www.opengl.org/wiki/Calculating_a_Surface_Normal">calculating a surface normal</a> <a href="https://web.archive.org/web/20160818003714/https://www.opengl.org/wiki/Calculating_a_Surface_Normal">Archived</a> 2016-08-18 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a> from either a triangle or polygon.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Ying Wu. "Radiometry, BRDF and Photometric Stereo" (PDF). Northwestern University. <a href="http://users.eecs.northwestern.edu/~yingwu/teaching/EECS432/Notes/lighting.pdf" target="_blank">http://users.eecs.northwestern.edu/~yingwu/teaching/EECS432/Notes/lighting.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"The Law of Reflection". The Physics Classroom Tutorial. Archived from the original on April 27, 2009. Retrieved 2008-03-31. <a href="https://www.physicsclassroom.com/class/refln/Lesson-1/The-Law-of-Reflection" target="_blank">https://www.physicsclassroom.com/class/refln/Lesson-1/The-Law-of-Reflection</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>