in this section we only use the upper 3 × 3 {\displaystyle 3\times 3} matrix, as translation is irrelevant to the calculation n = ∂ r ∂ s × ∂ r ∂ t . {\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}.}

Normal to surfaces in 3D space [ edit ] A curved surface showing the unit normal vectors (blue arrows) to the surface Calculating a surface normal [ edit ] The normal distance of a point Q to a curve or to a surface is the Euclidean distance between Q and its foot P. 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 cross product n = p × q . {\displaystyle \mathbf {n} =\mathbf {p} \times \mathbf {q} .} r ( s , t ) = r 0 + s p + t q , {\displaystyle \mathbf {r} (s,t)=\mathbf {r} _{0}+s\mathbf {p} +t\mathbf {q} ,}

For a plane given by the equation 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. When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals. If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector. 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} Since a surface does not have a tangent plane at a singular point, it has no well-defined normal at that point: for example, the vertex of a cone. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous.

English–Arabic English–Bengali English–Catalan English–Czech English–Danish English–Hindi English–Korean English–Malay English–Marathi English–Russian English–Tamil English–Telugu English–Thai English–Turkish English–Ukrainian English–VietnameseFor a convex polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of the polygon. 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).}

A normal vector may have length one (in which case it is a unit normal vector) or its length may represent the curvature of the object (a curvature vector).

This article is about the normal to 3D surfaces. For the normal to 3D curves, see Frenet–Serret formulas. A polygon and its two normal vectors 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. The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point P {\displaystyle P} is the set of vectors which are orthogonal to the tangent space at P . {\displaystyle P.}