Surface normal - Definition and Overview

A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. A normal to a non-flat surface at a point p on the surface is a vector which is perpendicular to the tangent plane to that surface at p.

A polygon and its normal

Calculating the surface normal

For a polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two edges of the polygon.

For a plane given by the equation <math>ax+by+cz=d<math>, the vector <math>(a, b, c)<math> is a normal.

If a (possibly non-flat) surface S is parametrized by a system of curvilinear coordinates x(s, t), with s and t real variables, then a normal is given by the cross product of the partial derivatives

<math>{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}.<math>

If a surface S is given implicitly, as the set of points <math>(x, y, z)<math> satisfying <math>F(x, y, z)=0<math>, then, a normal at a point <math>(x, y, z)<math> on the surface is given by the gradient

<math>\nabla F(x, y, z).<math>

If a surface does not have a tangent plane at a point, it does not have a normal at that point either. For example, a cone does not have a normal at its tip.

Uses

• Surface normals are essential in defining surface integrals of vector fields.

• Surface normals are commonly used in 3D computer graphics for lighting calculations, see Lambert's cosine law.

External link

• An explanation of normal vectors (http://msdn.microsoft.com/library/default.asp?url=/library/en-us/directx9_c/directx/graphics/programmingguide/GettingStarted/3DCoordinateSystems/facevertexnormalvectors.asp) from Microsoft's MSDN