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.

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 ax + by + cz = d, the vector (a,b,c) 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

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

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 from Microsoft's MSDN