# Surface normal

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

[itex]{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}.[itex]

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

[itex]\nabla F(x, y, z).[itex]

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

• 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 MSDNcs:Normála

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy