Surface normal

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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. The word normal is also used as an adjective as well as a noun with this meaning: a line normal to a plane, the normal component of a force, the normal vector, etc.

Image:SurfaceNormalDrawing.PNG
A polygon and its normal

1 Calculating a surface normal
2 Uniqueness of the normal
3 Uses
4 External link

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Calculating a 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 nor does it have a normal along the edge of its base. However, the normal to the cone is defined almost everywhere. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous.

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Uniqueness of the normal

A normal to a surface does not have a unique direction; the vector pointing in the opposite direction of a surface normal is also a surface normal. For an oriented surface, the surface normal is usually determined by the right-hand rule.

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