Surface normal

From Exampleproblems

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

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 $a x + b y + c z = d$ , 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

${\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}.$

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

$\nabla F(x, y, z).$

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.

Uniqueness of the normal

Surface normal at a point 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.