Chemistry - 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 $a x + b y + c z = d$ , 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.

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

Categories: Geometry | Multivariate calculus | Computer graphics