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.

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 $ax+by+cz=d$, 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

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

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

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

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 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.

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.

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

Geometry | Vector calculus | 3D computer graphics

Normála | Normalvektor | Normalenvektor | Normale | Normaalvector | Wektor normalny | Normalvektor