# Surface normal

File:Normal vectors2.svg
A polygon and two of its normal vectors.
File:Surface normal illustration.png
A normal to a surface at a point is the same as a normal to the tangent plane to that surface at that point.

A surface normal, or simply normal, to a flat surface is a vector which is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality.

## Calculating a surface normal

For a polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of the polygon.

For a plane given by the equation $ax+by+cz=d$ , the vector $(a,b,c)$ is a normal. For a plane given by the equation r = a + αb + βc, where a is a vector to get onto the plane and b and c are non-parallel vectors lying on the plane, the normal to the plane defined is given by b × c (the cross product of the vectors lying on the plane).

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 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.

## n-dimensional surfaces

The definition of a normal to a two-dimensional surface in three-dimensional space can be extended to $n-1$ -dimensional "surfaces" in $n$ -dimensional space. Such a hypersurface may be defined implicitly as the set of points $(x_{1},x_{2},\ldots ,x_{n})$ satisfying the equation $F(x_{1},x_{2},\ldots x_{n})=0$ . If $F$ is continuously differentiable, then the surface obtained is a differentiable manifold, and its surface normal is given by the gradient of $F$ ,

$\nabla F(x_{1},x_{2},\ldots ,x_{n})=\left({\tfrac {\partial F}{\partial x_{1}}},{\tfrac {\partial F}{\partial x_{2}}},\ldots ,{\tfrac {\partial F}{\partial x_{n}}}\right).$ ## Uniqueness of the normal

File:Surface normal.png
A vector field of normals to a surface.

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 a surface which is the topological boundary of a set in three dimensions, one can distinguish between the inward-pointing normal and outer-pointing normal, which can help define the normal in a unique way. For an oriented surface, the surface normal is usually determined by the right-hand rule. If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector.

## Normal in geometric optics

File:Reflection angles.svg
Diagram of specular reflection

The normal is an imaginary line perpendicular to the surface of an optical medium. The word normal is used here in the mathematical sense, meaning perpendicular. In reflection of light, the angle of incidence is the angle between the normal and the incident ray. The angle of reflection is the angle between the normal and the reflected ray. 