Normal Vector

The normal vector, often simply called the "normal," to a surface is a vector which is perpendicular to the surface at a given point. When normals are considered on closed surfaces, the inward-pointing normal (pointing towards the interior of the surface) and outward-pointing normal are usually distinguished.

The unit vector obtained by normalizing the normal vector (i.e., dividing a nonzero normal vector by its vector norm) is the unit normal vector, often known simply as the "unit normal." Care should be taken to not confuse the terms "vector norm" (length of vector), "normal vector" (perpendicular vector) and "normalized vector" (unit-length vector).

The normal vector is commonly denoted or , with a hat sometimes (but not always) added (i.e., and ) to explicitly indicate a unit normal vector.

The normal vector at a point on a surface is given by

where and are partial derivatives.

A normal vector to a plane specified by

is given by

where denotes the gradient. The equation of a plane with normal vector passing through the point is given by

For a plane curve, the unit normal vector can be defined by

where is the unit tangent vector and is the polar angle. Given a unit tangent vector

with , the normal is

For a plane curve given parametrically, the normal vector relative to the point is given by

To actually place the vector normal to the curve, it must be displaced by .

For a space curve, the unit normal is given by

where is the tangent vector, is the arc length, and is the curvature. It is also given by

where is the binormal vector (Gray 1997, p. 192).

For a surface with parametrization , the normal vector is given by

Given a three-dimensional surface defined implicitly by ,

If the surface is defined parametrically in the form

define the vectors

Then the unit normal vector is

Let be the discriminant of the metric tensor. Then