The normal vector, often simply called the "normal," to a surface is avector which isperpendicular 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.

Theunit vector obtained by normalizing the normal vector (i.e., dividing a nonzero normal vector by itsvector 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 ahat 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

(1)

where and arepartial derivatives.

A normal vector to aplane specified by

(2)

is given by

(3)

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

(4)

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

(5)

where is the unittangent vector and is thepolar angle. Given a unittangent vector

(6)

with, the normal is

(7)

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

			(8)
			(9)

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

For a space curve, the unit normal is given by

			(10)
			(11)
			(12)

where is thetangent vector, is thearc length, and is thecurvature. It is also given by

(13)

where is thebinormal vector (Gray 1997, p. 192).

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

(14)

Given a three-dimensional surface defined implicitly by,

(15)

If the surface is defined parametrically in the form

			(16)
			(17)
			(18)

define thevectors

(19)

(20)

Then the unit normal vector is

(21)

Let be the discriminant of themetric tensor. Then

(22)

See also

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More things to try:

• cross product
• dot product
• normal vector of a plane

References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Normal Vector." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/NormalVector.html

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