Editing Normal (geometry) (section)

{{Short description|Line or vector perpendicular to a curve or a surface}}
[[File:Normal vectors2.svg|thumb|A polygon and its two normal vectors|alt=]]
[[File:Surface normal illustration.svg|right|thumb|A normal to a surface at a point is the same as a normal to the tangent plane to the surface at the same point.]]

In [[geometry]], a '''normal''' is an [[mathematical object|object]] (e.g. a [[line (geometry)|line]], [[ray (geometry)|ray]], or [[Euclidean vector|vector]]) that is [[perpendicular]] to a given object. For example, the '''normal line''' to a [[plane curve]] at a given point is the infinite straight line perpendicular to the [[tangent line]] to the curve at the point.

A '''normal vector''' is a [[vector (geometry)|vector]] perpendicular to a given object at a particular point.
A normal [[unit vector|vector of length one]] is called a '''unit normal vector''' or '''normal direction'''. A [[curvature vector]] is a normal vector whose  length is the [[curvature]] of the object. 
Multiplying a normal vector by {{val|-1}} results in the [[opposite vector]], which may be used for indicating sides (e.g., interior or exterior).

In [[three-dimensional space]], a '''surface normal''', or simply '''normal''', to a [[surface (topology)|surface]] at point {{math|''P''}} is a vector perpendicular to the [[tangent plane]] of the surface at {{math|''P''}}. The [[vector field]] of normal directions to a surface is known as ''[[Gauss map]]''. The word "normal" is also used as an adjective: a line ''normal'' to a [[Euclidean plane|plane]], the ''normal'' component of a [[force]], etc. The concept of normality generalizes to [[orthogonality]] ([[right angle]]s).

The concept has been generalized to [[differentiable manifold]]s of arbitrary dimension embedded in a [[Euclidean space]]. The '''normal vector space''' or '''normal space''' of a manifold at point <math>P</math> is the set of vectors which are orthogonal to the [[tangent space]] at <math>P.</math>
Normal vectors are of special interest in the case of [[Differential geometry of curves|smooth curves]] and [[Differential geometry of surfaces|smooth surfaces]].

The normal is often used in [[3D computer graphics]] (notice the singular, as only one normal will be defined) to determine a surface's orientation toward a [[light source]] for [[flat shading]], or the orientation of each of the surface's corners ([[Vertex (geometry)|vertices]]) to mimic a curved surface with [[Phong shading]].

{{anchor|Foot}}The '''foot''' of a normal at a point of interest ''Q'' (analogous to the [[foot of a perpendicular]]) can be defined at the point ''P'' on the surface where the normal vector contains ''Q''.
The ''[[normal distance]]'' of a point ''Q'' to a curve or to a surface is the [[Euclidean distance]] between ''Q'' and its foot ''P''.