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In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, in two dimensions, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. A normal vector may have length one (a unit vector) or its length may represent the curvature of the object (a curvature vector); its algebraic sign may indicate sides (interior or exterior).

In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the 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 (right angles).

The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point P is the set of vectors which are orthogonal to the tangent space at P. Normal vectors are of special interest in the case of smooth curves and 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 (vertices) to mimic a curved surface with Phong shading.

Normal to surfaces in 3D space

A curved surface showing the unit normal vectors (blue arrows) to the surface

Calculating a surface normal

For a convex 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 ${\displaystyle ax+by+cz+d=0}$, the vector

In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the 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 (right angles).

The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point P is the set of vectors which are orthogonal to the tangent space at P. Normal vectors are of special interest in the case of smooth curves and 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 (vertices) to mimic a curved surface with Phong shading.

For a convex 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 ${\displaystyle ax+by+cz+d=0}$, the vector ${\displaystyle \mathbf {n} =(a,b,c)}$ is a normal.

For a plane whose equation is given in parametric form

${\displaystyle \mathbf {r} (s,t)=\mathbf {r} _{0}+s\mathbf {p} +t\mathbf {q} }$,

where r0 is a point on the plane and p, q are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both p and q, which can be found as the cross product For a plane given by the equation ${\displaystyle ax+by+cz+d=0}$, the vector ${\displaystyle \mathbf {n} =(a,b,c)}$ is a normal.

For a plane whose equation is given in parametric form

where r0 is a point on the plane and p, q are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both p and q, which can be found as the cross product ${\displaystyle \mathbf {n} =\mathbf {p} \times \mathbf {q} }$.

If a (possibly non-flat) surface S in 3-space R3 is parameterized by a system of curvilinear coordinates r(s, t) = (x(s,t), y(s,t), z(s,t)), with s and t real variables, then a normal to S is by definition a normal to a tangent plane, given by the cross product of the partial derivatives