In geometry, a normal is an object such as a line 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.

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. In the case of smooth curves, the curvature vector is a normal vector of special interest.

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 whose equation is given in parametric form

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

since the gradient at any point is perpendicular to the level set S.

Since a surface does not have a tangent plane at a singular point, it has no well-defined normal at that point: for example, the vertex of a cone. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous.

The normal to a (hyper)surface is usually scaled to have unit length, but it does not have a unique direction, since its opposite is also a unit 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. For an oriented surface, the normal is usually determined by the right-hand rule or its analog in higher dimensions.

If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector.

When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals.

Specifically, given a 3x3 transformation matrix M, we can determine the matrix W that transforms a vector n perpendicular to the tangent plane t into a vector n′ perpendicular to the transformed tangent plane M t, by the following logic:

Write n′ as W n. We must find W.

Therefore, one should use the inverse transpose of the linear transformation when transforming surface normals. The inverse transpose is equal to the original matrix if the matrix is orthonormal, i.e. purely rotational with no scaling or shearing.

The normal line is the one-dimensional subspace with basis {n}.

A differential variety defined by implicit equations in the n-dimensional space Rn is the set of the common zeros of a finite set of differentiable functions in n variables

The Jacobian matrix of the variety is the k×n matrix whose i-th row is the gradient of f**i. By the implicit function theorem, the variety is a manifold in the neighborhood of a point where the Jacobian matrix has rank k. At such a point P, the normal vector space is the vector space generated by the values at P of the gradient vectors of the f**i.

In other words, a variety is defined as the intersection of k hypersurfaces, and the normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point.

The normal (affine) space at a point P of the variety is the affine subspace passing through P and generated by the normal vector space at P.

These definitions may be extended verbatim to the points where the variety is not a manifold.

Let V be the variety defined in the 3-dimensional space by the equations

This variety is the union of the x-axis and the y-axis.

At a point (a, 0, 0), where a ≠ 0, the rows of the Jacobian matrix are (0, 0, 1) and (0, a, 0). Thus the normal affine space is the plane of equation x = a. Similarly, if b ≠ 0, the normal plane at (0, b, 0) is the plane of equation y = b.

At the point (0, 0, 0) the rows of the Jacobian matrix are (0, 0, 1) and (0, 0, 0). Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the z-axis.

The normal ray is the outward-pointing ray perpendicular to the surface of an optical medium at a given point.^[1] In reflection of light, the angle of incidence and the angle of reflection are respectively the angle between the normal and the incident ray (on the plane of incidence) and the angle between the normal and the reflected ray.