Normal (geometry)
lastmod 2026-09-14created 2026-09-141 edits
A normal vector is a vector perpendicular to a curve, surface, or higher-dimensional shape at a given point.
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In geometry, a normal is an object (e.g. a line, ray, or Euclidean vector) that is perpendicular to a given object. For example, the normal to a plane curve at a given point is the infinite straight line perpendicular to the tangent line to the curve at that point.
A normal vector is a vector perpendicular to a given object at a particular point. A normal vector of length 1 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 −1 gives the opposite vector, which can be used to indicate sides (e.g. inside or outside).
In three dimensions, a surface normal, or simply normal, to a surface at a point P is a vector perpendicular to the tangent plane of the surface at P. The vector field of normal directions to a surface is known as the Gauss map. The word "normal" is also used as an adjective, as in a line normal to a plane, the normal component of a force, and so on. 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 a point P {\displaystyle P} is the set of vectors that are orthogonal to the tangent space at P {\displaystyle P}. Normal vectors are of special interest in the case of smooth curves and smooth surfaces.
Normals are often used in 3D computer graphics 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 (note the singular, since only one normal is defined).
The foot of a normal from a point Q to a curve or surface can be defined as the point P where the normal to the curve or surface containing Q meets the curve or surface. The normal distance from a point Q to a curve or surface is the Euclidean distance between Q and its foot P.