Normal (geometry)

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 object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) 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).

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.

The normal distance of a point Q to a curve or to a surface is the Euclidean distance between Q and its perpendicular projection on the object (at the point P on the object where the normal contains Q). The normal distance is a type of perpendicular distance generalizing the distance from a point to a line and the distance from a point to a plane. It can be used for curve fitting and for defining offset surfaces.

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

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.

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.

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, that is, purely rotational with no scaling or shearing.

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

The normal ray is the outward-pointing ray perpendicular to the surface of an optical medium at a given point.[2] 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.