Normalfield

Normalfield is a term used in differential geometry to denote a vector field that assigns at each point of a surface a unit normal vector to the surface at that point. More generally, it refers to a smooth section of the normal bundle of an immersed submanifold.

On an oriented surface S ⊂ R^3, the normal field N is a map N: S → R^3 with N(p) ∈ T_pS⊥ and ||N(p)|| = 1. The map p ↦ N(p) is equivalent to the Gauss map, which records the orientation of the tangent plane. A globally defined normal field exists precisely when the surface is orientable.

The differential dN relates to the second fundamental form; the shape operator S_p = -dN_p maps the tangent space to itself and its eigenvalues are the principal curvatures. The mean curvature is the trace of the shape operator. The normal field thus encodes how the surface bends in ambient space.

In applications, normal fields are used in computer graphics for shading and lighting calculations, where normal vectors may be stored or perturbed via normal maps to simulate detail. In geometry processing and visualization, normal fields help convey surface orientation and curvature. In physics and engineering, normal fields describe flux across boundaries and constrain surface dynamics.

Examples: For a sphere of radius R centered at the origin, N(p) = p/||p|| is the outward normal. For a plane, N is constant. For a cylinder, N varies with the angular coordinate.

See also: Gauss map, tangent space, normal bundle, second fundamental form, mean curvature.