Enhetsnormalen

Enhetsnormalen, or the unit normal, is a normal vector of unit length associated with a smooth curve or surface in Euclidean space. It encodes the direction perpendicular to the tangent of a curve or to the surface itself and plays a central role in concepts such as curvature, orientation and shading.

In the plane, consider a parametric curve r(t) = (x(t), y(t)). The unit tangent is T = r'(t)/||r'(t)||. The unit normal, often called the principal normal, is N = (dT/ds)/||dT/ds||, where s denotes arc length. The curvature κ = ||dT/ds|| measures how quickly the curve turns. An alternative construction uses a 90-degree rotation of the unit tangent: N = R90(T), with the sign chosen to match the chosen orientation.

In space curves r(t) in R^3, the Frenet-Serret frame provides T = r'/||r'||, κ = ||dT/ds||, N = (dT/ds)/κ (the principal normal), and B = T × N (the binormal). The torsion τ is defined by dB/ds = -τ N. The unit normal N points in the direction in which the curve is turning, relative to its tangent.

For surfaces, the unit normal at a point on a surface F(x, y, z) = 0 is n = ∇F/||∇F||, oriented according to the sign of ∇F. For a parametrized surface X(u, v), a normal vector is ∂X/∂u × ∂X/∂v, and the unit normal is this cross product normalized.

Applications include differential geometry analyses, computer graphics shading, and physical contact modeling, where normals determine orientation, curvature directions, and lighting.