Normals

Normals are geometric constructs used in mathematics, computer science, and physics to describe directions perpendicular to a surface, curve, or interface at a given point. In three-dimensional space, a normal to a surface at a point is a vector perpendicular to the surface’s tangent plane; for a curve, a normal is a line or vector perpendicular to the curve’s tangent.

Surface normals can be computed in several ways. If a surface is given parametrically by r(u, v), the normal vector at a point is r_u × r_v, the cross product of the partial derivatives. If the surface is defined implicitly by F(x, y, z) = 0, the gradient ∇F is perpendicular to the level surface and thus serves as a normal vector. Normals are often expressed as unit normals, obtained by normalizing the vector to length one. The orientation of a normal (which way it points) matters in applications such as shading, where the sign influences lighting calculations.

In mesh geometry, normals are used to determine surface orientation for rendering. Vertex normals are commonly derived by averaging the normals of adjacent faces to create smooth shading. Normals also play a role in physics: the normal force is the contact force exerted by a surface on a body, acting perpendicular to the surface.

Normals extend to higher dimensions as well. For hypersurfaces in n-dimensional space, the gradient of a defining function provides a normal direction to the surface, with normalization yielding a unit normal.

In summary, normals describe perpendicular directions and underpin many techniques in geometry, computer graphics, and physics.