Normaalit

Normaalit are vectors perpendicular to a surface at a given point, used to describe the orientation of that surface. In three-dimensional geometry and computer graphics, a surface normal defines the direction perpendicular to the tangent plane and is essential for calculations involving lighting, shading, and interaction with the surface. Normals are typically expressed as three-component vectors (x, y, z).

For a polygonal surface such as a triangle with vertices p1, p2, p3, the face normal can be computed as the cross product of two edge vectors: n = (p2 − p1) × (p3 − p1). The resulting vector is often normalized to unit length. A mesh may have a normal for each face (face normal) and normals at each vertex (vertex normals). Vertex normals are commonly obtained by averaging the normals of adjacent faces and then normalizing the result. The direction of a normal depends on the order of the vertices used to define the surface; reversing the order flips the normal.

Normals have several properties: they are perpendicular to the surface at the point, their direction is not unique on curved surfaces, and their magnitude is often normalized for most computations, though non-unit normals can be used in some contexts.

Applications include shading models such as Lambertian and Phong, where normals determine how light interacts with the surface, as well as collision response, texture mapping, and geometric analysis. In differential geometry, the normal vector field of a surface provides information about orientation and curvature, and the Gauss map associates surface points with directions on the unit sphere.