Normalsvektor

Normalsvektor, or normal vector, is a vector perpendicular to a surface at a given point. In three-dimensional space, it describes the orientation of the surface locally. For a plane defined by Ax + By + Cz + D = 0, any nonzero multiple of the vector (A, B, C) is a normal vector to the plane. For a surface given implicitly by F(x, y, z) = 0, the gradient ∇F(x0, y0, z0) = (∂F/∂x, ∂F/∂y, ∂F/∂z) at a point (x0, y0, z0) is a normal vector to the surface at that point. For a parametric surface S(u, v) with tangent vectors Su and Sv, the cross product Su × Sv yields a normal vector to the surface at (u, v).

Unit normal vectors can be obtained by normalizing: n̂ = n / ||n||, where n is a normal vector and ||n|| is its length. Normal directions are not unique; any nonzero scalar multiple or its negation describes the same geometric normal, with orientation chosen by convention (for example, outward pointing normals on a closed surface).

Examples help illustrate the concept: the plane x + y + z − 1 = 0 has normal vector (1, 1, 1); a sphere x^2 + y^2 + z^2 = R^2 has normals that point radially outward, with at point (x0, y0, z0) the normal direction given by (x0, y0, z0).

Applications of normalsvektorer appear across disciplines. In computer graphics, normals are used for lighting calculations and shading, including normal mapping. In geometry processing and computer-aided design, they assist in mesh quality, collision detection, and surface reconstruction. In differential geometry, normals relate to gradients and the study of surface curvature.