normaalikartat

Normaalikartat is a Finnish term used to describe two related ideas that assign normal directions to points on a surface. In differential geometry, a normal map (normaalikartta) refers to a map that assigns to every point on a smooth surface in three-dimensional space its unit normal vector. This Gauss map, N: M -> S^2, encodes how the surface bends, and its differential is closely related to the shape operator and the second fundamental form. The Jacobian of the normal map equals the Gaussian curvature, linking local geometry to the global behavior of the surface. Normal maps thus convey information about orientation, curvature, and surface classification, and they are central to analyses of embedded surfaces.

In computer graphics and rendering, a normal map is a texture that stores normal vectors for each pixel rather than geometric detail. It is used to simulate fine surface relief and lighting effects without increasing the geometric complexity of the model. Normal maps typically encode vectors in tangent space and are read by shading pipelines to perturb the surface normal used in lighting calculations. The common encoding uses RGB values mapped from [0,1] to the range of −1 to 1 for each component, and it relies on a defined tangent-space basis derived from the surface geometry.

Beyond these, in more advanced areas of topology and geometry, the term can also refer to a normal map in the sense of a map between manifolds equipped with a stable isomorphism of their normal bundles, a concept encountered in surgery theory and related fields. Normaalikartat thus spans both rigorous geometric theory and practical computer-graphics applications.