About Curvatures

This section provides essential information about curvatures.

See Also
Creating an Associative Curvature Map
Defining Areas by Curvature Segmentation

There are five curvatures:

  • Maximum
  • Minimum
  • Mean
  • Gaussian
  • Absolute.

The geometric construction of the maximum and minimum curvatures is the following: Let be a plane containing the normal to the surface in a given point. This plane cuts the surface along a curve that has a given curvature in this point. If this plane rotates around the normal, the curvatures of the curves intersecting the surface vary between two utmost values: these two values are the maximum (KM) and the minimum (Km) curvatures.

The mean curvature is equal to (KM+Km)/2.

The utmost values appear where the surface is the most warped. The mean curvature is largely used to detect irregularities on the surface. A minimal surface is characterized by a null mean curvature.

The gaussian curvature is equal to KM.Km. It describes the local shape of a surface in one point:

  • If it is positive, the point is elliptic, that is, the surface has locally the shape of an ellipsoid around that point,
  • If it is negative, the surface is hyperbolic in this points, that is, the local shape is a horse saddle,
  • It is null, the surface is parabolic in this point, that is, one of the maximum or minimum curvatures is null in this point. The cone and the cylinder are two surfaces where all points are parabolic.

The absolute curvature is equal to |KM|+|Km|. It is used to detect the surface areas where the surface is locally almost flat (the absolute curvature is almost null).

The curvature radii are the inverse of the corresponding curvatures. Only the maximum and the minimum radii are relevant.