Curvatures (only Preprocessing)

Menu: View->Curvatures->Surfaces

Draw a contour fill of the curvature of all visible surfaces on each point and a scale of colors to represent its values

Definition of the surface curvature at a point:
For two-dimensional surfaces embedded in R3, consider the intersection of the surface with a plane containing the normal vector at a point and another arbitrary vector tangent to the surface. This intersection is a plane curve and has a curvature. This is the Normal curvature. The maximum and minimum values of the normal curvature at a point are called the principal curvatures, k1 and k2, and the extremal directions are called principal directions.

It is possible to draw some typical surface curvatures:

• Mean curvature: is equal to the mean of the two principal curvatures k1 and k2. Mean curvature is closely related to the first variation of surface area, in particular a minimal surface like a soap film has mean curvature zero and soap bubble has constant mean curvature.
• Gaussian curvature: is equal to the product of the principal curvatures, k1k2. Is positive for spheres, negative for one-sheet hyperboloids and zero for planes. It determines whether a surface is locally convex (when it is positive) or locally saddle (when it is negative).
• Maximum curvature: The greatest principal curvature k1
• Minimum curvature: The second principal curvature k2

Also the principal curvature directions can be shown as unitary vectors.

Menu: View->Curvatures->Lines

Draw by colors the curvature (the inverse of curvature radius) of all visible curves.