Curvature approximation with application to surface modeling

Abstract

When fair surfaces have to be designed, a uniformly distributed curvature is important. To achieve this designers try to minimize total curvature, or the variation of the curvature. This however is a very costly procedure. In the case of curves, the curvature can be easily approximated by quadratic functionals. This then allows optimization at interactive speed. The generalization of this method to surfaces is by no means straightforward.

In this note we describe how curvature can be approximated by quadratic functionals. Moreover, we apply the results to construct fairness functionals that are good approximations to the exact curvature functionals but are quadratic, hence relatively easy to minimize. We sketch how these functionals can be used to interpolate scattered data by fair tensor product B-spline functions.