A Variational Approach to Subdivision

In this paper a new class of interpolatory refinement schemes is presented which in every refinement step determine the new points by solving an optimization problem. In general, these schemes are global, i.e., every new point depends on all points of the polygon to be refined. By choosing appropriate quadratic functionals to be minimized iteratively during refinement, very efficient schemes producing limiting curves of high smoothness can be defined. The well known class of stationary interpolatory refinement schemes turns out to be a special case of these variational schemes.