Tensor product surfaces guided by minimal surface area triangulations

Abstract

We present a method for constructing tensor product Bezier surfaces from contour (cross-section) data. Minimal area triangulations are used to guide the surface construction, and the final surface reflects the optimality of the triangulation. The resulting surface differs from the initial triangulation in two important ways: it is smooth (as opposed to the piecewise planar triangulation), and it is in tensor product form (as opposed to the irregular triangular mesh). The surface reconstruction is efficient because we do not require an exact minimal surface. The triangulations are used as strong hints, but no more than that. The method requires the computation of both open and closed isoparametric curves of the surface, using triangulations as a guide. These isoparametric curves form a tensor product Bezier surface. We show how to control sampling density by filling and pruning isoparametric curves, for accuracy and economy. A rectangular grid of points is produced that is compatible with the expected format for a tensor product surface interpolation, so that a host of well-supported methods are available to generate and manipulate the surface.