Using Local Optimization in Surface Fitting To appear in Mathematical Methods for Curves and Surfaces

Local optimization is used to set the free parameters in a triangular surface tting scheme, resulting in surfaces with better shape. While some of the free parameters can be set to match curvature information , other free parameters are independent of this information. x1. Introduction A large number of local parametric triangular surface schemes have been developed over the past fteen years (see 8,9] for a survey of such schemes). These schemes are local in that changes to part of the data only aaect portions of the surface near the changed data. Surprisingly, all of these schemes exhibit similar shape defects. On closer inspection, it is seen that these schemes all have a large number of free parameters that are set using simple heuristics. By manually adjusting these parameters, one can improve the shape of the surfaces 9]. In this paper, I will investigate using local optimization to set the free parameters in a local, triangular, split-domain, polynomial surface interpolation scheme. More precisely, given a triangle of data, this scheme constructs three triangular parametric polynomial patches, each of which interpolates two of the positions and normals at the three vertices. Further, the patches meet each other with G 1 continuity, and if used to ll a triangular polyhedron, the resulting surface will also be G 1. Local optimization is used to set the remaining free parameters so as to minimize the error in interpolating second order data at the corners. Thus, our data is a set of three vertices with normals I will be working with a modiied version of Shirman-S equin's scheme 11,12]. Shirman and S equin created a split domain scheme that ts three quartic patches to a triangle of data. These triangles meet each other G 1 , and if used Mathematical Methods for Curves and Surfaces 1 M.