Generation of spline approximations to parametric tessellations

Abstract

In geometrical modeling, one is often provided a description of a surface that is defined in terms of a triangulation, which is supported by a discrete number of nodes in space. These faceted surface representations are defined to be C-0 continuous, and therefore in general have slope and curvature discontinuities at the triangle sides, unless the tessellation is planar. Unfortunately, analytical and computational methods often require a surface description that has well-defined and smoothly varying gradients and curvatures; in general spline surfaces possess such properties. Described herein is a process for generating a cubic spline surface that approximates, to within a user-specified tolerance, a given tessellated surface that may be non-convex or multiply connected. The method combines a local least-squares technique for specifying knot properties as well as an adaptation technique for selecting the necessary knot spacings. This new technique is first described along a curve for illustrative purposes. It is then expanded to the case of the general surface. A reparameterization technique that is required for surfaces with non-smooth parameterizations is described next. Computed results for two configurations are then shown.