Trimming Curve Approximation for Trimmed Surfaces

Abstract

A trimmed surface is defined to be a parametric surface together with trimming curves lying in the parametric space D of the surface. This paper investigates the interrelation between surface tessellation and trimming curve approximation, and shows that existing trimmed surface tessellation algorithms have some problems on trimming curve approximation. Several examples are proposed to show that a valid approximation of trimming curves in D together with the refinement imposed by surface tessellation does not necessarily generate a valid linear approximation in 3D space. Then we propose a novel step-length estimation method such that a piecewise linear interpolant of the trimming curve based on proposed step length will assure the 3D derivation tolerance. In this method, we exploit the triangle inequality and take the derivation tolerance in 3D space into account to compute the effective step length. Moreover, several empirical tests are given to demonstrate the correctness of our step length estimation.