Geometric hermite approximation of surface patch intersection curves

Abstract

This paper introduces two new tools for attacking the problem of approximating the intersection curve of two parametric surface patches: the Bézier clipping algorithm for curve⧸surface intersection, and geometric Hermite approximation of surface⧸surface intersection curves. The curve⧸surface intersection algorithm is used to compute a set of endpoint pairs for all components of the intersection curve. Thereafter, a G k piecewise parametric approximation of the intersection curve is directly computed with no further subdivision or marching. An error bound is determined directly from the approximation. If the error is too large, each unsatisfactory approximating curve is split in half and new approximations are made directly. This procedure is O( h 2 k+2 ) convergent, which means that each time the members of a G k sequence of approximating curves are split in half, the new error is 2 −(2 k+2) of the previous error (in the limit). Thus, doubling the number of approximating curves in a G 3 sequence reduces the error typically by two orders of magnitude (1⧸256).