Approximation of surface-to-surface intersection curves within a prescribed error bound satisfying G{sup}2 continuity

摘要

We present a new method of approximating the intersection curves of two parametric surfaces. Our approximation satisfies G{sup}2 continuity conditions, and is located within a prescribed distance from the exact intersection curve. First we identify the topology of the pre-image of the intersection in the domain of surface using extended [Grandine TA, Klein FW. A new approach to the surface intersection problem. Computer Aided Geometric Design 1997; 14(2): 111-34]'s topology determination method [Hur S, Oh M-J, Kim T-W. Classification and resolution of critical cases in Grandine-Klein's topology determination using a perturbation method. Computer Aided Geometric Design (2008). in press [doi:10.1016/j.cagd.2008.03.003]]. Then, we choose a segment of the pre-image, identify the tangential directions and the signed curvatures at its two end-points, and construct a G{sup}2 Hermite interpolation using a rational cubic Bezier curve. We go on to find the exact maximum error of the approximation, and the point at which that error occurs, using the Hausdorff distance function. If the error is larger than a prescribed error bound s, we subdivide the segment at the point of maximum error, and apply the G{sup}2 interpolation process to the two new segments. We continue this recursive process until we have an approximation of the pre-image with an error smaller than s. We also find the maximum error bound of the approximation in the model space M{sup}3; and the bound on the distance, in model space, between the approximations which come from the domain of each surface. We verify our method with several examples.