奇异情况下两个二次曲面间的求交

Computing the Singular Intersection Curves between Two Quadric Surfaces

曲面求交是几何造型系统的核心之一,奇异情况下求交算法的稳定性直接关系到后续的布尔运算乃至整个系统的稳定性.提出2种二次曲面间对应特征多项式有重根情形下鲁棒的求交算法.首先给出精确求解特征多项式重根的方法,若交线中存在奇异交点,则给出奇异交点关于特征多项式重根的显式表达式,从而稳定地求解出对应的奇异交点;同时给出一种交曲线有理参数化的构造性方法,可以弥补Farouki相应有理参数化方法中的缺陷.最后通过实例进一步说明了文中算法的求解稳定性及有理参数化的构造性方法的实用性.

Surface intersection problem is one of the key problems in the geometric modeling system. The robustness of the intersection algorithms in singular cases is very important to the stability of the whole modeling system. This paper presents a robust intersection algorithm between two quadric surfaces when the corresponding characteristic polynomial has a multiple root. Firstly, a method is given to solve the multiple root of the characteristic polynomial accurately. Secondly, it provides an explicit formula of the positions of the singular points on the intersection curves. The formula only depends on the multiple root of the characteristic polynomial and the singular points could be robustly solved. A constructive rational parameterization method is also presented, which works in the cases when the Farouki＇s rational parameterization method may fail. Examples illustrate the robustness of intersection calculation and the effectiveness of constructive rational parameterization method.