Computing the Singular Intersection Curves between Two Quadric Surfaces

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.