Algorithms for Intersecting Parametric and Algebraic Curves II: Multiple Intersections

The problem of computing the intersection of parametric and algebraic curves arises in many applications of computer graphics, geometric and solid modeling. Previous algorithms are based on techniques from elimination theory or subdivision and iteration and are typically limited to simple intersections of curves. Furthermore, algorithms based on elimination theory are restricted to low degree curves. This is mainly due to issues of e ciency and numerical stability. In this paper we use elimination theory and express the resultant of the equations of intersection as a matrix determinant. Using matrix computations the algorithm for intersection is reduced to computing eigenvalues and eigenvectors of matrices. We use techniques from linear algebra and numerical analysis to compute geometrically isolated higher order intersections of curves. Such intersections are obtained from tangential intersections, singular points etc. The main advantage of the algorithm lies in its e ciency and robustness. The numerical accuracy of the operations is well understood and we come up with tight bounds on the errors using 64 bit IEEE oating point arithmetic.