ALGORITHMS FOR INTERSECTING PARAMETRIC AND ALGEBRAIC-CURVES .2. MULTIPLE INTERSECTIONS

The problem of computing the intersection of parametric and algebraic curves arises in many applications of computer graphics and geometric and solid modeling. Previous algorithms are based on techniques from e limination theory or subdivision and iteration and are typically limit ed to simple intersections of curves. Furthermore, algorithms based on elimination theory are restricted to low degree curves. This is mainl y due to issues of efficiency and numerical stability. In this paper w e use elimination theory and express the resultant of the equations of intersection as a matrix determinant. Using matrix computations the a lgorithm for intersection is reduced to computing eigenvalues and eige nvectors of matrices. We use techniques from linear algebra and numeri cal analysis to compute geometrically isolated higher order intersecti ons of curves. Such intersections are obtained from tangential interse ctions, singular points, etc. The main advantage of the algorithm lies in its efficiency and robustness. The numerical accuracy of the opera tions is well understood and we come up with tight bounds on the error s using 64-bit IEEE floating point arithmetic. (C) 1995 Academic Press , Inc.