Algorithm for algebraic curve intersection

Abstract

A robust algorithm is presented for computing all real points at which two planar algebraic curves intersect within a specified area. The classical theory of polar curves is reviewed and applied to the problem of computing the double points on an algebraic curve. Using polar curves, the intersection algorithm can quickly and robustly compute all real double points on an algebraic curve, as well as silhouette points. Polar curves are shown to have a strikingly simple expression in Bernstein polynomial form. Hessian curves are also discussed. The inflection points of an algebraic curve can be found by intersecting the curve with its Hessian.