Calculating the self-intersections of Bézier curves

Abstract

A user-friendly “ divide-and-conquer” algorithm is presented for finding all the self-intersection points of a parametric curve in the Bernstein-Bézier representation. The underlying idea of the algorithm is to deal with the Bézier polygon instead of the curve description itself. By alternately subdividing the Bézier polygon and estimating the self-intersection regions the self-intersection points are finally approximated by straight line intersections of the refined Bézier polygons. The algorithm also calculates the parameter values of the self-intersection points. In addition to the convex hull and the approximation property of the Bézier polygon the working of the algorithm is based on a very intuitive angle criterion.