An Efficient Algorithm for the InCircle Predicate among Smooth Closed Curves

This paper concentrates on the InCircle predicate which is used for the computation of the Voronoi diagram of smooth closed curves. The predicate decides the position of a query object relative to the Voronoi circle of three given ones. We focus on (nonintersecting) ellipses but our method extends to arbitrary closed smooth curves, given in parametric representation. We describe an efficient algorithm for InCircle based on a certified numeric algorithm. The algorithm relies on a geometric preprocessing that guarantees a unique solution in a box of parametric space, where a customized subdivision-based method approximates the Voronoi circle tracing the bisectors. Our subdivision method achieves quadratic convergence by exploiting the geometric characteristics of the problem. The paper concludes with experiments showing that most instances run in less than 0.1 sec using floating-point arithmetic, on a 2.6GHz Pentium4.