Precise Voronoi cell extraction of free-form rational planar closed curves

Abstract

We present an algorithm for generating the Voronoi cells for a set of rational C1-continuous planar closed curves, which is precise up to machine precision. Initially, bisectors for pairs of curves, (C(t), Ci(r)), are generated symbolically and represented as implicit forms in the tr-parameter space. Then, the bisectors are properly trimmed after being split into monotone pieces. The trimming procedure uses the orientation of the original curves as well as their curvature fields, resulting in a set of trimmed-bisector segments represented as implicit curves in a parameter space. A lower-envelope algorithm is then used in the parameter space of the curve whose Voronoi cell is sought. The lower envelope represents the exact boundary of the Voronoi cell.