Computation of stationary points of distance functions

Abstract

This paper presents an algorithm for computation of the stationary points of the squared distance functions between two point sets. One point set consists of a single space point, a rational B-spline curve, or a rational B-spline surface. The problem is reformulated in terms of solution of n polynomial equations with n variables expressed in the tensor product Bernstein basis. The solution method is based on subdivision relying on the convex hull property of the n-dimensional Bernstein basis and minimization techniques. We also cover classification of the stationary points of these distance functions, and include a method for tracing curves of stationary points in case the solution set is not zerodimensional. The distance computation problem is shown to be equivalent to the geometrically intuitive problem of computing collinear normal points. Finally, examples illustrate the applicability of the method