Calculating the minimum distance between two NURBS curves

Abstract

A novel method is proposed to compute the minimum distance between two 2D or 3D NURBS curves using control polygons in an efficient and robust way. The first step is to decompose both of NURBS curves into their piecewise Bzier forms. The second step is to use a two level selection process to select a subset of all possible pairs. The first level selection uses upper-lower bounding boxes of Bzier subcurves to remove pairs. The second level selection is based on the spatial relationship test between a pair of Bzier subcurves. The third step is to use a multidimensional Newton-Raphson method to compute the approximate local minimum distances of pairs of Bzier subcurves. By comparing all local minimum distances between a pair of Bzier subcurves, it is able to find the global minimum distance. The final step is to use the multidimensional Newton-Raphson method to improve the accuracy.