Free-Form Surface Partition in 3-D

Abstract

We study the problem of partitioning a spherical representation S of a free-form surface F in 3-D, which is to partition a 3-D sphere S into two hemispheres such that a representative normal vector for each hemisphere optimizes a given global objective function. This problem arises in important practical applications, particularly surface machining in manufacturing. We model the spherical surface partition problem as processing multiple off-line sequences of insertions/deletions of convex polygons alternated with certain point queries on the common intersection of the polygons. Our algorithm combines nontrivial data structures, geometric observations, and algorithmic techniques. It takes $O(\min\{m^2n \log \log m + \frac{m^3 \log^2(mn) \log^2(\log m)}{\log^3 m}, m^3\log^2n+mn\})$ time, where m is the number of polygons, of size O(n) each, in one off-line sequence (generally, m ≤ n). This is a significant improvement over the previous best-known O(m ² n ²) time algorithm. As a by-product, our algorithm can process O(n) insertions/deletions of convex polygons (of size O(n) each) and queries on their common intersections in O(n ² loglogn) time, improving over the “standard” O(n ² logn) time solution for off-line maintenance of O(n ²) insertions/deletions of points and queries. Our techniques may be useful in solving other problems.