Efficient binary space partitions for hidden-surface removal and solid modeling

Abstract

We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such abinary space partition, or BSP, is naturally considered as a binary tree where each internal node corresponds to a division. The goal is to choose the hyperplanes properly so that the size of the BSP, i.e., the number of resulting fragments of the objects, is minimized. For the two-dimensional case, we construct BSPs of sizeO(n logn) forn edges, while in three dimensions, we obtain BSPs of sizeO(n 2) forn planar facets and prove a matching lower bound of Θ(n 2). Two applications of efficient BSPs are given. The first is anO(n 2)-sized data structure for implementing a hidden-surface removal scheme of Fuchset al. [6]. The second application is in solid modeling: given a polyhedron described by itsn faces, we show how to generate anO(n 2)-sized CSG (constructive-solid-geometry) formula whose literals correspond to half-spaces supporting the faces of the polyhedron. The best previous results for both of these problems wereO(n 3).