, Volume 5, Issue 1, pp 485-503

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

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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).

This research was done while M. S. Paterson was visiting the Xerox Palo Alto Research Center. This author is supported by a Senior Fellowship of the SERC and by the ESPRIT II BRA Program of the EC under Contract 3075 (ALCOM).