Efficient binary space partitions for hidden-surface removal and solid modeling
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- Paterson, M.S. & Yao, F.F. Discrete Comput Geom (1990) 5: 485. doi:10.1007/BF02187806
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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(n2) forn planar facets and prove a matching lower bound of Θ(n2). Two applications of efficient BSPs are given. The first is anO(n2)-sized data structure for implementing a hidden-surface removal scheme of Fuchset al. . The second application is in solid modeling: given a polyhedron described by itsn faces, we show how to generate anO(n2)-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(n3).