Discrete & Computational Geometry

, Volume 5, Issue 5, pp 485–503 | Cite as

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

  • Michael S. Paterson
  • F. Frances Yao


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. [6]. 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).


Internal Node Partition Tree Description Size Binary Partition Constructive Solid Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Michael S. Paterson
    • 1
  • F. Frances Yao
    • 2
  1. 1.Department of Computer ScienceUniversity of WarwickConventryEngland
  2. 2.Xerox Palo Alto Research CenterPalo AltoUSA

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