Discrete & Computational Geometry

, Volume 5, Issue 5, pp 485–503

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

  • Michael S. Paterson
  • F. Frances Yao

DOI: 10.1007/BF02187806

Cite this article as:
Paterson, M.S. & Yao, F.F. Discrete Comput Geom (1990) 5: 485. doi:10.1007/BF02187806


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).

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