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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
Article

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(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).

Keywords

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.

References

  1. 1.
    B. Chazelle, Intersecting is easier than sorting,Proc. 16th Ann. ACM Symp. on Theory of Computing, 1983, 125–134.Google Scholar
  2. 2.
    B. Chazelle, L. Guibas, and D. Lee, The power of geometric duality,BIT 25, 1985, 76–90.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. Dobkin, L. Guibas, J. Hershberger, and J. Snoeyink, An efficient algorithm for finding the CSG representation of a simple polygon,Computer Graphics 22, 1988, 31–40.CrossRefGoogle Scholar
  4. 4.
    H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, New York, 1987.CrossRefzbMATHGoogle Scholar
  5. 5.
    D. Eppstein, Private communication.Google Scholar
  6. 6.
    H. Fuchs, Z. Kedem, and B. Naylor, On visible surface generation by a priori tree structures,Computer Grahics (SIGGRAPH '80 Conference Proceedings),1980, 124–133.Google Scholar
  7. 7.
    E. Gilbert and E. Moore, Variable-length binary encoding,Bell System Technical Journal 38, 1959, 933–968.MathSciNetCrossRefGoogle Scholar
  8. 8.
    L. Guibas and F. Yao, On translating a set of rectangles, inAdvances in Computing Research, Vol. 1, edited by F. Preparata, JAI Press, Greenwich, CT, 1983, 61–77.Google Scholar
  9. 9.
    S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of a generalized path compression scheme,Combinatorica 6, 1986, 151–177.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. Knuth,The Art of Computer Programming. Vol. 3, Addison-Wesley, Reading, MA, 1973.Google Scholar
  11. 11.
    B. Naylor,A priori based techniques for determining visibility priority for 3-d scenes, Ph.D. dissertation, Univ. of Texas at Dallas, 1981.Google Scholar
  12. 12.
    M. Overmars and M. Sharir, Output-sensitive hidden surface removal,Proc. 30th IEEE Symp. on Foundations of Computer Science, 1989, 598–603.Google Scholar
  13. 13.
    M. Paterson and F. Yao, Optimal binary partitions with applications to hidden-surface removal and solid modelling,Proc. 5th Ann. ACM Symp. on Computational Geometry, 1989, 23–32 (also Dept. of Computer Science Research Report RR139, Univ. of Warwick, March 1989).Google Scholar
  14. 14.
    M. Paterson and F. Yao, Binary space partitions for orthogonal objects,Proc. 1st Annual ACM-SIAM Symp. on Discrete Algorithms, 1990, 100–106.Google Scholar
  15. 15.
    D. Peterson, Halfspace representations of extrusions, solids of revolution, and pyramids, SANDIA Report SAND84-0572, Sandia National Laboratories, 1984.Google Scholar
  16. 16.
    F. Preparata, A new approach to planar point location,SIAM Journal on Computing 10, 1981, 473–482.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    W. Thibault and B. Naylor, Set operations on polyhedra using binary space partitioning trees,Computer Graphics 21, 1987, 153–162.MathSciNetCrossRefGoogle Scholar

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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