Efficient binary space partitions for hidden-surface removal and solid modeling
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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. . 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).
- B. Chazelle, Intersecting is easier than sorting,Proc. 16th Ann. ACM Symp. on Theory of Computing, 1983, 125–134.
- B. Chazelle, L. Guibas, and D. Lee, The power of geometric duality,BIT 25, 1985, 76–90.
- 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.
- H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, New York, 1987.
- D. Eppstein, Private communication.
- 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.
- E. Gilbert and E. Moore, Variable-length binary encoding,Bell System Technical Journal 38, 1959, 933–968.
- 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.
- S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of a generalized path compression scheme,Combinatorica 6, 1986, 151–177.
- D. Knuth,The Art of Computer Programming. Vol. 3, Addison-Wesley, Reading, MA, 1973.
- B. Naylor,A priori based techniques for determining visibility priority for 3-d scenes, Ph.D. dissertation, Univ. of Texas at Dallas, 1981.
- M. Overmars and M. Sharir, Output-sensitive hidden surface removal,Proc. 30th IEEE Symp. on Foundations of Computer Science, 1989, 598–603.
- 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).
- M. Paterson and F. Yao, Binary space partitions for orthogonal objects,Proc. 1st Annual ACM-SIAM Symp. on Discrete Algorithms, 1990, 100–106.
- D. Peterson, Halfspace representations of extrusions, solids of revolution, and pyramids, SANDIA Report SAND84-0572, Sandia National Laboratories, 1984.
- F. Preparata, A new approach to planar point location,SIAM Journal on Computing 10, 1981, 473–482.
- W. Thibault and B. Naylor, Set operations on polyhedra using binary space partitioning trees,Computer Graphics 21, 1987, 153–162.
- Efficient binary space partitions for hidden-surface removal and solid modeling
Discrete & Computational Geometry
Volume 5, Issue 1 , pp 485-503
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Industry Sectors