Counting facets and incidences
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
We show thatm distinct cells in an arrangement ofn planes in ℝ3 are bounded byO(m 2/3 n+n 2) faces, which in turn yields a tight bound on the maximum number of facets boundingm cells in an arrangement ofn hyperplanes in ℝ d , for everyd≥3. In addition, the method is extended to obtain tight bounds on the maximum number of faces on the boundary of all nonconvex cells in an arrangement of triangles in ℝ3. We also present a simpler proof of theO(m 2/3 n d/3+n d−1) bound on the number of incidences betweenn hyperplanes in ℝ d andm vertices of their arrangement.
- B. Aronov, J. Matoušek and M. Sharir, On the sum of squares of cell complexities in hyperplane arrangements.Proceedings of the 7th Annual Symposium on Computational Geometry, 1991, pp. 307–313.
- B. Aronov and M. Sharir, Triangles in space or building (and analyzing) castles in the air,Combinatorica 10 (1990), 137–173.
- B. Aronov and M. Sharir, Castles in the air revisited.Proceedings of the 8th Annual Symposium on Computational Geometry, 1992, to appear.
- J. Beck, On the lattice property of the plane and some problems of Dirac, Motzkin and Erdös in combinatorial geometry,Combinatorica 3 (1983), 281–297.
- K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, and E. Welzl, Combinatorial complexity bounds for arrangements of curves and surfaces,Discrete and Computational Geometry 5 (1990), 99–160.
- H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.
- H. Edelsbrunner, L. Guibas, and M. Sharir. Complexity of many cells in arrangements of planes and related problems,Discrete and Computational Geometry 5 (1990), 197–216.
- H. Edelsbrunner and D. Haussler, The complexity of cells in three-dimensional arrangements,Discrete Mathematics 60 (1986), 139–146.
- H. Edelsbrunner and E. Welzl, On the maximum number of edges of many faces in an arrangement,Journal of Combinatorial Theory Ser. A 41 (1986), 159–166.
- J. Pach and M. Sharir, The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: combinatorial analysis,Discrete and Computational Geometry 4 (1989), 291–310.
- E. Szemerédi and W. Trotter, Jr., Extremal problems in discrete geometry,Combinatorica 3 (1983), 381–392.
- F. F. Yao, D. Dobkin, H. Edelsbrunner, and M. S. Paterson, Partitioning space for range queries,SIAM Journal on Computing 18 (1989), 371–384.
- Counting facets and incidences
Discrete & Computational Geometry
Volume 7, Issue 1 , pp 359-369
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Industry Sectors