Abstract
LetH be a collection ofn hyperplanes in ℝd, letA denote the arrangement ofH, and let σ be a (d−1)-dimensional algebraic surface of low degree, or the boundary of a convex set in ℝd. Thezone of σ inA is the collection of cells ofA crossed by σ. We show that the total number of faces bounding the cells of the zone of σ isO(n d−1 logn). More generally, if σ has dimensionp, 0≤p<d, this quantity isO(n [(d+p)/2]) ford−p even andO(n [(d+p)/2] logn) ford−p odd. These bounds are tight within a logarithmic factor.
Article PDF
Similar content being viewed by others
References
P. K. Agarwal and J. Matoušek, On range searching with semialgebraic sets,Proc. 17th Symp. on Mathematical Foundations of Computer Science, 1992, pp. 1–13. Lecture Notes in Computer Science, Vol. 629, Springer-Verlag, Berlin.
B. Aronov, J. Matoušek, and M. Sharir, On the sum of squares of cell complexities in hyperplane arrangements,Proc. 7th Symp. on Computational Geometry, 1991, pp. 307–313.
B. Aronov and M. Sharir, On the zone of a surface in a hyperplane arrangement,Proc. 2nd Workshop on Algorithms and Data Structures, Ottawa, 1991, pp. 13–19.
B. Aronov and M. Sharir, Castles in the air revisited,Proc. 8th Symp. on Computational Geometry, 1992, pp. 146–156.
M. Bern, D. Eppstein, P. Plassman, and F. Yao, Horizon theorems for lines and polygons. InDiscrete and Computational Geometry: Papers from the DIMACS Special Year, J. Goodman, R. Pollack, and W. Steiger, eds., pp. 45–66. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 6, American Mathematical Society, Providence RI, 1991.
M. de Berg, D. Halperin, M. Overmars, J. Snoeyink, and M. van Kreveld, Efficient ray shooting and hidden surface removal,Proc. 7th Symp. on Computational Geometry, 1991, pp. 21–30.
H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.
H. Edelsbrunner, L. Guibas, J. Pach, R. Pollack, R. Seidel, and M. Sharir, Arrangements of curves in the plane: Topology, combinatorics, and algorithms,Proc. 15th Internat. Colloq. on Automata, Languages and Programming, 1988, pp. 214–229.
H. Edelsbrunner, R. Seidel, and M. Sharir, On the zone theorem for hyperplane arrangements,SIAM J. Computing, to appear.
B. Grünbaum,Convex Polytopes, Wiley, New York, 1967.
M. E. Houle and T. Tokuyama, On zones of flats in hyperplane arrangements, Technical Report 92-3, Department of Information Science, Faculty of Science, University of Tokyo, April 1992.
J. Milnor, On the Betti numbers of real varieties,Proc. Amer. Math. Soc. 15 (1964), 275–280.
M. Pellegrini, Combinatorial and algorithmic analysis of stabbing and visibility problems in 3-dimensional space, Ph.D. Thesis, Courant Institute, New York University, February 1991.
M. Pellegrini, Ray-shooting and isotopy classes of lines in 3-dimensional space,Proc. 2nd Workshop on Algorithms and Data Structures, Ottawa, 1991, pp. 20–31.
M. Pellegrini, On the zone of a codimensionp surface in a hyperplane arrangement,Proc. 3rd Canadian Conference on Computational Geometry, Vancouver, 1991, pp. 233–238.
Author information
Authors and Affiliations
Additional information
This paper is the union of two conference proceedings papers [3], [15]. Work on this paper by M. Pellegrini and M. Sharir has been supported by NSF Grant CCR-8901484. Work on this paper by M. Sharir has also been supported by ONR Grant N00014-90-J-1284 and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F. (the German-Israeli Foundation for Scientific Research and Development), and the Fund for Basic Research administered by the Israeli Academy of Sciences. M. Pellegrini's current address is Department of Computing, King's College, Strand, London WC2R 2LS, England.
Rights and permissions
About this article
Cite this article
Aronov, B., Pellegrini, M. & Sharir, M. On the zone of a surface in a hyperplane arrangement. Discrete Comput Geom 9, 177–186 (1993). https://doi.org/10.1007/BF02189317
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02189317