article

On the zone of a surface in a hyperplane arrangement

Authors:

Marco Pellegrini, and

Micha SharirAuthors Info & Claims

Discrete & Computational Geometry, Volume 9, Issue 2

Pages 177 - 186

https://doi.org/10.1007/BF02189317

Published: 01 December 1993 Publication History

Abstract

LetH be a collection ofn hyperplanes in d, letA denote the arrangement ofH, and let be a (d 1)-dimensional algebraic surface of low degree, or the boundary of a convex set in d. Thezone of inA is the collection of cells ofA crossed by . We show that the total number of faces bounding the cells of the zone of isO(nd 1 logn). More generally, if has dimensionp, 0≤p

References

[1]

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.

[2]

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.

[3]

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.

[4]

B. Aronov and M. Sharir, Castles in the air revisited,Proc. 8th Symp. on Computational Geometry, 1992, pp. 146---156.

[5]

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.

[6]

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.

[7]

H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.

[8]

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.

[9]

H. Edelsbrunner, R. Seidel, and M. Sharir, On the zone theorem for hyperplane arrangements,SIAM J. Computing, to appear.

[10]

B. Grünbaum,Convex Polytopes, Wiley, New York, 1967.

[11]

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.

[12]

J. Milnor, On the Betti numbers of real varieties,Proc. Amer. Math. Soc.15 (1964), 275---280.

[13]

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.

[14]

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.

[15]

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.

Cited By

Rubin N(2022)An Improved Bound for Weak Epsilon-nets in the PlaneJournal of the ACM10.1145/355598569:5(1-35)Online publication date: 27-Oct-2022
https://dl.acm.org/doi/10.1145/3555985
Nivasch G(2017)On the zone of a circle in an arrangement of linesDiscrete Mathematics10.1016/j.disc.2017.02.022340:7(1535-1552)Online publication date: 1-Jul-2017
https://dl.acm.org/doi/10.1016/j.disc.2017.02.022
Raz O(2015)On the zone of the boundary of a convex bodyComputational Geometry: Theory and Applications10.1016/j.comgeo.2014.12.00148:4(333-341)Online publication date: 1-May-2015
https://dl.acm.org/doi/10.1016/j.comgeo.2014.12.001
Show More Cited By

Index Terms

On the zone of a surface in a hyperplane arrangement
1. Mathematics of computing
  1. Discrete mathematics
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

Index terms have been assigned to the content through auto-classification.

Recommendations

Cutting hyperplane arrangements

We consider a collectionH ofn hyperplanes in Ed (where the dimensiond is fixed). An -cutting forH is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all Ed and such that the interior of any simplex is ...
Read More
Cutting hyperplane arrangements

We consider a collectionH ofn hyperplanes in Ed (where the dimensiond is fixed). An -cutting forH is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all Ed and such that the interior of any simplex is ...
Read More
Cutting hyperplane arrangements
SCG '90: Proceedings of the sixth annual symposium on Computational geometry

We will consider an arrangement H of n hyperplanes in E^d (where the dimension d is fixed). An ε-cutting for H will be a collection of (possibly unbounded) d-dimensional simplices with disjoint interiors, which cover all E^d and such that the interior of ...
Read More

Comments

Information & Contributors

Information

Published In

cover image Discrete & Computational Geometry

Discrete & Computational Geometry Volume 9, Issue 2

February 1993

102 pages

ISSN:0179-5376

Issue’s Table of Contents

Copyright © Copyright © 1993 Springer-Verlag New York Inc.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 December 1993

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

22
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Other Metrics

View Author Metrics

Citations

Cited By

Rubin N(2022)An Improved Bound for Weak Epsilon-nets in the PlaneJournal of the ACM10.1145/355598569:5(1-35)Online publication date: 27-Oct-2022
https://dl.acm.org/doi/10.1145/3555985
Nivasch G(2017)On the zone of a circle in an arrangement of linesDiscrete Mathematics10.1016/j.disc.2017.02.022340:7(1535-1552)Online publication date: 1-Jul-2017
https://dl.acm.org/doi/10.1016/j.disc.2017.02.022
Raz O(2015)On the zone of the boundary of a convex bodyComputational Geometry: Theory and Applications10.1016/j.comgeo.2014.12.00148:4(333-341)Online publication date: 1-May-2015
https://dl.acm.org/doi/10.1016/j.comgeo.2014.12.001
Kaplan HRubin NSharir M(2007)Linear data structures for fast ray-shooting amidst convex polyhedraProceedings of the 15th annual European conference on Algorithms10.5555/1778580.1778609(287-298)Online publication date: 8-Oct-2007
https://dl.acm.org/doi/10.5555/1778580.1778609
Das GGunopulos DKoudas NSarkas NKlas WNeuhold E(2007)Ad-hoc top-k query answering for data streamsProceedings of the 33rd international conference on Very large data bases10.5555/1325851.1325875(183-194)Online publication date: 23-Sep-2007
https://dl.acm.org/doi/10.5555/1325851.1325875
Agarwal PAronov BKoltun V(2006)Efficient algorithms for bichromatic separabilityACM Transactions on Algorithms10.1145/1150334.11503382:2(209-227)Online publication date: 1-Apr-2006
https://dl.acm.org/doi/10.1145/1150334.1150338
Feldman SSharir M(2005)An Improved Bound for Joints in Arrangements of Lines in SpaceDiscrete & Computational Geometry10.5555/3115476.311579333:2(307-320)Online publication date: 1-Feb-2005
https://dl.acm.org/doi/10.5555/3115476.3115793
Aronov BSharir M(2004)Cell Complexities in Hyperplane ArrangementsDiscrete & Computational Geometry10.5555/3115957.311603132:1(107-115)Online publication date: 1-May-2004
https://dl.acm.org/doi/10.5555/3115957.3116031
Matousek JWagner U(2004)New Constructions of Weak ź-NetsDiscrete & Computational Geometry10.5555/3115953.311600132:2(195-206)Online publication date: 1-Jul-2004
https://dl.acm.org/doi/10.5555/3115953.3116001
Koltun VSharir MFortune S(2003)Curve-sensitive cuttingsProceedings of the nineteenth annual symposium on Computational geometry10.1145/777792.777814(136-143)Online publication date: 8-Jun-2003
https://dl.acm.org/doi/10.1145/777792.777814
Show More Cited By

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents