Discrete & Computational Geometry

, Volume 10, Issue 2, pp 215–232 | Cite as

On ray shooting in convex polytopes

  • Jiří Matoušek
  • Otfried Schwarzkopf


LetP be a convex polytope withn facets in the Euclidean space of a (small) fixed dimensiond. We consider themembership problem forP (given a query point, decide whether it lies inP) and theray shooting problem inP (given a query ray originating insideP, determine the first facet ofP hit by it). It was shown in [AM2] that a data structure for the membership problem satisfying certain mild assumptions can also be used for the ray shooting problem, with a logarithmic overhead in query time. Here we show that some specific data structures for the membership problem can be used for ray shooting in a more direct way, reducing the overhead in the query time and eliminating the use of parametric search.

We also describe an improved static solution for the membership problem, approaching the conjectured lower bounds more tightly.


Query Point Query Time Convex Polytope Membership Problem Partition Tree 
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.


  1. [AESW] P. K. Agarwal, H. Edelsbrunner, O. Schwarzkopf, and E. Welzl. Euclidean minimum spanning trees and bichromatic closest pairs.Discrete & Computational Geometry, 6:407–422, 1991.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [AM1] P. K. Agarwal and J. Matoušek. Dynamic half-space range reporting and its applications. Technical Report CS-1991-43, Duke University, 1991.Google Scholar
  3. [AM2] P. K. Agarwal and J. Matoušek. Ray shooting and parametric search. InProc. 24th ACM Symposium on Theory of Computing, pages 517–526, 1992. Full version as Technical Report CS-1991-22, Duke University, 1991.Google Scholar
  4. [BS] J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Static-to-dynamic transformation.Journal of Algorithms, 1:301–358, 1980.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [BC] H. Brönnimann and B. Chazelle. How hard is halfspace range searching? InProc. 8th Symposium on Computational Geometry, pages 271–275, 1992.Google Scholar
  6. [CEGS] B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir. A singly-exponential stratification scheme for real semi-algebraic varieties and its applications. InProc. 16th International Colloquium on Automata, Languages and Programming, pages 179–192. Lecture Notes in Computer Science, vol. 372. Springer-Verlag, Berlin, 1989.CrossRefGoogle Scholar
  7. [CF] B. Chazelle and J. Friedman. Point location among hyperplanes and vertical ray shooting.Computational Geometry: Theory and Applications (to appear).Google Scholar
  8. [Ch1] B. Chazelle. How to search in history.Information and Control, 64:77–99, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [Ch2] B. Chazelle. Lower bounds on the complexity of polytope range searching.Journal of the American Mathematical Society, 2(4):637–666, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [Ch3] B. Chazelle. Cutting hyperplanes for divide-and-conquer.Discrete & Computational Geometry, 9:145–158, 1993.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [Cl1] K. L. Clarkson. New applications of random sampling in computational geometry.Discrete & Computational Geometry, 2:195–222, 1987.zbMATHMathSciNetCrossRefGoogle Scholar
  12. [Cl2] K. L. Clarkson. A randomized algorithm for closest-point queries.SIAM Journal on Computing, 17:830–847, 1988.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [CS] K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II.Discrete & Computational Geometry, 4:387–421, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [CSW] B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems.Algorithmica, 8:407–429, 1992.zbMATHMathSciNetCrossRefGoogle Scholar
  15. [dBH+] M. de Berg, D. Halperin, M. Overmars, J. Snoeyink, and M. van Kreveld. Efficient ray shooting and hidden surface removal.Algorithmica (to appear).Google Scholar
  16. [E] H. Edelsbrunner.Algorithms in Combinatorial Geometry. Springer-Verlag, New York, 1987.zbMATHCrossRefGoogle Scholar
  17. [HW] D. Haussler and E. Welzl. ∈-nets and simplex range queries.Discrete & Computational Geometry, 2:127–151, 1987.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [Ma1] J. Matoušek. Efficent partition trees.Discrete & Computational Geometry, 8:315–334, 1992.zbMATHMathSciNetCrossRefGoogle Scholar
  19. [Ma2] J. Matoušek. Reporting points in halfspaces.Computational Geometry: Theory & Applications, 2:169–186, 1992.zbMATHCrossRefGoogle Scholar
  20. [Ma3] J. Matoušek. On vertical ray shooting in arrangements.Computational Geometry: Theory & Applications (to appear).Google Scholar
  21. [Me] N. Megiddo. Applying parallel computation algorithms in the design of serial algorithms.Journal of the ACM, 30:852–865, 1983.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [MS] J. Matoušek and O. Schwarzkopf. Linear optimization queries. InProc. 8th ACM Symposium on Computational Geometry, pages 16–25, 1992.Google Scholar
  23. [Mu] K. Mulmuley. Randomized multidimensional search trees: further results in dynamic sampling. InProc. 32nd Symposium on Foundations of Computer Science, pages 216–227, 1991.Google Scholar
  24. [S] R. Seidel. Low dimensional linear programming and convex hulls made easy.Discrete & Computational Geometry, 6(5):423–434, 1991.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Jiří Matoušek
    • 1
    • 2
  • Otfried Schwarzkopf
    • 3
  1. 1.Katedra aplikované matematikyUniversita KarlovaPraha 1Czech Republic
  2. 2.Institut für InformatikFreie Universität BerlinBerlin 33Federal Republic of Germany
  3. 3.Vakgroep InformaticaUniversiteit UtrechtUtrechtThe Netherlands

Personalised recommendations