Discrete & Computational Geometry

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

On ray shooting in convex polytopes

  • Jiří Matoušek
  • Otfried Schwarzkopf
Article

Abstract

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.

Keywords

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.

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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

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