On ray shooting for triangles in 3-space and related problems

Esther Ezra, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We consider several problems that involve lines in three dimensions, and present improved algorithms for solving them. The problems include (i) ray shooting amid triangles in ℝ3, (ii) reporting intersections between query lines (segments, or rays) and input triangles, as well as approximately counting the number of such intersections, (iii) computing the intersection of two nonconvex polyhedra, (iv) detecting, counting, or reporting intersections in a set of lines in ℝ3, and (v) output-sensitive construction of an arrangement of triangles in three dimensions. Our approach is based on the polynomial partitioning technique. For example, our ray-shooting algorithm processes a set of n triangles in R3 into a data structure for answering ray shooting queries amid the given triangles, which uses O(n3/2+ε) storage and preprocessing, and answers a query in O(n1/2+ε) time, for any ε > 0. This is a significant improvement over known results, obtained more than 25 years ago, in which, with this amount of storage, the query time bound is roughly n5/8. The algorithms for the other problems have similar performance bounds, with similar improvements over previous results. We also derive a nontrivial improved tradeoff between storage and query time. Using it, we obtain algorithms that answer m queries on n objects in max {O(m2/3n5/6+ε + n1+ε), O(m5/6+εn2/3 + m1+ε) time, for any ε > 0, again an improvement over the earlier bounds.

Original languageEnglish
Title of host publication37th International Symposium on Computational Geometry, SoCG 2021
EditorsKevin Buchin, Eric Colin de Verdiere
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771849
StatePublished - 1 Jun 2021
Event37th International Symposium on Computational Geometry, SoCG 2021 - Virtual, Buffalo, United States
Duration: 7 Jun 202111 Jun 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference37th International Symposium on Computational Geometry, SoCG 2021
Country/TerritoryUnited States
CityVirtual, Buffalo

Bibliographical note

Publisher Copyright:
© Esther Ezra and Micha Sharir; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).


  • Polynomial partitioning
  • Ray shooting
  • Three dimensions
  • Tradeoff


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