Intersection Searching Amid Tetrahedra in 4-Space and Efficient Continuous Collision Detection

Esther Ezra, Micha Sharir

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

1 Scopus citations

Abstract

We develop data structures for intersection detection queries in four dimensions that involve segments, triangles and tetrahedra. Specifically, we study two main problems: (i) Preprocess a set of n tetrahedra in R4 into a data structure for answering segment-intersection queries amid the given tetrahedra (referred to as segment-tetrahedron intersection queries), and (ii) Preprocess a set of n triangles in R4 into a data structure that supports triangle-intersection queries amid the input triangles (referred to as triangle-triangle intersection queries). As far as we can tell, these problems have not been previously studied. For problem (i), we first present a "standard" solution which, for any prespecified value n ≤ s ≤ n6 of a so-called storage parameter s, yields a data structure with O*(s) storage and expected preprocessing, which answers an intersection query in O*(n/s1/6) time (here and in what follows, the O*( ) notation hides subpolynomial factors). For problem (ii), using similar arguments, we present a solution that has the same asymptotic performance bounds. We then improve the solution for problem (i), and present a more intricate data structure that uses O*(n2) storage and expected preprocessing, and answers a segment-tetrahedron intersection query in O*(n1/2) time. Using the parametric search technique of Agarwal and Matoušek [3], we can obtain data structures with similar performance bounds for the ray-shooting problem amid tetrahedra in R4. Unfortunately, so far we do not know how to obtain a similar improvement for problem (ii). Our algorithms are based on a primal-dual technique for range searching with semi-algebraic sets, based on recent advances in this area [2, 11]. As this is a result of independent interest, we spell out the details of this technique. As an application, we present a solution to the problem of "continuous collision detection" amid moving tetrahedra in 3-space. That is, the workspace consists of n tetrahedra, each moving at its own fixed velocity, and the goal is to detect a collision between some pair of moving tetrahedra. Using our solutions to problems (i) and (ii), we obtain an algorithm that detects a collision in O*(n12/7) expected time. We also present further applications, including an output-sensitive algorithm for constructing the arrangement of n tetrahedra in R4 and an output-sensitive algorithm for constructing the intersection or union of two or several nonconvex polyhedra in R4.

Original languageEnglish
Title of host publication30th Annual European Symposium on Algorithms, ESA 2022
EditorsShiri Chechik, Gonzalo Navarro, Eva Rotenberg, Grzegorz Herman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772471
DOIs
StatePublished - 1 Sep 2022
Event30th Annual European Symposium on Algorithms, ESA 2022 - Berlin/Potsdam, Germany
Duration: 5 Sep 20229 Sep 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume244
ISSN (Print)1868-8969

Conference

Conference30th Annual European Symposium on Algorithms, ESA 2022
Country/TerritoryGermany
CityBerlin/Potsdam
Period5/09/229/09/22

Bibliographical note

Publisher Copyright:
© 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

Keywords

  • Computational geometry
  • Intersection queries in R4
  • Polynomial partitioning
  • Range searching
  • Ray shooting
  • Semi-algebraic sets
  • Tetrahedra in R4
  • Tradeoff

Fingerprint

Dive into the research topics of 'Intersection Searching Amid Tetrahedra in 4-Space and Efficient Continuous Collision Detection'. Together they form a unique fingerprint.

Cite this