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DOI: 10.4230/LIPIcs.ESA.2022.51
URN: urn:nbn:de:0030-drops-169895
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16989/
Ezra, Esther ;
Sharir, Micha
Intersection Searching Amid Tetrahedra in 4-Space and Efficient Continuous Collision Detection
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 {ℝ}⁴ 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 {ℝ}⁴ 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/s^{1/6}) 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^*(n^{1/2}) time. Using the parametric search technique of Agarwal and Matoušek [P. K. Agarwal and J. Matoušek, 1993], we can obtain data structures with similar performance bounds for the ray-shooting problem amid tetrahedra in {ℝ}⁴. 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 [P. K. Agarwal et al., 2021; J. Matoušek and Z. Patáková, 2015]. 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^*(n^{12/7}) 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.
BibTeX - Entry
@InProceedings{ezra_et_al:LIPIcs.ESA.2022.51,
author = {Ezra, Esther and Sharir, Micha},
title = {{Intersection Searching Amid Tetrahedra in 4-Space and Efficient Continuous Collision Detection}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {51:1--51:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16989},
URN = {urn:nbn:de:0030-drops-169895},
doi = {10.4230/LIPIcs.ESA.2022.51},
annote = {Keywords: Computational geometry, Ray shooting, Tetrahedra in \{\mathbb{R}\}⁴, Intersection queries in \{\mathbb{R}\}⁴, Polynomial partitioning, Range searching, Semi-algebraic sets, Tradeoff}
}
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Keywords: |
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Computational geometry, Ray shooting, Tetrahedra in {ℝ}⁴, Intersection queries in {ℝ}⁴, Polynomial partitioning, Range searching, Semi-algebraic sets, Tradeoff |
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Collection: |
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30th Annual European Symposium on Algorithms (ESA 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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01.09.2022 |
