When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.4
URN: urn:nbn:de:0030-drops-160126
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16012/
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Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems

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Abstract

Let 𝒯 be a set of n planar semi-algebraic regions in ℝ³ of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess 𝒯 into a data structure so that for a query object γ, which is also a plate, we can quickly answer various intersection queries, such as detecting whether γ intersects any plate of 𝒯, reporting all the plates intersected by γ, or counting them. We focus on two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree algebraic arcs in ℝ³ (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in ℝ³. These interesting special cases form the building blocks for the general case.
By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we obtain a variety of results with different storage and query-time bounds, depending on the complexity of the input and query objects. For example, if 𝒯 is a set of plates and the query objects are arcs, we obtain a data structure that uses O^*(n^{4/3}) storage (where the O^*(⋅) notation hides subpolynomial factors) and answers an intersection query in O^*(n^{2/3}) time. Alternatively, by increasing the storage to O^*(n^{3/2}), the query time can be decreased to O^*(n^{ρ}), where ρ = (2t-3)/3(t-1) < 2/3 and t ≥ 3 is the number of parameters needed to represent the query arcs.

BibTeX - Entry

```@InProceedings{agarwal_et_al:LIPIcs.SoCG.2022.4,
author =	{Agarwal, Pankaj K. and Aronov, Boris and Ezra, Esther and Katz, Matthew J. and Sharir, Micha},
title =	{{Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems}},
booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
pages =	{4:1--4:14},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-227-3},
ISSN =	{1868-8969},
year =	{2022},
volume =	{224},
editor =	{Goaoc, Xavier and Kerber, Michael},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},