Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems

Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, Micha Sharir

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

1 Scopus citations

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 ?, which is also a plate, we can quickly answer various intersection queries, such as detecting whether ? intersects any plate of T, reporting all the plates intersected by ?, 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*(n4/3) storage (where the O*(·) notation hides subpolynomial factors) and answers an intersection query in O*(n2/3) time. Alternatively, by increasing the storage to O*(n3/2), the query time can be decreased to O*(n?), where ? = (2t - 3)/3(t - 1) < 2/3 and t = 3 is the number of parameters needed to represent the query arcs.

Original languageEnglish
Title of host publication38th International Symposium on Computational Geometry, SoCG 2022
EditorsXavier Goaoc, Michael Kerber
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772273
DOIs
StatePublished - 1 Jun 2022
Event38th International Symposium on Computational Geometry, SoCG 2022 - Berlin, Germany
Duration: 7 Jun 202210 Jun 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume224
ISSN (Print)1868-8969

Conference

Conference38th International Symposium on Computational Geometry, SoCG 2022
Country/TerritoryGermany
CityBerlin
Period7/06/2210/06/22

Keywords

  • Collision detection
  • Cylindrical algebraic decomposition
  • Intersection searching
  • Multilevel partition trees
  • Point-enclosure queries
  • Polynomial partitions
  • Ray-shooting queries
  • Semi-algebraic range searching

ASJC Scopus subject areas

  • Software

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