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

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

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Intersection searching
  • Semi-algebraic range searching
  • Point-enclosure queries
  • Ray-shooting queries
  • Polynomial partitions
  • Cylindrical algebraic decomposition
  • Multi-level partition trees
  • Collision detection

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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.

Cite As Get BibTex

Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, and Micha Sharir. Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.4

Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Boris Aronov
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Esther Ezra
  • School of Computer Science, Bar Ilan University, Ramat Gan, Israel
Matthew J. Katz
  • Department of Computer Science, Ben Gurion University, Beer Sheva, Israel
Micha Sharir
  • School of Computer Science, Tel Aviv University, Tel Aviv, Israel

Funding

  • Agarwal, Pankaj K.: Work partially supported by NSF grants IIS-18-14493 and CCF-20-07556.
  • Aronov, Boris: Work partially supported by NSF Grants CCF-15-40656 and CCF-20-08551, and by Grant 2014/170 from the US-Israel Binational Science Foundation.
  • Ezra, Esther: Work partially supported by NSF CAREER under Grant CCF:AF-1553354 and by Grant 824/17 from the Israel Science Foundation.
  • Katz, Matthew J.: Work partially supported by Grant 1884/16 from the Israel Science Foundation, and by Grant 2019715/CCF-20-08551 from the US-Israel Binational Science Foundation/US National Science Foundation.
  • Sharir, Micha: Work partially supported by Grant 260/18 from the Israel Science Foundation.

Acknowledgements

We thank Peyman Afshani for sharing with us problems that have motivated our study of segment-intersection searching amid spherical caps, Ovidiu Daescu for suggesting the collision-detection application that motivated some aspects of our work, and the reviewers of this work for their helpful comments.

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