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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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


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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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • 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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