Counting and representing intersections among triangles in three dimensions

Esther Ezra*, Micha Sharir

*Corresponding author for this work

Research output: Contribution to conferencePaperpeer-review

1 Scopus citations


We present an algorithm that efficiently counts all intersecting triples among a collection T of triangles in ℝ3 in nearly-quadratic time. This solves a problem posed by Pellegrini. Using a variant of the technique, one can represent the set of all κ triple intersections, in compact form, as the disjoint union of complete tripartite hypergraphs, which requires nearly-quadratic construction time and storage. Our approach also applies to any collection of convex planar objects of constant description complexity in ℝ3, with the same performance bounds. We also prove that this counting problem belongs to the 3SUM-hard family, and thus our algorithm is likely to be nearly optimal (since it is believed that 3SUM-hard problems cannot be solved in subquadratic time).

Original languageEnglish
Number of pages10
StatePublished - 2004
EventProceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04) - Brooklyn, NY, United States
Duration: 9 Jun 200411 Jun 2004


ConferenceProceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04)
Country/TerritoryUnited States
CityBrooklyn, NY


FundersFunder number
Israel Science Fund
Israeli Academy of Sciences
National Science FoundationCCR-00-98246, CCR-97-32101
Anacostia Community Museum
United States-Israel Binational Science Foundation
Tel Aviv University


    • 3SUM-hard problems
    • Arrangements
    • Counting intersections
    • Curve-sensitive cuttings
    • Triangles in three dimensions


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