Counting and representing intersections among triangles in three dimensions

Authors:
Esther Ezra;Micha Sharir
Affiliations:
Tel Aviv University, Tel Aviv, Israel;Tel Aviv University, Tel Aviv, Israel and New York University, New York, NY
Venue:
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Year:
2004

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Abstract

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, [18]. 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).