The Number of Unit-Area Triangles in the Plane: Theme and Variations

Orit E. Raz, Micha Sharir

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

4 Scopus citations

Abstract

We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n20/9), improving the earlier bound O(n9/4) of Apfelbaum and Sharir [2]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if S consists of points on three lines, the number of unit-area triangles that S spans can be ω(n2), for any triple of lines (it is always O(n2) in this case). (ii) We show that if S is a convex grid of the form A × B, where A, B are convex sets of n1/2 real numbers each (i.e., the sequences of differences of consecutive elements of A and of B are both strictly increasing), then S determines O(n31/14) unit-area triangles.

Original languageAmerican English
Title of host publication31st International Symposium on Computational Geometry, SoCG 2015
EditorsJanos Pach, Janos Pach, Lars Arge
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages569-583
Number of pages15
ISBN (Electronic)9783939897835
DOIs
StatePublished - 1 Jun 2015
Externally publishedYes
Event31st International Symposium on Computational Geometry, SoCG 2015 - Eindhoven, Netherlands
Duration: 22 Jun 201525 Jun 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume34
ISSN (Print)1868-8969

Conference

Conference31st International Symposium on Computational Geometry, SoCG 2015
Country/TerritoryNetherlands
CityEindhoven
Period22/06/1525/06/15

Keywords

  • Combinatorial geometry
  • Incidences
  • Repeated configurations

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