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

Orit E. Raz; Micha Sharir

doi:https://doi.org/10.1007/s00493-016-3440-8

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

Orit E. Raz, Micha Sharir

School of Computer Science

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n^20/9), improving the earlier bound O(n^9/4) of Apfelbaum and Sharir [2]. We also 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).

Original language	English
Pages (from-to)	1221-1240
Number of pages	20
Journal	Combinatorica
Volume	37
Issue number	6
DOIs	https://doi.org/10.1007/s00493-016-3440-8
State	Published - 1 Dec 2017

All Science Journal Classification (ASJC) codes

Discrete Mathematics and Combinatorics
Computational Mathematics

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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 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).",

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N2 - 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 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).

AB - 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 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).

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JF - Combinatorica

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ER -

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

Abstract

All Science Journal Classification (ASJC) codes

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