On the number of classes of triangles determined by N points in \R²

2012/05/22 by Misha Rudnev, Rudnev, Misha · 2 citations
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Mathematics and Applications #Point processes and geometric inequalities #math.CO #math.NT #msc:11B75 #msc:68R05

paper · pdf · doi:10.48550/arxiv.1205.4865

6pp

arxiv created 2012/05/26 · arxiv updated 2012/05/29

Abstract

Let P be a set of N points in the Euclidean plane, where a positive proportion of points lies off a single straight line. This note points out two facts concerning the number of equivalence classes of triangles that P determines, namely that (i) P determines Ω(N²) different equivalence classes of congruent triangles, and (ii) P determines Ω((N²)/(log N)) different equivalence classes of similar triangles. The first fact follows from the recent theorem by Guth-Katz on point-line incidences in \R³. The second one, perhaps not so well known, is due to Solymosi and Tardos.

Cited by

• On the number of hinges defined by a point set in \mathbb R²
• Generalised Erdős distance theory on graphs

Related

• Congruence classes of triangles in \mathbbF_p²
• Triangles with one fixed side-length, a Furstenberg type problem, and incidences in finite vector spaces
• The number of unit-area triangles in the plane: Theme and variations
• Three-point configurations determined by subsets of \mathbbF_q² via the Elekes-Sharir paradigm
• Incidences with curves in three dimensions