On the number of hinges defined by a point set in \mathbb R²

2019/02/15 by Misha Rudnev, Rudnev, Misha · 1 citation
Computer Science · Decision Sciences · Engineering · #11B75 #68R05 #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.1902.05791

openalex publication_date 2019/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is shown that the number of distinct types of three-point hinges, defined by a real plane set of n points is ≫ n²log^-3 n, where a hinge is identified by fixing two pair-wise distances in a point triple. This is achieved via strengthening (modulo a log n factor) of the Guth-Katz estimate for the number of pair-wise intersections of lines in \mathbb R³, arising in the context of the plane Erd\H os distinct distance problem, to a second moment incidence estimate. This relies, in particular, on the generalisation of the Guth-Katz incidence bound by Solomon and Sharir.

Citations

• On the number of classes of triangles determined by N points in \R²
• A short proof of Rudnev's point-plane incidence bound

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• Angle chains and pinned variants

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