Distinct distances on algebraic curves in the plane

TL;DR

This paper establishes lower bounds on the number of distinct distances determined by points on algebraic curves in the plane, with exceptions for special geometric configurations like lines and circles.

Contribution

It generalizes previous results by providing bounds for points on algebraic curves and between two such curves, identifying specific geometric exceptions.

Findings

At least c_d n^{4/3} distinct distances unless the curve contains a line or circle.

Lower bound c_d' min(m^{2/3}n^{2/3}, m^2, n^2) for distances between points on two algebraic curves.

Results extend and generalize prior work on distances between lines.

Abstract

Let $P$ be a set of $n$ points in the real plane contained in an algebraic curve $C$ of degree $d$. We prove that the number of distinct distances determined by $P$ is at least $c_d n^{4/3}$, unless $C$ contains a line or a circle. We also prove the lower bound $c_d' \min(m^{2/3}n^{2/3}, m^2, n^2)$ for the number of distinct distances between $m$ points on one irreducible plane algebraic curve and $n$ points on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer, and Solymosi in arXiv:1302.3081.