Distinct Distances on Algebraic Curves in the Plane

Source abstract

Let S be a set of n points in ${\mathbb R}^{2}$ contained in an algebraic curve C of degree d . We prove that the number of distinct distances determined by S 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 [19].