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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 cdn4/3, unless C contains a line or a circle.

We also prove the lower bound cd′ min{m2/3n2/3, m2, n2} 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].

Let p1, p2, p3 be three noncollinear points in the plane, and let P be a set of n other points in the plane. We show that the number of distinct distances between p1, p2, p3 and the points of P is Ω(n6/11), improving the lower bound Ω(n0.502) of Elekes and Szabó [4] (and considerably simplifying the analysis).

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Distinct Distances on Algebraic Curves in the Plane

Distinct Distances from Three Points

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2 results

Distinct Distances on Algebraic Curves in the Plane

Distinct Distances from Three Points