Distinct distances on algebraic curves in the plane
Distinct distances on algebraic curves in the plane
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DOI：
10.1145/2582112.2582135
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发表时间：
2013-08
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影响因子：
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通讯作者：
J. Pach;Frank de Zeeuw
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文献类型：
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作者：
J. Pach;Frank de Zeeuw
设S是包含在d次代数曲线C中的R2中的n个点的集合。我们证明了由S确定的不同距离的数目至少是cdn 4/3，除非C包含一条直线或一个圆。我们还证明了下界c'd min{m2/3 n2/3，m2，n2}的数量不同的距离之间的m点在一个不可约的平面代数曲线和n点在另一个，除非这两条曲线是平行线，正交线，或同心圆。这推广了[18]中Sharir、Sheffer和Solymosi关于直线间距离的一个结果。
Let S be a set of n points in R2 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 c'd 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 [18].