Distinct Distances on Algebraic Curves in the Plane
2016 ◽
Vol 26
(1)
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pp. 99-117
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LetSbe a set ofnpoints in${\mathbb R}^{2}$contained in an algebraic curveCof degreed. We prove that the number of distinct distances determined bySis at leastcdn4/3, unlessCcontains a line or a circle.We also prove the lower boundcd′ min{m2/3n2/3,m2,n2} for the number of distinct distances betweenmpoints on one irreducible plane algebraic curve andnpoints 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].
2001 ◽
Vol 11
(04)
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pp. 439-453
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Keyword(s):
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2011 ◽
Vol 07
(04)
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pp. 921-931
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Keyword(s):
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2007 ◽
Vol 2007
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pp. 1-11
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