Distinct Distances on Non-Ruled Surfaces and Between Circles

Mathialagan, Surya; Sheffer, Adam

doi:10.1007/s00454-022-00449-x

Cited by 3 publications

(8 citation statements)

References 23 publications

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“…The following lemma is a generalization of a lemma of Mathialagan and Sheffer [12], which in turn adapts a result of Raz [15]. Lemma 1.7.…”

Section: Introductionmentioning

confidence: 80%

“…Proof. (a) To see that parallel lines, orthogonal lines, and aligned circles span few distances, see for example [12].…”

Section: First Distinct Distances Boundsmentioning

confidence: 99%

“…Our technique. Our analysis begins with the proof of Mathialagan and Sheffer [12], but significantly generalizes this proof. We now briefly describe one of the main tools of our analysis.…”

Section: Introductionmentioning

confidence: 99%

“…In other words, for any two curves that do not span many distances, there exist point sets that span a linear number of distinct distances. Mathialagan and Sheffer [12] studied the case where C 1 and C 2 are circles in R 3 . The axis of a circle C ⊂ R 3 is the line passing through the center of C and orthogonal to the plane that contains C. Two circles are aligned if they share a common axis.…”

Section: Introductionmentioning

confidence: 99%

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Distinct Distances in $R^3$ Between Quadratic and Orthogonal Curves

Aldape¹,

Liu²,

Pylypovych³

et al. 2023

Preprint

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We study the minimum number of distinct distances between point sets on two curves in R 3 . Assume that one curve contains m points and the other n points. Our main results:(a) When the curves are conic sections, we characterize all cases where the number of distances is O(m + n). This includes new constructions for points on two parabolas, two ellipses, and one ellipse and one hyperbola. In all other cases, the number of distances is Ω(min{m 2/3 n 2/3 , m 2 , n 2 }).(b) When the curves are not necessarily algebraic but smooth and contained in perpendicular planes, we characterize all cases where the number of distances is O(m + n). This includes a surprising new construction of non-algebraic curves that involve logarithms. In all other cases, the number of distances is Ω(min{m 2/3 n 2/3 , m 2 , n 2 }).

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“…The following lemma is a generalization of a lemma of Mathialagan and Sheffer [12], which in turn adapts a result of Raz [15]. Lemma 1.7.…”

Section: Introductionmentioning

confidence: 80%

“…Proof. (a) To see that parallel lines, orthogonal lines, and aligned circles span few distances, see for example [12].…”

Section: First Distinct Distances Boundsmentioning

confidence: 99%

“…Our technique. Our analysis begins with the proof of Mathialagan and Sheffer [12], but significantly generalizes this proof. We now briefly describe one of the main tools of our analysis.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Distinct Distances in $R^3$ Between Quadratic and Orthogonal Curves

Aldape¹,

Liu²,

Pylypovych³

et al. 2023

Preprint

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“…A number of recent papers (see for example Sharir and Solomon (2016) and Mathialagan and Sheffer (2023)) study the question of the minimum number of distinct distances on varieties of degree 2 in R 3 .…”

Section: Distinct Angles On Surfacesmentioning

confidence: 99%

Distinct Angles and Angle Chains in Three Dimensions

Ruben¹,

Livia²,

Duke³

et al. 2023

Discrete Mathematics &Amp; Theoretical Computer Science

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In 1946, Erd\H{o}s posed the distinct distance problem, which seeks to find the minimum number of distinct distances between pairs of points selected from any configuration of $n$ points in the plane. The problem has since been explored along with many variants, including ones that extend it into higher dimensions. Less studied but no less intriguing is Erd\H{o}s' distinct angle problem, which seeks to find point configurations in the plane that minimize the number of distinct angles. In their recent paper "Distinct Angles in General Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum number of distinct angles in the plane in general position, which prohibits three points on any line or four on any circle. We consider the question of distinct angles in three dimensions and provide bounds on the minimum number of distinct angles in general position in this setting. We focus on pinned variants of the question, and we examine explicit constructions of point configurations in $\mathbb{R}^3$ which use self-similarity to minimize the number of distinct angles. Furthermore, we study a variant of the distinct angles question regarding distinct angle chains and provide bounds on the minimum number of distinct chains in $\mathbb{R}^2$ and $\mathbb{R}^3$.

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