Incidences between points and curves with almost two degrees of freedom

Sharir, Micha; Solomon, Noam; Zlydenko, Oleg

doi:10.48550/arxiv.2003.02190

Cited by 3 publications

(6 citation statements)

References 12 publications

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“…Aside from the log factor, this bound generalizes the recent result of Sharir and Zlydenko [12] (see also Sharir, Solomon, and Zlydenko [10]) on incidences between so-called directed points and circles. A directed point is a pair (p, u) where p is a point in the plane and u is a direction, and (p, u) is incident to a circle c if p ∈ c and u is the direction of the tangent to c at p. The bound in [10,12] is O(m 3/5 n 3/5 + m + n) which is similar, albeit slightly sharper, than the bound in Theorem 10. The two setups are indeed related, as a directed point of degree at least two is a limiting case of a lens, and the resulting infinitesimal limit lenses are clearly pairwise non-overlapping.…”

Section: Circle-lens Incidence Boundssupporting

confidence: 82%

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On rich lenses in planar arrangements of circles and related problems

Ezra,

Raz,

Sharir

et al. 2020

Preprint

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We show that the maximum number of pairwise non-overlapping k-rich lenses (lenses formed by at least k circles) in an arrangement of n circles in the plane is O n 3/2 log (n/k 3 )+ n k , and the sum of the degrees of the lenses of such a family (where the degree of a lens is the number of circles+ n . Two independent proofs of these bounds are given, each interesting in its own right (so we believe). We then show that these bounds lead to the known bound of [1,8] on the number of point-circle incidences in the plane. Extensions to families of more general algebraic curves and some other related problems are also considered.

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Section: Circle-lens Incidence Boundssupporting

confidence: 82%

“…Combining the bounds in (10), (11), adding the overhead O(D 2 n), and making the constants in the O(•) notation explicit, bounding all of them by the same constant A, we obtain the recurrence asserted in the lemma.…”

Section: Second Proof Of Theorem 1: Reduction To Large Kmentioning

confidence: 96%

On rich lenses in planar arrangements of circles and related problems

Ezra,

Raz,

Sharir

et al. 2020

Preprint

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“…Very few earlier works have used this kind of restriction; see Elekes et al [5] for one of the few exceptions. Similar bounds have recently been obtained by the authors for other special cases of the incidence problem [26,27], using related but different approaches.…”

Section: Sketch Of Proofsupporting

confidence: 79%

On rich points and incidences with restricted sets of lines in 3-space

Sharir¹,

Solomon²

2020

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Let L be a set of n lines in R 3 that is contained, when represented as points in the four-dimensional Plücker space of lines in R 3 , in an irreducible variety T of constant degree which is non-degenerate with respect to L (see below). We show:(1) If T is two-dimensional, the number of r-rich points (points incident to at least r lines of L) is O(n 4/3+ε /r 2 ), for r ≥ 3 and for any ε > 0, and, if at most n 1/3 lines of L lie on any common regulus, there are at most O(n 4/3+ε ) 2-rich points. For r larger than some sufficiently large constant, the number of r-rich points is also O(n/r).As an application, we deduce (with an ε-loss in the exponent) the bound obtained by Pach and de Zeeuw [16] on the number of distinct distances determined by n points on an irreducible algebraic curve of constant degree in the plane that is not a line nor a circle.(2) If T is two-dimensional, the number of incidences between L and a set of m points in R 3 is O(m + n).(3) If T is three-dimensional and nonlinear, the number of incidences between L and a set of m points in R 3 is O m 3/5 n 3/5 + (m 11/15 n 2/5 + m 1/3 n 2/3 )s 1/3 + m + n , provided that no plane contains more than s of the points. When s = O(min{n 3/5 /m 2/5 , m 1/2 }), the bound becomes O(m 3/5 n 3/5 + m + n).As an application, we prove that the number of incidences between m points and n lines in R 4 contained in a quadratic hypersurface (which does not contain a hyperplane) is O(m 3/5 n 3/5 + m + n).The proofs use, in addition to various tools from algebraic geometry, recent bounds on the number of incidences between points and algebraic curves in the plane.

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“…As already mentioned, the results of this paper have recently been used in Sharir et al [42] for bounding incidences between points and curves with almost two degrees of freedom.…”

Section: Discussionmentioning

confidence: 89%

“…(3) The results and techniques in this paper have recently been applied in Sharir et al [42] to incidence problems in three dimensions with curves that have 'almost two degrees of freedom' (a notion defined in that paper).…”

Section: Remarks (1)mentioning

confidence: 99%

Incidences with curves in three dimensions

Sharir,

Solomon

2020

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We study incidence problems involving points and curves in R 3 . The current (and in fact only viable) approach to such problems, pioneered by Guth and Katz [23,24], requires a variety of tools from algebraic geometry, most notably (i) the polynomial partitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recent studies [25], by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for pointcurve incidence problems in R 3 .Incidences of this kind have been considered in several previous studies, starting with Guth and Katz's work on points and lines [24]. Our results, which are based on the work of Guth and Zahl [25] concerning surfaces that are doubly ruled by curves, provide a grand generalization of most of the previous results. We reconstruct the bound for points and lines, and improve, in certain signifcant ways, recent bounds involving points and circles (in [36]), and points and arbitrary constant-degree algebraic curves (in [35]). While in these latter instances the bounds are not known (and are strongly suspected not) to be tight, our bounds are, in a certain sense, the best that can be obtained with this approach, given the current state of knowledge.As an application of our point-curve incidence bound, we show that the number of triangles spanned by a set of n points in R 3 and similar to a given triangle is O(n 15/7 ), which improves the bound of Agarwal et al. [1]. Our results are also related to a study by Guth et al. (work in progress), and have been recently applied in Sharir et al. [42] to related incidence problems in three dimensions.

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Incidences between points and curves with almost two degrees of freedom

Cited by 3 publications

References 12 publications

On rich lenses in planar arrangements of circles and related problems

On rich lenses in planar arrangements of circles and related problems

On rich points and incidences with restricted sets of lines in 3-space

Incidences with curves in three dimensions

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