Non-Degenerate Spheres in Three Dimensions

Apfelbaum, Roel; Sharir, Micha

doi:10.1017/s0963548311000010

Cited by 11 publications

(29 citation statements)

References 13 publications

(41 reference statements)

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“…For the residual subgraph G 0 (P, S), we derive a sharp bound on the actual number of incidences that it encodes (namely, the number of its edges). This generalizes previous results in which one had to require that G(P, S) does not contain some fixed-size complete bipartite graph, or (only for spheres or planes) that the surfaces in S be "non-degenerate" ( [5,26]; see below).…”

Section: The Setupssupporting

confidence: 80%

“…Initial partial results go back to Chung [22] and to Clarkson et al [23], and continue with the work of Aronov et al [7]. Later, Agarwal et al [1] have bounded the number of non-degenerate spheres with respect to a given point set; their bound was subsequently improved by Apfelbaum and Sharir [5]. 2 The aforementioned recent work of Zahl [69] can be applied in the case of spheres if one assumes that no three, or any larger but constant number, of the spheres intersect in a common circle.…”

Section: Introductionmentioning

confidence: 79%

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Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances

Sharir¹,

Solomon²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study a wide spectrum of incidence problems involving points and curves or points and surfaces in R 3 . The current (and in fact the only viable) approach to such problems, pioneered by Guth and Katz [38,39], 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 [40], by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for numerous incidence problems in R 3 .In broad terms, we consider two kinds of problems, those involving points and constant-degree algebraic curves, and those involving points and constant-degree algebraic surfaces. In some variants we assume that the points lie on some fixed constant-degree algebraic variety, and in others we consider arbitrary sets of points in 3-space.The case of points and curves has been considered in several previous studies, starting with Guth and Katz's work on points and lines [39]. Our results, which are based on a recent work of Guth and Zahl [40] concerning surfaces that are doubly ruled by curves, provide a grand generalization of all previous results. We reconstruct the bound for points and lines, and improve, in certain signifcant ways, recent bounds involving points and circles (in [54]), and points and arbitrary constant-degree algebraic curves (in [53]). 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.In the case of points and surfaces, the incidence graph between them can contain large complete bipartite graphs, each involving points on some curve and surfaces containing this curve (unlike earlier studies, we do not rule out this possibility, which makes our approach more general). Our bounds estimate the total size of the vertex sets in such a complete bipartite graph decomposition of the incidence graph. In favorable cases, our bounds translate into actual incidence bounds. Overall, here too our results provide a "grand generalization" of most of the previous studies of (special instances of) this problem.As an application of our point-curve incidence bound, we consider the problem of bounding the number of similar triangles spanned by a set of n points in R 3 . We obtain the bound O(n 15/7 ), which improves the bound of Agarwal et al. [1].As applications of our point-surface incidence bounds, we consider the problems of distinct and repeated distances determined by a set of n points in R 3 , two of the most celebrated open problems in combinatorial geometry. We obtain new and improved bounds for two special cases, one in which the points lie on some algebraic variety of constant degree, and one involving incidences between pairs in P 1 × P 2 , where P 1 is contained in a variety and P 2 is arbitrary.Recently, Sharir and Zahl [60] have considered general families of con...

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Section: The Setupssupporting

confidence: 80%

Section: Introductionmentioning

confidence: 79%

Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances

Sharir¹,

Solomon²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…As already mentioned in the introduction, this slightly improves a previous bound in [6] (see also [1]). …”

Section: Remarkssupporting

confidence: 75%

“…The problem of obtaining good bounds for F (m) is motivated by questions in exact pattern matching, and has been studied in several previous works (see [1,4,6,10]). Theorem 1.2 implies the bound F (m) = O(m 15/7 ), which slightly improves upon the previous bound of O * (m 58/27 ) from [6] (in the previous bound, the O * () notation only hides polylogarithmic factors); see also [1]. The new bound is an almost immediate corollary of Theorem 1.2, while the previous bound requires a more complicated analysis.…”

Section: Introductionmentioning

confidence: 99%

Improved bounds for incidences between points and circles

Sharir

Sheffer

Zahl

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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We establish an improved upper bound for the number of incidences between m points and n arbitrary circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, is O * ( m 2/3 n 2/3 + m 6/11 n 9/11 + m + n )(where the O * (·) notation hides sub-polynomial factors). Since all the points and circles may lie on a common plane or sphere, it is impossible to improve the bound in R 3 without first improving it in the plane. Nevertheless, we show that if the set of circles is required to be "truly three-dimensional" in the sense that no sphere or plane contains more than q of the circles, for some q ≪ n, then the bound can be improved to O * ( m 3/7 n 6/7 + m 2/3 n 1/2 q 1/6 + m 6/11 n 15/22 q 3/22 + m + n ) . For various ranges of parameters (e.g., when m = Θ(n) and q = o(n 7/9 )), this bound is smaller than the best known twodimensional worst-case lower bound Ω * (m 2/3 n 2/3 + m + n).We present several extensions and applications of the new bound: (i) For the special case where all the circles have the same radius, we obtain the improved bound O * ( m 5/11 n 9/11 + m 2/3 n 1/2 q 1/6 + m + n ). (ii) We present an improved analysis that removes the subpolynomial factors from the bound when m = O(n 3/2−ε ) for any fixed ε > 0. (iii) We use our results to obtain the improved bound * O(m 15/7 ) for the number of mutually similar triangles determined by any set of m points in R 3 .Our result is obtained by applying the polynomial partitioning technique of Guth and Katz using a constant-degree partitioning polynomial (as was also recently used by Solymosi and Tao). We also rely on various additional tools from analytic, algebraic, and combinatorial geometry.

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“…We say that S is α-non-degenerate with respect to a set of points P if for every circle C ⊂ S we have |C ∩ S ∩ P| ≤ α|S ∩ P|. Theorem 2.5 (Apfelbaum and Sharir [1]). Let P be a set of m points in R 3 .…”

Section: Existing Incidence Bounds For Points Circles and Spheresmentioning

confidence: 99%