Surfaces containing two circles through each point

Skopenkov, Mikhail; Krasauskas, Rimvydas

doi:10.1007/s00208-018-1739-z

Cited by 19 publications

(70 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We recall that the only surfaces that are infinitely ruled by lines are planes (see, e.g., Fuchs and Tabachnikov [31,Corollary 16.2]), and that the only surfaces that are infinitely ruled by circles are spheres and planes (see, e.g., Lubbes [45,Theorem 3] and Schicho [52]; see also Skopenkov and Krasauskas [62] for recent work on celestials, namely surfaces doubly ruled by circles, and Nilov and Skopenkov [48], proving that a surface that is ruled by a line and a circle through each point is a quadric). It should be noted that, in general, for this definition to make sense, it is important to require that the degree E of the ruling curves be much smaller than deg(V ).…”

Section: Our Resultsmentioning

confidence: 99%

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

View full text Add to dashboard Cite

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...

show abstract

Section: Our Resultsmentioning

confidence: 99%

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

View full text Add to dashboard Cite

show abstract

“…In fact, we can replace the requirement that at most B circles lie in a variety of degree ≤ 400 with the requirement that at most B circles lie in a surface that is doubly ruled by circles. The set of all such surfaces has been classified in [19]. However this classification will not be relevant for our proof.…”

Section: Eliminating Depth Cycles Of Type 2 3 Andmentioning

confidence: 99%

Breaking the 3/2 Barrier for Unit Distances in Three Dimensions

Zahl

2018

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…This surface is characterized by the third row of Theorem 1. There has been recent interest in the classification of surfaces that contain at least two circles through each point [12,15]. Surfaces that contain infinitely many circles 2020/08/27 13:32…”

Section: Problem Classify Up To Möbius Equivalence Real Surfaces Tmentioning

confidence: 99%