{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article13670","name":"Incidences Between Points and Curves with Almost Two Degrees of Freedom","abstract":"We study incidences between points and (constant-degree algebraic) curves in three dimensions, taken from a family C of curves that have almost two degrees of freedom, meaning that (i) every pair of curves of C intersect in O(1) points, (ii) for any pair of points p, q, there are only O(1) curves of C that pass through both points, and (iii) a pair p, q of points admit a curve of C that passes through both of them if and only if F(p,q)=0 for some polynomial F of constant degree associated with the problem. (As an example, the family of unit circles in \u211d\u00b3 that pass through some fixed point is such a family.)\r\nWe begin by studying two specific instances of this scenario. The first instance deals with the case of unit circles in \u211d\u00b3 that pass through some fixed point (so called anchored unit circles). In the second case we consider tangencies between directed points and circles in the plane, where 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 tangent to a circle \u03b3 if p \u2208 \u03b3 and u is the direction of the tangent to \u03b3 at p. A lifting transformation due to Ellenberg et al. maps these tangencies to incidences between points and curves (\"lifted circles\") in three dimensions. In both instances we have a family of curves in \u211d\u00b3 with almost two degrees of freedom.\r\nWe show that the number of incidences between m points and n anchored unit circles in \u211d\u00b3, as well as the number of tangencies between m directed points and n arbitrary circles in the plane, is O(m^(3\/5)n^(3\/5)+m+n) in both cases.\r\nWe then derive a similar incidence bound, with a few additional terms, for more general families of curves in \u211d\u00b3 with almost two degrees of freedom, under a few additional natural assumptions.\r\nThe proofs follow standard techniques, based on polynomial partitioning, but they face a critical novel issue involving the analysis of surfaces that are infinitely ruled by the respective family of curves, as well as of surfaces in a dual three-dimensional space that are infinitely ruled by the respective family of suitably defined dual curves. We either show that no such surfaces exist, or develop and adapt techniques for handling incidences on such surfaces.\r\nThe general bound that we obtain is O(m^(3\/5)n^(3\/5)+m+n) plus additional terms that depend on how many curves or dual curves can lie on an infinitely-ruled surface.","keywords":["Incidences","Polynomial partition","Degrees of freedom","Infinitely-ruled surfaces","Three dimensions"],"author":[{"@type":"Person","name":"Sharir, Micha","givenName":"Micha","familyName":"Sharir","email":"mailto:michas@tau.ac.il","affiliation":"School of Computer Science, Tel Aviv University, Israel","funding":"Partially supported by ISF Grant 260\/18, by grant 1367\/2016 from the German-Israeli Science Foundation (GIF), and by Blavatnik Research Fund in Computer Science at Tel Aviv University."},{"@type":"Person","name":"Zlydenko, Oleg","givenName":"Oleg","familyName":"Zlydenko","email":"mailto:zlydenko@gmail.com","affiliation":"School of Computer Science, Tel Aviv University, Israel"}],"position":66,"pageStart":"66:1","pageEnd":"66:14","dateCreated":"2020-06-08","datePublished":"2020-06-08","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Sharir, Micha","givenName":"Micha","familyName":"Sharir","email":"mailto:michas@tau.ac.il","affiliation":"School of Computer Science, Tel Aviv University, Israel","funding":"Partially supported by ISF Grant 260\/18, by grant 1367\/2016 from the German-Israeli Science Foundation (GIF), and by Blavatnik Research Fund in Computer Science at Tel Aviv University."},{"@type":"Person","name":"Zlydenko, Oleg","givenName":"Oleg","familyName":"Zlydenko","email":"mailto:zlydenko@gmail.com","affiliation":"School of Computer Science, Tel Aviv University, Israel"}],"copyrightYear":"2020","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2020.66","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["https:\/\/doi.org\/10.1017\/S0305004115000468","http:\/\/arxiv.org\/abs\/1302.6710","https:\/\/doi.org\/10.1016\/j.jcta.2005.07.002","http:\/\/arxiv.org\/abs\/2003.02190"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6367","volumeNumber":164,"name":"36th International Symposium on Computational Geometry (SoCG 2020)","dateCreated":"2020-06-08","datePublished":"2020-06-08","editor":[{"@type":"Person","name":"Cabello, Sergio","givenName":"Sergio","familyName":"Cabello","email":"mailto:sergio.cabello@fmf.uni-lj.si","sameAs":"https:\/\/orcid.org\/0000-0002-3183-4126","affiliation":"University of Ljubljana, Ljubljana, Slovenia"},{"@type":"Person","name":"Chen, Danny Z.","givenName":"Danny Z.","familyName":"Chen","email":"mailto:dchen@nd.edu","sameAs":"https:\/\/orcid.org\/0000-0001-6565-2884","affiliation":"University of Notre Dame, Indiana, USA"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article13670","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6367"}}}