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Register a new userOn rich lenses in planar arrangements of circles and related problems
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Added by
Micha Sharir
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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 $Oleft(frac{n^{3/2}log{(n/k^3)}}{k^{5/2}} + frac{n}{k} right)$, and the sum of the degrees of the lenses of such a family (where the degree of a lens is the number of circles that form it) is $Oleft(frac{n^{3/2}log{(n/k^3)}}{k^{3/2}} + nright)$. 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 Agarwal et al. (JACM 2004) and Marcus and Tardos (JCTA 2006) 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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In this paper, we study arrangements of orthogonal circles, that is, arrangements of circles where every pair of circles must either be disjoint or intersect at a right angle. Using geometric arguments, we show that such arrangements have only a linear number of faces. This implies that orthogonal circle intersection graphs have only a linear number of edges. When we restrict ourselves to orthogonal unit circles, the resulting class of intersection graphs is a subclass of penny graphs (that is, contact graphs of unit circles). We show that, similarly to penny graphs, it is NP-hard to recognize orthogonal unit circle intersection graphs.
Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as points in the four-dimensional Plucker space of lines in $R^3$, in an irreducible variety $T$ of constant degree which is emph{non-degenerate} with respect to $L$ (see below). We show: medskip oindent{bf (1)} If $T$ is two-dimensional, the number of $r$-rich points (points incident to at least $r$ lines of $L$) is $O(n^{4/3+epsilon}/r^2)$, for $r ge 3$ and for any $epsilon>0$, and, if at most $n^{1/3}$ lines of $L$ lie on any common regulus, there are at most $O(n^{4/3+epsilon})$ $2$-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 $epsilon$-loss in the exponent) the bound obtained by Pach and de Zeeuw (2107) 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. medskip oindent{bf (2)} If $T$ is two-dimensional, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $O(m+n)$. medskip oindent{bf (3)} If $T$ is three-dimensional and nonlinear, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $Oleft(m^{3/5}n^{3/5} + (m^{11/15}n^{2/5} + m^{1/3}n^{2/3})s^{1/3} + m + n right)$, 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.
Motivated by the successful application of geometry to proving the Harary-Hill Conjecture for pseudolinear drawings of $K_n$, we introduce pseudospherical drawings of graphs. A spherical drawing of a graph $G$ is a drawing in the unit sphere $mathbb{S}^2$ in which the vertices of $G$ are represented as points -- no three on a great circle -- and the edges of $G$ are shortest-arcs in $mathbb{S}^2$ connecting pairs of vertices. Such a drawing has three properties: (1) every edge $e$ is contained in a simple closed curve $gamma_e$ such that the only vertices in $gamma_e$ are the ends of $e$; (2) if $e e f$, then $gamma_ecapgamma_f$ has precisely two crossings; and (3) if $e e f$, then $e$ intersects $gamma_f$ at most once, either in a crossing or an end of $e$. We use Properties (1)--(3) to define a pseudospherical drawing of $G$. Our main result is that, for the complete graph, Properties (1)--(3) are equivalent to the same three properties but with precisely two crossings in (2) replaced by at most two crossings. The proof requires a result in the geometric transversal theory of arrangements of pseudocircles. This is proved using the surprising result that the absence of special arcs ( coherent spirals) in an arrangement of simple closed curves characterizes the fact that any two curves in the arrangement have at most two crossings. Our studies provide the necessary ideas for exhibiting a drawing of $K_{10}$ that has no extension to an arrangement of pseudocircles and a drawing of $K_9$ that does extend to an arrangement of pseudocircles, but no such extension has all pairs of pseudocircles crossing twice.
We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a `truly $d$-dimensional configuration of points $v_1,ldots,v_n in mathbb{C}^d$. More formally, we show that, if the number of $r$-rich lines is significantly larger than $n^2/r^d$ then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor $r^d$ can be replaced with a tight $r^{d+1}$. If true, this would generalize the classic Szemeredi-Trotter theorem which gives a bound of $n^2/r^3$ on the number of $r$-rich lines in a planar configuration. This conjecture was shown to hold in $mathbb{R}^3$ in the seminal work of Guth and Katz cite{GK10} and was also recently proved over $mathbb{R}^4$ (under some additional restrictions) cite{SS14}. For the special case of arithmetic progressions ($r$ collinear points that are evenly distanced) we give a bound that is tight up to low order terms, showing that a $d$-dimensional grid achieves the largest number of $r$-term progressions. The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree $r-2$ Veronese embedding takes $r$-collinear points to $r$ linearly dependent images. Hence, each collinear $r$-tuple of points, gives us a dependent $r$-tuple of images. We then use the design-matrix method of cite{BDWY12} to convert these local linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set.
The ErdH{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Less well known is ErdH{o}s distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the plane. The standard problem is already well understood. However, it admits many of the same variants as the distinct distance problem, many of which are unstudied. We provide upper and lower bounds on a broad class of distinct angle problems. We show that the number of distinct angles formed by $n$ points in general position is $O(n^{log_2(7)})$, providing the first non-trivial bound for this quantity. We introduce a new class of asymptotically optimal point configurations with no four cocircular points. Then, we analyze the sensitivity of asymptotically optimal point sets to perturbation, yielding a much broader class of asymptotically optimal configurations. In higher dimensions we show that a variant of Lenzs construction admits fewer distinct angles than the optimal configurations in two dimensions. We also show that the minimum size of a maximal subset of $n$ points in general position admitting only unique angles is $Omega(n^{1/5})$ and $O(n^{log_2(7)/3})$. We also provide bounds on the partite variants of the standard distinct angle problem.
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