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تسجيل مستخدم جديدOn rich lenses in planar arrangements of circles and related problems
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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 line
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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 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 belo
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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{
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