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16
Lenses in arrangements of pseudocircles and their applications
 J. ACM
, 2004
"... Abstract. A collection of simple closed Jordan curves in the plane is called a family of pseudocircles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct Work on this article by P. Agarwal, J. Pach, and M. Sharir has been supported by a joint grant ..."
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Cited by 24 (10 self)
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Abstract. A collection of simple closed Jordan curves in the plane is called a family of pseudocircles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct Work on this article by P. Agarwal, J. Pach, and M. Sharir has been supported by a joint grant from the U.S.–Israel Binational Science Foundation. Work by P. Agarwal has also been supported by NSF grants EIA9870724, EIA9972879, ITR333
Cutting Circles into Pseudosegments and Improved Bounds for Incidences
 Geom
, 2000
"... We show that n arbitrary circles in the plane can be cut into O(n 3/2+# ) arcs, for any # > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m poi ..."
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Cited by 23 (12 self)
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We show that n arbitrary circles in the plane can be cut into O(n 3/2+# ) arcs, for any # > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree. 1 Introduction Let P be a finite set of points in the plane and C a finite set of circles. Let I = I(P, C) denote the number of incidences between the points and the circles. Let I(m, n) denote the maximum value of I(P, C), taken over all sets P of m points and sets C of n circles, and let I # (m, n, X) denote the maximum value of I(P, C), taken over all sets P of m points and sets C of n circles with at most X intersecting pairs. In this paper we derive improved upper bounds for I(m, n) and I # (m, n, X). The previous best upper bounds were I(m, n) = O(m 3/5 n 4/5 ...
Pseudoline arrangements: Duality. algorithms and applications
 SIAM J. Comput
, 2001
"... A collection L of xmonotone unbounded Jordan curves in the plane is called a family of pseudolines if every pair of curves intersects in at most one point, and the two curves cross each other there. Let P be a set of points in R 2. We define a duality transform that maps L to a set L of points in R ..."
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Cited by 12 (5 self)
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A collection L of xmonotone unbounded Jordan curves in the plane is called a family of pseudolines if every pair of curves intersects in at most one point, and the two curves cross each other there. Let P be a set of points in R 2. We define a duality transform that maps L to a set L of points in R 2 and P to a set
Incidences between points and circles in three and higher dimensions
 Geom
, 2002
"... We show that the number of incidences between m distinct points and n distinct circles in R d, for any d ≥ 3, is O(m 6/11 n 9/11 κ(m 3 /n)+m 2/3 n 2/3 +m+n), where κ(n) = (log n) O(α2 (n)) and where α(n) is the inverse Ackermann function. The bound coincides with the recent bound of Aronov and Shar ..."
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Cited by 11 (7 self)
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We show that the number of incidences between m distinct points and n distinct circles in R d, for any d ≥ 3, is O(m 6/11 n 9/11 κ(m 3 /n)+m 2/3 n 2/3 +m+n), where κ(n) = (log n) O(α2 (n)) and where α(n) is the inverse Ackermann function. The bound coincides with the recent bound of Aronov and Sharir, or rather with its slight improvement by Agarwal et al., for the planar case. We also show that the number of incidences between m points and n unrestricted convex (or boundeddegree algebraic) plane curves, no two in a common plane, is O(m 4/7 n 17/21 + m 2/3 n 2/3 + m + n), in any dimension d ≥ 3. Our results improve the upper bound on the number of congruent copies of a fixed tetrahedron in a set of n points in 4space and the lower bound for the number of distinct distances in a set of n points in 3space. Another application is an improved bound for the number of incidences (or, rather, containments) between lines and reguli in three dimensions. The latter result has already been applied by Feldman and Sharir to obtain a new bound on the number of joints in an arrangement of lines in three dimensions.
Algorithms for Drawing Media
"... We describe algorithms for drawing media, systems of states, tokens and actions that have state transition graphs in the form of partial cubes. Our algorithms are based on two principles: embedding the state transition graph in a lowdimensional integer lattice and projecting the lattice onto the p ..."
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Cited by 10 (4 self)
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We describe algorithms for drawing media, systems of states, tokens and actions that have state transition graphs in the form of partial cubes. Our algorithms are based on two principles: embedding the state transition graph in a lowdimensional integer lattice and projecting the lattice onto the plane, or drawing the medium as a planar graph with centrally symmetric faces.
On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences
 In Proc. 44th IEEE Sympos. Found. Comput. Sci
, 2003
"... We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the klevel has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which provided a ..."
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Cited by 9 (2 self)
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We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the klevel has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which provided a weaker bound for a restricted class of curves (graphs of degrees polynomials) only. When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved klevel results for most of the curve families studied earlier, including a nearO(n ) bound for parabolas.
On the complexity of many faces in arrangements of pseudosegments and
 of circles, in Discrete and Computational Geometry: The GoodmanPollack Festschrift
"... We obtain improved bounds on the complexity of m distinct faces in an arrangement of n pseudosegments, n circles, or n unit circles. The bounds are worstcase optimal for unit circles; they are also worstcase optimal for the case of pseudosegments, except when the number of faces is very small, i ..."
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Cited by 9 (5 self)
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We obtain improved bounds on the complexity of m distinct faces in an arrangement of n pseudosegments, n circles, or n unit circles. The bounds are worstcase optimal for unit circles; they are also worstcase optimal for the case of pseudosegments, except when the number of faces is very small, in which case our upper bound is a polylogarithmic factor from the bestknown lower bound. For general circles, the bounds nearly coincide with the bestknown bounds for the number of incidences between m points and n circles, recently obtained in [9]. 1
Extremal Configurations and Levels in Pseudoline Arrangements
"... This paper studies a variety of problems involving certain types of extreme con gurations in arrangements of (xmonotone) pseudolines. For example, we obtain a very simple proof of the bound O(nk ) on the maximum complexity of the kth level in an arrangement of n pseudolines, which becomes ev ..."
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Cited by 4 (0 self)
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This paper studies a variety of problems involving certain types of extreme con gurations in arrangements of (xmonotone) pseudolines. For example, we obtain a very simple proof of the bound O(nk ) on the maximum complexity of the kth level in an arrangement of n pseudolines, which becomes even simpler in the case of lines. We thus simplify considerably previous proofs by Dey and by Tamaki and Tokuyama.
Staying in the middle: Exact and approximate medians in R 1 and R 2 for moving points, manuscript
 In Proc. of the Canadian Conference on Computational Geometry
, 2003
"... Many divideandconquer based geometric algorithms and orderstatistics problems ask for a point that lies “in the middle ” of a given point set. We study several fundamental problems of this type for moving points in one and two dimensions. In particular, we show how to kinetically maintain the med ..."
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Cited by 2 (0 self)
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Many divideandconquer based geometric algorithms and orderstatistics problems ask for a point that lies “in the middle ” of a given point set. We study several fundamental problems of this type for moving points in one and two dimensions. In particular, we show how to kinetically maintain the median of a set of n points moving on the real line, and a center point of a set of n points moving in the plane, that is, a point such that any line through it has at most 2n/3 on either side of it. Since the maintenance of exact medians and center points can be quite expensive, we also show how to maintain εapproximate medians and center points and argue that the latter can be made to be much more stable under motion. These results are based on a new algorithm to maintain an εapproximation of a range space under insertions and deletions, which is of independent interest. All our approximation algorithms run in nearlinear time. 1