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 ...

Abstract. A collection of simple closed Jordan curves in the plane is called a family of pseudo-circles 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 EIA-98-70724, EIA-99-72879, ITR-333-

A collection L of x-monotone 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

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 k-level 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 degree-s polynomials) only. When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved k-level results for most of the curve families studied earlier, including a near-O(n ) bound for parabolas.

We obtain improved bounds on the complexity of m distinct faces in an arrangement of n pseudo-segments, n circles, or n unit circles. The bounds are worst-case optimal for unit circles; they are also worst-case optimal for the case of pseudo-segments, except when the number of faces is very small, in which case our upper bound is a polylogarithmic factor from the best-known lower bound. For general circles, the bounds nearly coincide with the best-known bounds for the number of incidences between m points and n circles, recently obtained in [9].

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 low-dimensional integer lattice and projecting the lattice onto the plane, or drawing the medium as a planar graph with centrally symmetric faces.

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 bounded-degree 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 4-space and the lower bound for the number of distinct distances in a set of n points in 3-space. 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.

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This paper studies a variety of problems involving certain types of extreme con gurations in arrangements of (x-monotone) pseudo-lines. For example, we obtain a very simple proof of the bound O(nk ) on the maximum complexity of the k-th level in an arrangement of n pseudo-lines, which becomes even simpler in the case of lines. We thus simplify considerably previous proofs by Dey and by Tamaki and Tokuyama.

Many divide-and-conquer based geometric algorithms and order-statistics 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 ε-approximate 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 ε-approximation of a range space under insertions and deletions, which is of independent interest. All our approximation algorithms run in near-linear time. 1