P. K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir, and S. Smorodinsky. Lenses in arrangements of pseudo-circles and their applications. J. ACM, to appear. (Preliminary version in Proc. 18th ACM Sympos.Comput. Geom., pages 123-- 132, 2002.)  Home/Search   Document Details and Download   Summary   Related Articles   Check

....techniques. Roughly the same bound holds for polynomial curve segments. Although polynomials are the most natural instances, the above proof clearly works for any curve families whose (s 1) th derivatives obey the pseudo line property. 7 An improvement Recently, Agarwal et al. [5] have improved Tamaki and Tokuyama s theorem [48] in various special cases; in particular, they have proved that every family of n parabolas can be cut into ) pseudo segments. Their result immediately implies an improvement to our Theorem 6.2. The improvement though is small, because only ....

....n) the rst improvement over Katoh, Tokuyama, and Iwano s O(n ) bound [36] Tamaki and Tokuyama [48] were unable to obtain such an improvement because their result on multiple chains for pseudo parabolas was too weak. By Agarwal et al. s recently improved cut theorem for parabolas [5], the bound can be further reduced to O(n 8=3 For algorithms on the parametric kinetic minimum spanning tree problem in both its graph and geometric settings, see [3, 13] Very recently, our Theorem 3.3 was used by Agarwal, Aronov, and Sharir [2] to bound the combinatorial complexity of ....

P. K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir, and S. Smorodinsky. Lenses in arrangements of pseudo-circles and their applications. Manuscript, 2001.

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P.K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir and S. Smorodinsky, Lenses in arrangements of pseudocircles and their applications, in preparation.

....into O(N log N) subarcs, each of which can be extended into an unbounded x monotone curve, so that these curves constitute a family of pseudo lines. One can then use the close relationship between line and pseudo line arrangements to solve a variety of problems involving arrangements of arcs; see [2, 7, 9, 11, 30]. In this paper, we focus on algorithmic problems involving arrangements of pseudo lines in the plane, problems that are much less studied than the corresponding combinatorial problems. Of course, one has to assume a reasonable representation of the given pseudo lines, in order to develop ecient ....

....the problem of splitting the curves in such a family L into arcs (pseudo segments) so that each pair of arcs intersect in at most one point. This work started with the paper of Tamaki and Tokuyama [30] and has continued with recent papers of Aronov and Sharir [9] Chan [11] and Agarwal et al. [7]. Since the resulting set of arcs is a collection of pseudo segments, one can obtain bounds on the complexity of various substructures in arrangements of pseudo circles by applying the known results for pseudosegment arrangements. This approach has recently been used to obtain, among other ....

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P. K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir, and S. Smorodinsky, Lenses in arrangements of pseudo-circles and their applications, manuscript, 2001.

....approach. It transforms the sets P and L to a planar con guration involving points and circles, where circles are tangent to each other at the given points, and where the goal is to bound the number of such tangencies. This is handled using recent tools developed for arrangements of circles in [1, 9]. 2 Prerequisites We recall some of the tools we need for our proofs. On the way, we show that many lines in a common plane or in a common regulus are counter productive to having many incidences between the lines and their joints. Szemer edi Trotter Bound [16] Given a set L of n lines and a ....

....fully contain) the convex hulls of more than cr sets, for some absolute constant c 0. Let C i denote the subset of C consisting of circles whose representing points are projected to points in P i , for i = 1; 2; q. As shown by Aronov and Sharir [1] and later slightly improved in [9], a set of N circles in the plane can be cut into O(N (N) arcs, where (N) is as de ned above, so that each pair of arcs intersect at most once. We adapt the analysis of [1] to the case at hand, which is highly degenerate due to the multitude of tangencies. Since a tangency is a degenerate ....

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E. Nevo, J. Pach, R. Pinchasi, M. Sharir and S. Smorodinsky, Lenses in arrangements of pseudocircles and their applications, Proceedings of the 18th Ann. ACM Symp. on Computational Geometry (2002), 123-132. 22

....plus the number of elementary lenses that lie in the interior of the cell Delta of Xi, so P Delta k Delta k. We also have T (m; 0) O(m log 3 m) Hence, the same analysis as in [20, 25] implies that T (m; k) O(m 1 p k) The following lemma is a re statement of a recent result in [5]. Lemma 6.3 The number of elementary bichromatic lenses formed by A and B is O(m) Hence, k = O(m) therefore the total time spent in cutting the bichromatic lenses formed by A and B is O(m 3=2 ) Repeating this procedure to all bipartite graphs A i Theta B i , and adding up the resulting ....

P. K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir, and S. Smorodinsky, Lenses in arrangements of pseudocircles and their applications, manuscript, 2001.

No context found.

P. K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir, and S. Smorodinsky. Lenses in arrangements of pseudo-circles and their applications. J. ACM, to appear. (Preliminary version in Proc. 18th ACM Sympos.Comput. Geom., pages 123-- 132, 2002.)

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