TY - JOUR

T1 - Lenses in arrangements of pseudo-circles and their applications

AU - Agarwal, Pankaj

AU - Nevo, Eran

AU - Pach, János

AU - Pinchasi, Rom

AU - Pinchasi, Rom

AU - Smorodinsky, Shakhar

PY - 2002

Y1 - 2002

N2 - 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 sub-arcs of distinct pseudo-circles is said to be an empty lens if it does not intersect any other member of the family. We establish a linear upper bound on the number of empty lenses in an arrangement of n pseudo-circles with the property that any two curves intersect precisely twice. Enhancing this bound in several ways, and combining it with the technique of Tamaki and Tokuyama [16], we show that any collection of n pseudo-circles can be cut into O(n3/2(log n)O(α(s)(n))) arcs so that any two intersect at most once, provided that the given pseudo-circles are x-monotone and admit an algebraic representation by three real parameters; here α(n) is the inverse Ackermann function, and s is a constant that depends on the algebraic degree of the representation of the pseudo-circles (s = 2 for circles and parabolas). For arbitrary collections of pseudo-circles, any two of which intersect twice, the number of necessary cuts reduces to O(n4/3). As applications, we obtain improved bounds for the number of point-curve incidences, the complexity of a single level, and the complexity of many faces in arrangements of circles, pairwise intersecting pseudo-circles, parabolas, and families of homothetic copies of a fixed convex curve. We also obtain a variant of the Gallai-Sylvester theorem for arrangements of pairwise intersecting pseudo-circles, and a new lower bound for the number of distinct distances among n points in the plane under any simply-defined norm or convex distance function.

AB - 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 sub-arcs of distinct pseudo-circles is said to be an empty lens if it does not intersect any other member of the family. We establish a linear upper bound on the number of empty lenses in an arrangement of n pseudo-circles with the property that any two curves intersect precisely twice. Enhancing this bound in several ways, and combining it with the technique of Tamaki and Tokuyama [16], we show that any collection of n pseudo-circles can be cut into O(n3/2(log n)O(α(s)(n))) arcs so that any two intersect at most once, provided that the given pseudo-circles are x-monotone and admit an algebraic representation by three real parameters; here α(n) is the inverse Ackermann function, and s is a constant that depends on the algebraic degree of the representation of the pseudo-circles (s = 2 for circles and parabolas). For arbitrary collections of pseudo-circles, any two of which intersect twice, the number of necessary cuts reduces to O(n4/3). As applications, we obtain improved bounds for the number of point-curve incidences, the complexity of a single level, and the complexity of many faces in arrangements of circles, pairwise intersecting pseudo-circles, parabolas, and families of homothetic copies of a fixed convex curve. We also obtain a variant of the Gallai-Sylvester theorem for arrangements of pairwise intersecting pseudo-circles, and a new lower bound for the number of distinct distances among n points in the plane under any simply-defined norm or convex distance function.

KW - Arrangements

KW - Distinct distances

KW - Incidences

KW - Lenses

KW - Levels

KW - Pseudo-circles

UR - http://www.scopus.com/inward/record.url?scp=0036367546&partnerID=8YFLogxK

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AN - SCOPUS:0036367546

SP - 123

EP - 132

JO - Proceedings of the Annual Symposium on Computational Geometry

JF - Proceedings of the Annual Symposium on Computational Geometry

ER -