Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Transversal numbers for hypergraphs arising in geometry
Advances in Applied Mathematics
On levels in arrangements of curves
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
On the Complexity of Many Faces in Arrangements of Circles
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Point-line incidences in space
Proceedings of the eighteenth annual symposium on Computational geometry
Incidences between points and circles in three and higher dimensions
Proceedings of the eighteenth annual symposium on Computational geometry
Topological graphs with no self-intersecting cycle of length 4
Proceedings of the nineteenth annual symposium on Computational geometry
Point–Line Incidences in Space
Combinatorics, Probability and Computing
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(MATH) 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 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 $\bx$ 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 $\alpha(n)$ 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(n 4/3). 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.