Lenses in Arrangements of Pseudo-circles and Their Applications£

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 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 ofÒpseudo-circles with the property that any two curves intersect precisely twice. This bound implies that any collection ofÒÜ-monotone pseudo-circles can be cut intoÇÒ���arcs so that any two intersect at most once; this improves a previous bound ofÇÒ��due to Tamaki and Tokuyama. If, in addition, the given collection admits an algebraic representation by three real parameters that satisfies some simple conditions, then the number of cuts can be further reduced toÇÒ�ÐÓ�ÒÇ«×Ò, where«Òis the inverse Ackermann function, and×is a constant that depends on the the representation of the pseudo-circles. For arbitrary collections of pseudocircles, any two of which intersect exactly twice, the number of necessary cuts reduces still further toÇÒ��. As applications, we obtain improved bounds for the number of incidences, the complexity of a single level, and the complexity of many faces in arrangements of circles