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 the closed Jordan region that it bounds 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. We use this bound to show that any collection of n x -monotone pseudo-circles can be cut into O ( n 8/5 ) arcs so that any two intersect at most once; this improves a previous bound of O ( n 5/3 ) 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 to O ( n 3/2 (log n ) O (α( s ( n )) ), where α( n ) is the inverse Ackermann function, and s is a constant that depends on the the representation of the pseudo-circles. For arbitrary collections of pseudo-circles, any two of which intersect exactly twice, the number of necessary cuts reduces still further to O ( n 4/3 ). 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, of pairwise intersecting pseudo-circles, of arbitrary x -monotone pseudo-circles, of parabolas, and of homothetic copies of any fixed simply shaped convex curve. We also obtain a variant of the Gallai--Sylvester theorem for arrangements of pairwise intersecting pseudo-circles, and a new lower bound on the number of distinct distances under any well-behaved norm.