Lenses in arrangements of pseudo-circles and their applications

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(n^3/2(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(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.