Lenses in arrangements of pseudo-circles and their applications Academic Article uri icon


  • 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 …

publication date

  • January 1, 2004