On the Number of Radial Orderings of Colored Planar Point Sets

Abstract

Let n be an even natural number and let S be a set of n red and n blue points in general position in the plane. Let p ∉ S be a point such that S ∪ {p} is in general position. A radial ordering of S with respect to p is a circular ordering of the elements of S by angle around p. A colored radial ordering is a radial ordering of S in which only the colors of the points are considered. We show that: the number of distinct radial orderings of S is at most O(n⁴) and at least Ω(n²); the number of colored radial orderings of S is at most O(n⁴) and at least Ω(n); there exists sets of points with Θ(n⁴) colored radial orderings and sets of points with only O(n²) colored radial orderings.