On the numbers of radial orderings of planar point sets

José Miguel Dıaz-Banez,Pablo Pérez-Lantero,Ruy Fabila-Monroy

doi:10.46298/dmtcs.2087

José Miguel Dıaz-Banez, Pablo Pérez-Lantero + Show 1 more

Open Access

https://doi.org/10.46298/dmtcs.2087

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Abstract

Combinatorics Given a set S of n points in the plane, a radial ordering of S with respect to a point p (not in S) is a clockwise circular ordering of the elements in S by angle around p. If S is two-colored, a colored radial ordering is a radial ordering of S in which only the colors of the points are considered. In this paper, we obtain bounds on the number of distinct non-colored and colored radial orderings of S. We assume a strong general position on S, not three points are collinear and not three lines 14;each passing through a pair of points in S 14;intersect in a point of ℝ2 S. In the colored case, S is a set of 2n points partitioned into n red and n blue points, and n is even. We prove that: the number of distinct radial orderings of S is at most O(n4) and at least Ω(n3); the number of colored radial orderings of S is at most O(n4) and at least Ω(n); there exist sets of points with Θ(n4) colored radial orderings and sets of points with only O(n2) colored radial orderings.

Highlights

Let S be a set of n points in the plane
To prove the lower bound on g(n), we construct a two-colored set, P, of 2n points, in strong general position, with at most O(n2) colored radial orderings
We proved an upper bound of O(n4) and a lower bound of Ω(n3) on the number of radial orderings that every set of n points in strong general position in the plane must have

Summary

Introduction

Let S be a set of n points in the plane. We say that S is in strong general position if it is in general position (not three of its points are collinear) and every time that three lines—each passing through a pair of points in S—intersect, they do so in a point in S. Let ρcol(S) be the number of distinct colored radial orderings of S with respect to every observation point in the plane. For a bi-colored point set, a radial sweeping algorithm requires the ordering as an initial step, so it could be useful to know bounds on the number of different colored radial ordering of S from points in the plane. The existence of balanced—each part having an equal number of red an blue points—k-fans for colored point sets has been studied in recent papers [5, 6] but, as far as we know, the number of different monochromatic partitions induced by k-fans has not yet been considered. A preliminary version of this paper appeared in [9]

Uncolored Case

Colored Case

Conclusions