On the number of radial orderings of planar point sets

TL;DR

This paper investigates the maximum and minimum number of distinct radial orderings, both colored and uncolored, for planar point sets under certain general position assumptions, providing bounds and constructions.

Contribution

It establishes bounds on the number of radial orderings and constructs point sets achieving these bounds, advancing understanding of geometric orderings.

Findings

Maximum uncolored radial orderings are O(n^4).

Minimum uncolored radial orderings are Ω(n^3).

Colored radial orderings range from Ω(n) to Θ(n^4).

Abstract

Given a set $S$ of $n$ points in the plane, a \emph{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 \emph{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---each passing through a pair of points in $S$---intersect in a point of $\R^2\setminus S$. 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 $O(n^4)$ and at least $\Omega(n^3)$; the number of colored radial orderings of $S$ is at most $O(n^4)$ and at least $\Omega(n)$; there exist sets of points with $\Theta(n^4)$ colored radial orderings and sets of points with only $O(n^2)$ colored radial orderings.