Separating dots with circles

TL;DR

This paper investigates the combinatorial properties of circles that separate points in a plane or sphere, providing explicit formulas and analyzing how these configurations change continuously, with implications for Voronoi decompositions and cluster algebras.

Contribution

It introduces explicit formulas for counting separating circles of two types and analyzes their invariance and changes under continuous point configuration variations.

Findings

Number of separating circles is configuration-independent

Explicit formulas for counts of separating circles

Voronoi decomposition changes via local moves

Abstract

Given a finite set of points in general position in the plane or sphere, we count the number of ways to separate those points using two types of circles: circles through three of the points, and circles through none of the points (up to an equivalence). In each case, we show the number of circles which separate the points into subsets of size k and l is independent of the configuration of points, and we provide an explicit formula in each case. We also consider how the circles change as the configuration of dots varies continuously. We show that an associated higher order Voronoi decomposition of the sphere changes by a sequence of local `moves'. As a consequence, an associated cluster algebra is independent of the configuration of dots, and only depends on the number of dots and the order of the Voronoi decomposition.