The betweenness relation distinguishes non-similar pairs of concentric circles

TL;DR

This paper characterizes when two unions of concentric circles are betweenness isomorphic, showing they are so if and only if they are similar, and that all such isomorphisms are scaled isometries.

Contribution

It provides a complete characterization of betweenness isomorphism classes for unions of concentric circles, linking them to similarity transformations.

Findings

Betweenness isomorphism coincides with similarity for unions of concentric circles.

There are continuum many distinct betweenness isomorphism classes in this family.

All betweenness isomorphisms are restrictions of scaled isometries.

Abstract

Two subsets $A, B$ of the plane are betweenness isomorphic if there is a bijection $f\colon A\to B$ such that, for every $x,y,z\in A$, the point $f(z)$ lies on the line segment connecting $f(x)$ and $f(y)$ if and only if $z$ lies on the line segment connecting $x$ and $y$. In general, it is quite difficult to tell whether two given subsets of the plane are betweenness isomorphic. We concentrate on the case when the sets $A,B$ belong to the family $ \mathcal A_c$ of unions of pairs of concentric circles in the plane. We prove that $A, B \in \mathcal A_c$ are betweenness isomorphic if and only if they are similar. In particular, there are continuum many betweenness isomorphism classes in $ \mathcal A_c$, and each of these classes consists exactly of all scaled translations of an arbitrary representative of the class. Furthermore, we show that every betweenness isomorphism between sets $A,B\in \mathcal A_c$ is exactly the restriction of a scaled isometry of the plane.