Betweenness isomorphisms in the plane — the case of a circle and points.

Two subsets A, B of the plane are betweenness isomorphic if there is a bijection f : A → B such that, for every x, y, z ∈ 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 each of the sets A, B is of the form C ∪ D where C is a circle and D is a finite set. We fully characterize the betweenness isomorphism classes in the family of all circles with three collinear points inside. In particular, we show that there are only countably many isomorphism classes, for which we provide an algebraic description. [ABSTRACT FROM AUTHOR]