Inscribable order types

TL;DR

This paper studies which order types can be realized with all extreme points on a circle, proving inscribability for simple types with few interior points, and constructing uninscribable types using Möbius transformations.

Contribution

It characterizes inscribability for simple order types with limited interior points and introduces an infinite family of uninscribable order types, advancing understanding of geometric realizability.

Findings

All simple order types with up to 2 interior points are inscribable.

Number of such inscribable order types is Θ(4^n / n^{3/2}).

Constructed an infinite family of minimally uninscribable order types.

Abstract

We call an order type inscribable if it is realized by a point configuration where the extreme points are all on a circle. In this paper, we investigate inscribability of order types. We first show that every simple order type with at most 2 interior points is inscribable, and that the number of such order types is $\Theta(\frac{4^n}{n^{3/2}})$. We further construct an infinite family of minimally uninscribable order types. The proof of uninscribability mainly uses M\"obius transformations. We also suggest open problems around inscribability.