Geodesic Order Types

TL;DR

This paper extends the concept of order types to geodesic paths within polygons, demonstrating realizability of certain abstract types and providing a method to construct polygons that realize specified geodesic order types.

Contribution

It introduces geodesic order types as a generalization of Euclidean order types and shows how to realize them through polygon constructions and dual arrangements.

Findings

Any point set with an ordered subset of at least four points can have a polygon constructed to realize a specified geodesic hull.

Certain abstract order types, such as those from the dual of the Pappus arrangement, can be realized as geodesic order types.

The concept broadens the understanding of geometric configurations within polygonal environments.

Abstract

The geodesic between two points $a$ and $b$ in the interior of a simple polygon~$P$ is the shortest polygonal path inside $P$ that connects $a$ to $b$. It is thus the natural generalization of straight line segments on unconstrained point sets to polygonal environments. In this paper we use this extension to generalize the concept of the order type of a set of points in the Euclidean plane to geodesic order types. In particular, we show that, for any set $S$ of points and an ordered subset $\mathcal{B} \subseteq S$ of at least four points, one can always construct a polygon $P$ such that the points of $\mathcal{B}$ define the geodesic hull of~$S$ w.r.t.~$P$, in the specified order. Moreover, we show that an abstract order type derived from the dual of the Pappus arrangement can be realized as a geodesic order type.