Constructing Order Type Graphs using an Axiomatic Approach

TL;DR

This paper introduces OT-graphs, a new class of geometric graphs based on axiomatic order types, with algorithms for recognition and minimal representation, supported by experimental analysis up to nine points.

Contribution

It presents OT-graphs derived from axiomatic order types, offering a novel visualization tool and algorithms for recognition and minimal graph computation.

Findings

OT-graphs correspond uniquely to order types.

Efficient algorithms for recognition and minimal OT-graph computation.

Experimental validation on order types up to nine points.

Abstract

A given order type in the plane can be represented by a point set. However, it might be difficult to recognize the orientations of some point triples. Recently, Aichholzer \etal \cite{abh19} introduced exit graphs for visualizing order types in the plane. We present a new class of geometric graphs, called {\em OT-graphs}, using abstract order types and their axioms described in the well-known book by Knuth \cite{k92}. Each OT-graph corresponds to a unique order type. We develop efficient algorithms for recognizing OT-graphs and computing a minimal OT-graph for a given order type in the plane. We provide experimental results on all order types of up to nine points in the plane including a comparative analysis of exit graphs and OT-graphs.