Convex-Arc Drawings of Pseudolines

TL;DR

This paper explores various geometric representations of pseudoline arrangements, including convex arc drawings, and characterizes which arrangements can be represented by different convex curves and polygons.

Contribution

It introduces new geometric representations for pseudoline arrangements, especially for outerplanar and simple arrangements, expanding understanding of their visualizations.

Findings

Outerplanar arrangements can be represented by circle chords, convex chains, or hyperbolic lines.

Simple non-outerplanar arrangements can be represented by convex polygonal chains or smooth convex curves.

The paper characterizes the types of arrangements that admit these convex arc representations.

Abstract

A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar---each crossing is incident to an unbounded face---and simple---each crossing point is the crossing of only two curves. We show that these arrangements can be represented by chords of a circle, by convex polygonal chains with only two bends, or by hyperbolic lines. Simple but non-outerplanar arrangements (non-weak) can be represented by convex polygonal chains or convex smooth curves of linear complexity.