Geodesics in planar Poisson roads random metric

TL;DR

This paper analyzes the structure of shortest paths in a fractal random road network model, proving that geodesics do not pause on roads and characterizing their local structure around roads.

Contribution

It proves that geodesics in Kendall's fractal Poisson roads model do not pause en route and provides a detailed description of local geodesic structures.

Findings

Geodesics do not pause en route, even on small speed limit roads.

The geodesic frame consists solely of points on roads.

Leaving a road off-road is never part of a geodesic.

Abstract

We study the structure of geodesics in the fractal random metric constructed by Kendall from a self-similar Poisson process of roads (i.e, lines with speed limits) in $\mathbb{R}^2$. In particular, we prove a conjecture of Kendall stating that geodesics do not pause en route, i.e, use roads of arbitrary small speed except at their endpoints. It follows that the geodesic frame of $\left(\mathbb{R}^2,T\right)$ is the set of points on roads. We also consider geodesic stars and hubs, and give a complete description of the local structure of geodesics around points on roads. Notably, we prove that leaving a road by driving off-road is never geodesic.