Asymptotic of geometrical navigation on a random set of points of the plane

TL;DR

This paper investigates the asymptotic behavior of geometric navigation strategies on dense, non-uniform Poisson point sets in the plane, analyzing path lengths, trajectory geometry, and related graph properties.

Contribution

It establishes uniform convergence results for navigation path metrics and geometry on dense Poisson point processes, extending to Yao-graphs and θ-graphs.

Findings

Path lengths converge uniformly as point density increases.

Traveller trajectories exhibit predictable geometric patterns.

Stretch factors of random Yao-graphs and θ-graphs are characterized asymptotically.

Abstract

A navigation on a set of points $S$ is a rule for choosing which point to move to from the present point in order to progress toward a specified target. We study some navigations in the plane where $S$ is a non uniform Poisson point process (in a finite domain) with intensity going to $+\infty$. We show the convergence of the traveller path lengths, the number of stages done, and the geometry of the traveller trajectories, uniformly for all starting points and targets, for several navigations of geometric nature. Other costs are also considered. This leads to asymptotic results on the stretch factors of random Yao-graphs and random $\theta$-graphs.