Full and Half Gilbert Tessellations with Rectangular Cells

TL;DR

This paper analyzes ray-length distributions in two rectangular Gilbert tessellations, developing analytical and simulation methods, and discovers a surprising distribution coincidence between the full and half models.

Contribution

It introduces new analytical series expansions and simulation techniques for rectangular Gilbert tessellations, revealing a distribution coincidence and providing improved moment calculations.

Findings

Distributions for full and half models appear identical at certain intensities.

Analytical series expansion for the half model's ray-length distribution.

Exact expressions for the first and second moments of ray length.

Abstract

We investigate the ray-length distributions for two different rectangular versions of Gilbert's tessellation. In the full rectangular version, lines extend either horizontally (with east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a Poisson point process, each ray stopping when another ray is met. In the half rectangular version, east and south growing rays do not interact with west and north rays. For the half rectangular tessellation we compute analytically, via recursion, a series expansion for the ray-length distribution, whilst for the full rectangular version we develop an accurate simulation technique, based in part on the stopping-set theory of Zuyev, to accomplish the same. We demonstrate the remarkable fact that plots of the two distributions appear to be identical when the intensity of seeds in the half model is twice that in the full model. Our paper explores this coincidence mindful of the fact that, for one model, our results are from a simulation (with inherent sampling error). We go on to develop further analytic theory for the half-Gilbert model using stopping-set ideas once again, with some novel features. Using our theory, we obtain exact expressions for the first and second moment of ray length in the half-Gilbert model. For all practical purposes, these results can be applied to the full-Gilbert model as much better approximations than those provided by Mackissack and Miles.