## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 45, Number 1 (2013), 1-19.

### Full- and half-Gilbert tessellations with rectangular cells

James Burridge, Richard Cowan, and Isaac Ma

#### Abstract

We investigate the ray-length distributions for two different
rectangular versions of Gilbert's tessellation (see Gilbert (1967)). In
the *full* rectangular version, lines extend either horizontally
(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
for Poisson processes (see Zuyev (1999)), 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. In this paper we explore 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 moments of the 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 Mackisack and Miles (1996).

#### Article information

**Source**

Adv. in Appl. Probab., Volume 45, Number 1 (2013), 1-19.

**Dates**

First available in Project Euclid: 15 March 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1363354100

**Digital Object Identifier**

doi:10.1239/aap/1363354100

**Mathematical Reviews number (MathSciNet)**

MR3077538

**Zentralblatt MATH identifier**

1281.60011

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05B45: Tessellation and tiling problems [See also 52C20, 52C22]

Secondary: 52C17: Packing and covering in $n$ dimensions [See also 05B40, 11H31] 60G55: Point processes 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]

**Keywords**

Random tessellation point process crack formation fragmentation division of space

#### Citation

Burridge, James; Cowan, Richard; Ma, Isaac. Full- and half-Gilbert tessellations with rectangular cells. Adv. in Appl. Probab. 45 (2013), no. 1, 1--19. doi:10.1239/aap/1363354100. https://projecteuclid.org/euclid.aap/1363354100