## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 48, Number 2 (2016), 574-584.

### Planar tessellations that have the half-Gilbert structure

James Burridge and Richard Cowan

#### Abstract

In the *full* rectangular version of Gilbert's planar tessellation (see
Gilbert (1967), Mackisack and Miles (1996), and Burridge *et al.* (2013)),
lines extend either horizontally (with east- and west-growing rays) or
vertically (north- and south-growing rays) from seed points which form a
stationary Poisson point process, each ray stopping when it meets another ray
that has blocked its path. In the *half-Gilbert* rectangular version,
east- and south-growing rays, whilst having the potential to block each other,
do not interact with west and north rays, and vice versa. East- and
south-growing rays have a *reciprocality of blocking*, as do west and
north. In this paper we significantly expand and simplify the half-Gilbert
analytic results that we gave in Burridge *et al.* (2013). We also show
how the idea of *reciprocality of blocking* between rays can be used in a
much wider context, with rays not necessarily orthogonal and with seeds that
produce more than two rays.

#### Article information

**Source**

Adv. in Appl. Probab., Volume 48, Number 2 (2016), 574-584.

**Dates**

First available in Project Euclid: 9 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1465490763

**Mathematical Reviews number (MathSciNet)**

MR3511776

**Zentralblatt MATH identifier**

1344.60014

**Subjects**

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

Secondary: 60G55: Point processes 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]

**Keywords**

Random tessellation point process crack formation division of space

#### Citation

Burridge, James; Cowan, Richard. Planar tessellations that have the half-Gilbert structure. Adv. in Appl. Probab. 48 (2016), no. 2, 574--584. https://projecteuclid.org/euclid.aap/1465490763