Advances in Applied Probability

Full- and half-Gilbert tessellations with rectangular cells

James Burridge, Richard Cowan, and Isaac Ma

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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


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References

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