  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.
