Stochastic Geometry - Line and Hyper-flat Processes

Line and Hyper-flat Processes

Suppose we are concerned no longer with compact objects, but with objects which are spatially extended: lines on the plane or flats in 3-space. This leads to consideration of line processes, and of processes of flats or hyper-flats. There can no longer be a preferred spatial location for each object; however the theory may be mapped back into point process theory by representing each object by a point in a suitable representation space. For example, in the case of directed lines in the plane one may take the representation space to be a cylinder. A complication is that the Euclidean motion symmetries will then be expressed on the representation space in a somewhat unusual way. Moreover calculations need to take account of interesting spatial biases (for example, line segments are less likely to be hit by random lines to which they are nearly parallel) and this provides an interesting and significant connection to the hugely significant area of stereology, which in some respects can be viewed as yet another theme of stochastic geometry. It is often the case that calculations are best carried out in terms of bundles of lines hitting various test-sets, rather than by working in representation space.

Line and hyper-flat processes have their own direct applications, but also find application as one way of creating tessellations dividing space; hence for example one may speak of Poisson line tessellations. A notable recent result proves that the cell at the origin of the Poisson line tessellation is approximately circular when conditioned to be large. Tessellations in stochastic geometry can of course be produced by other means, for example by using Voronoi and variant constructions, and also by iterating various means of construction.