Homogeneous line-segment processes

Abstract

SummaryA number of formulae of an integral geometric character are presented for the most general planar, homogeneous line-segment process, one in which a given segment length and orientation may depend upon (a) the point process of segment midpoints and (b) the lengths and orientations of other segments. The sense in which these formulae have a probabilistic/statistical interpretation is made precise. For the general process, two interpretations are given; one requires the theory of Palm distributions whilst the other depends upon ergodic results. When additional structure for the process is assumed, the integral geometric formulae lead to interesting, non-intuitive sampling formulae.