Two lilypond systems of finite line-segments

The paper discusses two models for non-overlapping finite line-segments constructed via the lilypond protocol, operating here on a given array of points P = {P-i} in R-2 with which are associated directions {theta(i)}. At time zero, for each and every i, a line-segment L-i starts growing at unit rate around the point P i in the direction theta(i), the point P-i remaining at the centre of L-i; each line-segment, under Model 1, ceases growth when one of its ends hits another line, while under Model 2, its growth ceases either when one of its ends hits another line or when it is hit by the growing end of some other line. The paper shows that these procedures are well defined and gives constructive algorithms to compute the half-lengths R-i of all L-i. Moreover, it specifies assumptions under which stochastic versions, i.e. models based on point processes, exist. Afterwards, it deals with the question as to whether there is percolation in Model 1. The paper concludes with a section containing several conjectures and final remarks.