  Consider an improper Poisson line process, marked by positive speeds so as to
satisfy a scale-invariance property (actually, scale-equivariance). The line
process can be characterized by its intensity measure, which belongs to a
one-parameter family if scale and Euclidean invariance are required. This paper
investigates a proposal by Aldous, namely that the line process could be used
to produce a scale-invariant random spatial network (SIRSN) by means of
connecting up points using paths which follow segments from the line process at
the stipulated speeds. It is shown that this does indeed produce a
scale-invariant network, under suitable conditions on the parameter; indeed
that this produces a parameter-dependent random geodesic metric for
d-dimensional space ($d\geq2$), where geodesics are given by minimum-time
paths. Moreover in the planar case it is shown that the resulting geodesic
metric space has an almost-everywhere-unique-geodesic property, that geodesics
are locally of finite mean length, and that if an independent Poisson point
process is connected up by such geodesics then the resulting network places
finite length in each compact region. It is an open question whether the result
is a SIRSN (in Aldous' sense; so placing finite mean length in each compact
region), but it may be called a pre-SIRSN.
