  We study graphs that are formed by independently-positioned needles (i.e.,
line segments) in the unit square. To mathematically characterize the graph
structure, we derive the probability that two line segments intersect and
determine related quantities such as the distribution of intersections, given a
certain number of line segments $N$. We interpret intersections between line
segments as nodes and connections between them as edges in a spatial network
that we refer to as random-line graph (RLG). Using methods from the study of
random-geometric graphs, we show that the probability of RLGs to be connected
undergoes a sharp transition if the number of lines exceeds a threshold $N^*$.
