Some properties of line segment processes

Abstract

This paper formulates the random process of line-segments in the Euclidean plane. Under conditions more general than Poisson, expressions are obtained, for Borel A ⊂ R 2 , for the first moments of M ( A ), the number of segment mid-points in A ; N ( A ), the number of segments which intersect with convex A ; S ( A ), the total length within A of segments crossing A ; and C ( A ) the number of segment-segment crossings within A . In the case of Poisson mid-points, the distribution of the r th nearest line-segment to a given point is found.