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1 results

  • On the distribution of random lines

  • Peter Hall
  • Journal:
    Journal of Applied Probability / Volume 18 / Issue 3 / September 1981
    Published online by Cambridge University Press:
    14 July 2016, pp. 606-616
    Print publication:
    September 1981

  • Draw two lines, L 1 and L 2, in a plane. Along L 1 place the points of a Poisson process, and through each point draw a line, the angles of intersection with L 1 being distributed independently and uniformly on (0, π) . The intercepts of these random lines with L 2 form a new process, which is Poisson if and only if L 1 and L 2 are parallel. Curiously, if L 1 and L 2 are inclined then, with probability 1, the new process forms a dense subset of L 2. It is not really even a point process. In the present paper we shall investigate this anomaly and some of its generalizations.