  We study the angular process related to random walks in the Euclidean and in
the non-Euclidean space where steps are Cauchy distributed.
  This leads to different types of non-linear transformations of Cauchy random
variables which preserve the Cauchy density. We give the explicit form of these
distributions for all combinations of the scale and the location parameters.
  Continued fractions involving Cauchy random variables are analyzed. It is
shown that the $n$-stage random variables are still Cauchy distributed with
parameters related to Fibonacci numbers. This permits us to show the
convergence in distribution of the sequence to the golden ratio.
