  Let Z be a so-called well-behaved percolation, i.e. a certain random closed
set in the hyperbolic plane, whose law is invariant under all isometries; for
example the covered region in a Poisson Boolean model. The Hausdorff-dimension
of the set of directions is determined in terms of the $\alpha$-value of Z in
which visibility from a fixed point to the ideal boundary of the hyperbolic
plane is possible within Z. Moreover, the Hausdorff-dimension of the set of
(hyperbolic) lines through a fixed point contained in Z is calculated. Thereby
several conjectures raised by Benjamini, Jonasson, Schramm and Tykesson are
confirmed.
