The distribution of geodesic excursions into the neighborhood of a cone   singularity on a hyperbolic 2-orbifold
  A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an infinite sequence of penetrations into a neighborhood of a cone singularity, so that the sequence of depths of maximal penetration has a limiting distribution. The distribution function is the same for all such surfaces and is described by a fairly simple formula. 
Andrew Haas