According to the discussion in [Coxeter (1968)](http://link.springer.com/article/10.1007%2FBF01817563), the tangent points lie asymptotically on a [concho-spiral](http://en.wikipedia.org/wiki/Conchospiral), so the distribution is not uniform on the sphere, but is uniform on a circle.

By the way, the circle is in general not a great circle.  In fact, the process can be bidirectional, so there will be two accumulation point.  The tangent points are like on a "tube in hyperbolic space" but not on a plane.