I'm currently reading McShane's paper "Simple geodesics and a series constant over Teichmuller space". The most difficult part of the argument seems to be finding a sequence of simple geodesics on a once-punctured torus which approximate a given simple geodesic with one end up the cusp, whose other end spirals to a minimal lamination which is not a closed geodesic.
I can't seem to find a concrete example or a picture of such a geodesic on a hyperbolic once-punctured torus (that has one end up the cusp and the other spiralling to a minimal lamination which is not a closed geodesic).
Is there an easy construction of such a geodesic?