  We classify, in terms of topology of highest arcs, low height non-simple
geodesics on the modular hyperbolic punctured sphere with three elliptic fixed
points of order two. Of eight possible types, exactly one consists of geodesics
that form a bigon about the cusp; we express all such geodesics in terms of
Markoff triples. We also show that any geodesic joining two elliptic fixed
points has locally isolated height.
