  Any surface that is intrinsically polyhedral can be represented by a
collection of simple polygons (fragments), glued along pairs of equally long
oriented edges, where each fragment is endowed with the geodesic metric arising
from its Euclidean metric. We refer to such a representation as a portalgon,
and we call two portalgons equivalent if the surfaces they represent are
isometric. We analyze the complexity of shortest paths in portalgons. We call a
fragment happy if any shortest path on the portalgon visits it at most a
constant number of times. A portalgon is happy if all of its fragments are
happy. We present an efficient algorithm to compute shortest paths on happy
portalgons. The number of times that a shortest path visits a fragment is
unbounded in general. We contrast this by showing that the intrinsic Delaunay
triangulation of any polyhedral surface corresponds to a happy portalgon. Since
computing the intrinsic Delaunay triangulation may be inefficient, we provide
an efficient algorithm to compute happy portalgons for a restricted class of
portalgons.
