  Gordon and Wilton recently proved that the double D of a free group F
amalgamated along a cyclic subgroup C of F contains a surface group if a
generator w of C satisfies a certain 3-manifold theoretic condition, called
virtually geometricity. Wilton and the author defined the polygonality of w
which also guarantees the existence of a surface group in D. In this paper,
virtual geometricity is shown to imply polygonality up to descending to a
finite-index subgroup F' and applying an automorphism on F'. That the converse
does not hold will follow from an example formerly considered by Manning.
