  We show that the hyperbolic structure on a closed, orientable, hyperbolic
3-manifold can be constructed from a solution to the hyperbolic gluing
equations using any triangulation with essential edges. The key ingredients in
the proof are Thurston's spinning construction and a volume rigidity result
attributed by Dunfield to Thurston, Gromov and Goldman. As an application, we
show that this gives a new algorithm to detect hyperbolic structures on closed
3-manifolds.
