  By demanding consistency of the Legendre transform construction of
hyperk\"ahler metrics in projective superspace, we derive the expression for
the Darboux coordinates on the hyperk\"ahler manifold. We apply these results
to study the Coulomb branch moduli space of 4D, N=2 super-Yang-Mills theory
(SYM) on R^3 x S^1, recovering the results by GMN. We also apply this method to
study the electric corrections to the moduli space of 5D, N=1 SYM on R^3 x T^2
and give the Darboux coordinates explicitly.
