  We consider the generalized Kahler structures (g,J_+,J_-) that arise on a
hyperkahler manifold (M,g,I,J,K) when we choose J_+ and J_- from the twistor
space of M. We find a relation between semichiral and arctic superfields which
can be used to determine the generalized Kahler potential for hyperkahler
manifolds whose description in projective superspace is fully understood. We
use this relation to determine an S^2-family of generalized Kahler potentials
for Euclidean space and for the Eguchi-Hanson geometry. Cotangent bundles of
Hermitian symmetric spaces constitute a class of hyperkahler manifolds where
our method can be applied immediately since the necessary results from
projective superspace are already available. As a non-trivial
higher-dimensional example, we determine the generalized potential for T*CP^n,
which generalizes the Eguchi-Hanson result.
