  This is the first in a series of papers in which we develop a twistor-based method of constructing hyperkaehler metrics from holomorphic functions and elliptic curves. As an application, we revisit the Atiyah-Hitchin manifold and derive in an explicit holomorphic coordinate basis closed-form formulas for, among other things, the metric, the holomorphic symplectic form and all three Kaehler potentials. 
Elliptic constructions of hyperkaehler metrics I: The Atiyah-Hitchin   manifold