  The classical Euler's problem on optimal configurations of elastic rod in the
plane with fixed endpoints and tangents at the endpoints is considered. The
global structure of the exponential mapping that parameterises extremal
trajectories is described. It is proved that open domains cut out by Maxwell
strata in the preimage and image of the exponential mapping are mapped
diffeomorphically. As a consequence, computation of globally optimal elasticae
with given boundary conditions is reduced to solving systems of algebraic
equations having unique solutions in the open domains. For certain special
boundary conditions, optimal elasticae are presented.
