  We present an exact solution of the equations for orbit determination of a
two body system in a hyperbolic or parabolic motion. In solving this problem,
we extend the method employed by Asada, Akasaka and Kasai (AAK) for a binary
system in an elliptic orbit. The solutions applicable to each of elliptic,
hyperbolic and parabolic orbits are obtained by the new approach, and they are
all expressed in an explicit form, remarkably, only in terms of elementary
functions. We show also that the solutions for an open orbit are recovered by
making a suitable transformation of the AAK solution for an elliptic case.
