  As shown by Johannes Kepler in 1609, in the two-body problem, the shape of
the orbit, a given ellipse, and a given non-vanishing constant angular momentum
determines the motion of the planet completely.
  Even in the three-body problem, in some cases, the shape of the orbit,
conservation of the centre of mass and a constant of motion (the angular
momentum or the total energy) determines the motion of the three bodies. We
show, by a geometrical method, that choreographic motions, in which equal mass
three bodies chase each other around a same curve, will be uniquely determined
for the following two cases. (i) Convex curves that have point symmetry and
non-vanishing angular momentum are given. (ii) Eight-shaped curves which are
similar to the curve for the figure-eight solution and the energy constant are
given.
  The reality of the motion should be tested whether the motion satisfies an
equation of motion or not.
  Extensions of the method for generic curves are shown. The extended methods
are applicable to generic curves which does not have point symmetry. Each body
may have its own curve and its own non-vanishing masses.
