  We consider the equations of motion of $n$ vortices of equal circulation in
the plane, in a disk and on a sphere. The vortices form a polygonal equilibrium
in a rotating frame of reference. We use numerical continuation in a boundary
value setting to determine the Lyapunov families of periodic orbits that arise
from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit
and the frequency of the rotating frame have a rational relationship then the
orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits,
with frequencies satisfying a diophantine equation, corresponds to
choreographies of the $n$ vortices. We include numerical results for all cases,
for various values of $n$, and we provide key details on the computational
approach.
