  The closedness condition for real geodesics on n-dimensional ellipsoids is in
general transcendental in the parameters (semiaxes of the ellipsoid and
constants of motion). We show that it is algebraic in the parameters if and
only if both the real and the imaginary geodesics are closed and we
characterize such double--periodicity condition via real hyperelliptic
tangential coverings. We prove the density of algebraically closed geodesics on
n-dimensional ellipsoids with respect to the natural topology in the
(2n)-dimensional real parameter space. In particular, the approximating
sequence of algebraic closed geodesics on the approximated ellipsoids may be
chosen so to share the same values of the length and of the real period vector
as the limiting closed geodesic on the limiting ellipsoid.
  Finally, for real doubly-periodic geodesics on triaxial ellipsoids, we show
how to evaluate algebraically the period mapping and we present some explicit
examples of families of algebraically closed geodesics.
