  We consider the Keplerian distance $d$ in the case of two elliptic orbits,
i.e. the distance between one point on the first ellipse and one point on the
second one, assuming they have a common focus. The absolute minimum $d_{\rm
min}$ of this function, called MOID or orbit distance in the literature, is
relevant to detect possible impacts between two objects following approximately
these elliptic trajectories. We revisit and compare two different approaches to
compute the critical points of $d^2$, where we squared the distance $d$ to
include crossing points among the critical ones. One approach uses
trigonometric polynomials, the other uses ordinary polynomials. A new way to
test the reliability of the computation of $d_{\rm min}$ is introduced, based
on optimal estimates that can be found in the literature. The planar case is
also discussed: in this case we present an estimate for the maximal number of
critical points of $d^2$, together with a conjecture supported by numerical
tests.
