  We derive necessary and sufficient conditions for periodic and for elliptic
periodic trajectories of billiards within an ellipse in the Minkowski plane in
terms of an underlining elliptic curve. We provide several examples of periodic
and elliptic periodic trajectories with small periods. We observe relationship
between Cayley-type conditions and discriminantly separable and factorizable
polynomials. Equivalent conditions for periodicity and elliptic periodicity are
derived in terms of polynomial-functional equations as well. The corresponding
polynomials are related to the classical extremal polynomials. In particular,
the light-like periodic trajectories are related to the classical Chebyshev
polynomials. The similarities and differences with respect to previously
studied Euclidean case are indicated.
