  For billiards in an ellipse with an ellipse as caustic, there exist canonical
coordinates such that the billiard transformation from vertex to vertex is
equivalent to a shift of coordinates. A kinematic analysis of billiard motions
paves the way to an explicit canonical parametrization of the billiard and even
of the associated Poncelet grid. This parametrization uses Jacobian elliptic
functions to the numerical eccentricity of the caustic as modulus.
