On the motion of billiards in ellipses

For billiards in an ellipse e with an ellipse as caustic, there exist canonical coordinates on e such that the billiard transformation from vertex to vertex is equivalent to a shift of coordinates. A kinematic analysis of billiard motions offers a new approach to canonical parametrizations of billiards and associated Poncelet grids. This parametrization uses Jacobian elliptic functions with the modulus equal to the numerical eccentricity of the caustic and is the basis for proving a few invariants of periodic billiards.