  Billiards in ellipses have a confocal ellipse or hyperbola as caustic. The
goal of this paper is to prove that for each billiard of one type there exists
an isometric counterpart of the other type. Isometry means here that the
lengths of corresponding sides are equal. The transition between these two
isometric billiard can be carried out continuosly via isometric focal billiards
in a fixed ellipsoid. The extended sides of these particular billiards in an
ellipsoid are focal axes, i.e., generators of confocal hyperboloids. This
transition enables to transfer properties of planar billiards to focal
billiards, in particular billiard motions and canonical parametrizations. A
periodic planar billiard and its associated Poncelet grid give rise to periodic
focal billiards and spatial Poncelet grids. If the sides of a focal billiard
are materialized as thin rods with spherical joints at the vertices and other
crossing points between different sides, then we obtain Henrici's hyperboloid,
which is flexible between the two planar limits.
