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

Isometric Billiards in Ellipses and Focal Billiards in Ellipsoids