Geodesic
Isometric Billiards in Ellipses and Focal Billiards in Ellipsoids
Journal for geometry and graphics, Tome 25 (2021) no. 1, pp. 97-118 Cet article a éte moissonné depuis la source Heldermann Verlag

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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.
Classification : 51M04, 53A05, 53A17, 37C83
Mots-clés : Billiard in ellipse, caustic, Poncelet grid, confocal conics, confocal quadrics, focal axis, focal billiard in ellipsoids, billiard motion, canonical parametrization 
@article{JGG_2021_25_1_JGG_2021_25_1_a8,
     author = {H. Stachel },
     title = {Isometric {Billiards} in {Ellipses} and {Focal} {Billiards} in {Ellipsoids}},
     journal = {Journal for geometry and graphics},
     pages = {97--118},
     year = {2021},
     volume = {25},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JGG_2021_25_1_JGG_2021_25_1_a8/}
}
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H. Stachel . Isometric Billiards in Ellipses and Focal Billiards in Ellipsoids. Journal for geometry and graphics, Tome 25 (2021) no. 1, pp. 97-118. http://geodesic.mathdoc.fr/item/JGG_2021_25_1_JGG_2021_25_1_a8/