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Billiards in ellipses revisited
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In this article, the authors proved that the sum of cosines of the angles of a periodic billiard polygon remains constant in the one-parameter family of such polygons (that exist due to the Poncelet porism).Abstract:
We prove some recent experimental observations of D. Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the one-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.read more
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Dan Reznik’s identities and more
Misha Bialy,Serge Tabachnikov +1 more
TL;DR: In this paper, a non-standard generating function for the billiard ball map is used to obtain some conserved quantities associated with periodic billiard trajectories in ellipses.
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Can the Elliptic Billiard Still Surprise Us
TL;DR: It is shown that Monge's Orthoptic Circle's close relation to 4-periodic Billiard trajectories is well-known and its geometry provided clues with which to generalize 3- periodic invariants to trajectories of an arbitrary number of edges.
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Eighty New Invariants of N-Periodics in the Elliptic Billiard
TL;DR: 50+ new invariants manifested by the dynamic geometry of N -periodics in the Elliptic Billiard, detected with an experimental/interactive toolbox are introduced.
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New Properties of Triangular Orbits in Elliptic Billiards
TL;DR: Some of the proofs omitted from the introduction of new invariants in the one-dimensional family of 3-periodic orbits in the elliptic billiard are presented as well as a few new related facts.
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Related by Similarity II: Poncelet 3-Periodics in the Homothetic Pair and the Brocard Porism.
Dan Reznik,Ronaldo Garcia +1 more
TL;DR: Two new Poncelet 3-periodic families are studied: a first one interscribed in a pair of concentric, homothetic ellipses, and a second non-concentric one known as the Brocard porism: fixed circumcircle and Brocard inellipse.
References
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Book
Geometry and billiards
TL;DR: In this article, the authors discuss the existence and non-existence of caustics and periodic trajectories of billiards inside conics and quadrics, as well as in polygons.
Book
Poncelet Porisms and Beyond: Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics
TL;DR: In this paper, Poncelet-Darboux curves and Cayley's condition are used to define the Jacobians of Hyper-Elliptic curves and their Jacobians.