Billiards in ellipses revisited
Arseniy Akopyan,Richard Evan Schwartz,Serge Tabachnikov +2 more
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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 1-parameter family of such polygons (that exist due to the Poncelet porism).Abstract:
We prove some recent experimental observations of Dan 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 1-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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New Properties of Triangular Orbits in Elliptic Billiards
TL;DR: In this article, new invariants in the family of 3-periodics in an elliptic billiard were introduced, stemming from both experimental and theoretical work, including relationships between radii, angles and areas of triangular members of the family, as well as a special stationary circle.
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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.
Journal ArticleDOI
Fifty New Invariants of N-Periodics in the Elliptic Billiard
TL;DR: In this article, 50+ new invariants manifested by the dynamic geometry of N-periodics in the Elliptic Billiard are detected with an experimental/interactive toolbox.
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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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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.
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Bounds for Minkowski Billiard Trajectories in Convex Bodies
TL;DR: In this paper, the Ekeland-Hofer-Zehnder capacity was used to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body.
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From symplectic measurements to the Mahler conjecture
TL;DR: In this paper, the authors link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains and Mahler's conjecture on the volume product of centrally symmetric convex bodies.