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Fifty New Invariants of N-Periodics in the Elliptic Billiard

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Abstract

We introduce 50+ new invariants manifested by the dynamic geometry of N-periodics in the Elliptic Billiard, detected with an experimental/interactive toolbox. These involve sums, products and ratios of distances, areas, angles, etc. Though curious in their manifestation, said invariants do all depend upon the two fundamental conserved quantities in the Elliptic Billiard: perimeter and Joachimsthal’s constant. Several proofs have already been contributed (references are provided); these have mainly relied on algebraic geometry. We very much welcome new proofs and contributions.

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Notes

  1. Antipedals can be self-intersecting.

  2. J. Steiner (following a similar result by J. Sturm in 1823 for triangles) proved in 1825 that the area of pedal polygons of a polygon R with respect to points on any given circumference centered on K [25] is invariant.

  3. The evolute of a smooth curve is the envelope of the normals [27, Evolute]. The perpendicular bisector is its discrete version.

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Acknowledgements

We would like to thank Olga Romaskevitch, Sergei Tabachnikov, Richard Schwartz, Arseniy Akopyan, Hellmuth Stachel, Alexey Glutsyuk, Corentin Fierobe, Maxim Arnold, and Pedro Roitman for useful discussions and insights.

The second author is fellow of CNPq and coordinator of Project PRONEX/CNPq/FAPEG 2017 10 26 7000 508.

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Authors and Affiliations

  1. Data Science Consulting, Rio de Janeiro, Brazil

    Dan Reznik

  2. Math and Statistics Institute, Federal University of Goiás, Goiânia, Brazil

    Ronaldo Garcia

  3. Federal University of Juiz de Fora, Juiz de Fora, Brazil

    Jair Koiller

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  1. Dan Reznik
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  2. Ronaldo Garcia
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  3. Jair Koiller
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Correspondence to Dan Reznik.

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Appendix

Appendix

Table 11 Symbols used in the invariants. Note \(i=1,...,N\) and \(j=1,2\)
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Reznik, D., Garcia, R. & Koiller, J. Fifty New Invariants of N-Periodics in the Elliptic Billiard. Arnold Math J. 7, 341–355 (2021). https://doi.org/10.1007/s40598-021-00174-y

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