Abstract
On an elliptic billiard, we study the set of the circumcenters of all triangular orbits and we show that this is an ellipse. This article follows Romaskevich (L’Enseig Math 60:247–255, 2014), which proves the same result with the incenters, and Glutsyuk (Moscow Math J 14:239–289, 2014), which among others, introduces the theory of complex reflection in the complex projective plane. The result we present was found at the same time in Garcia (Amer Math Monthly 126:491–504, 2019). His proof uses completely different methods of real differential calculus.
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Acknowledgements
This article could not be possible without the precious help, advice, and support of Alexey Glutsyuk and Olga Romaskevich, who also suggested to me to work on this topic. This article is an adaptation of Olga’s proof, and many ideas come from her article [16]. I am also really grateful to Anastasia Kozyreva for her help.
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Fierobe, C. On the Circumcenters of Triangular Orbits in Elliptic Billiard. J Dyn Control Syst 27, 693–705 (2021). https://doi.org/10.1007/s10883-021-09537-2
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DOI: https://doi.org/10.1007/s10883-021-09537-2
Keywords
- Billiard
- Elliptic billiard
- Periodic orbits
- Triangular orbits
- Complex reflection law
- Circumcenters
- Ellipse
- Conic