More About Areas and Centers of Poncelet Polygons
Tóm tắt
We study the locus of the Circumcenter of Mass of Poncelet polygons, and the limit of the Center of Mass (when we consider the polygon as a “homogeneous lamina”) for degenerate Poncelet polygons. We also provide a proof for one of Dan Reznik invariants for billiard trajectories. Plus, we take a look at how the scene looks like when we shift to spherical geometry.
Tài liệu tham khảo
Akopyan, A., Schwartz, R., Tabachnikov, S.: Billiards in ellipses revisited. arXiv preprintarXiv:2001.02934, (2020)
Berger, M.: Geometry revealed: a Jacob’s ladder to modern higher geometry. Springer, Berlin (2010)
Bialy, M., Tabachnikov, S.: Dan Reznik’s identities and more. arXiv preprint arXiv:2001.08469, (2020)
Fierobe, C.: On the circumcenters of triangular orbits in elliptic billards. arXiv:1807.11903, (2018)
Flatto, L.: Poncelet’s theorem. American Mathematical Society, Providence, Rhode Island (2009)
Garcia, R., Reznik, D., Koiller, J.: Loci of 3-periodics in an Elliptic Billiard: why so many ellipses? arXiv:2001.08041, (2020)
Garcia, R., Reznik, D., Koiller, J.: New Properties of triangular orbits in Elliptic Billiards. arXiv:2001.08054, (2020)
Griffiths, P., Harris, J.: On Cayley’s explicit solution to Poncelet’s porism. Enseign. Math 24(1–2), 31–40 (1978)
Izmestiev, I.: Spherical and hyperbolic conics. In: Papadopoulos, A., Alberge, V. (eds.) Eighteen Essays in Non-Euclidean Geometry, Chap. 15, pp. 263–320. European Mathematical Society Publishing House, Zürich (2019)
Reznik, D., Garcia, R., Koiller, J.: The Ballet of triangle centers on the Elliptic Billiard. arXiv:2002.00001, (2020)
Reznik, D., Garcia, R., Koiller, J.: Can the elliptic billiard still surprise us? Math. Intell. 42(1), 6–17 (2020)
Romaskevich, O.: On the incenters of triangular orbits on elliptic billiards. L’Enseignement Mathématique 60(3), 247–255 (2015)
Schwartz, R., Tabachnikov, S.: Centers of mass of Poncelet polygons, 200 years after. Math. Intell. 38, 29–34 (2016)
Tabachnikov, S., Tsukerman, E.: Circumcenter of mass and generalized euler line. Discrete Comput. Geom. 51(4), 815–836 (2014)