Universal measure for Poncelet-type theorems

Avksentyev, Evgeny; Protasov, Vladimir

doi:10.1090/proc/13838

Universal measure for Poncelet-type theorems
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by Evgeny A. Avksentyev and Vladimir Yu. Protasov PDF

Proc. Amer. Math. Soc. 146 (2018), 4843-4854 Request permission

Abstract:

We give a simple proof of the Emch closing theorem by introducing a new invariant measure on the circle. Special cases of that measure are well known and have been used in the literature to prove Poncelet’s and the zigzag theorems. Some further generalizations are also obtained by applying the new measure.

References

E. A. Avksent′ev, A universal measure for a pencil of conics and the great Poncelet theorem, Mat. Sb. 205 (2014), no. 5, 3–22 (Russian, with Russian summary); English transl., Sb. Math. 205 (2014), no. 5-6, 613–632. MR 3242629, DOI 10.1070/sm2014v205n05abeh004390
E. A. Avksent′ev, The great Emch closure theorem and combinatorial proof of Poncelet’s theorem, Mat. Sb. 206 (2015), no. 11, 3–18 (Russian, with Russian summary); English transl., Sb. Math. 206 (2015), no. 11-12, 1509–1523. MR 3438567, DOI 10.4213/sm8404
W. Barth and Th. Bauer, Poncelet theorems, Exposition. Math. 14 (1996), no. 2, 125–144. MR 1395253
Marcel Berger, Géométrie. Vol. 1, CEDIC, Paris; Nathan Information, Paris, 1977 (French). Actions de groupes, espaces affines et projectifs. [Actions of groups, affine and projective spaces]. MR 536870
W. L. Black, H. C. Howland, and B. Howland, A theorem about zig-zags between two circles, Amer. Math. Monthly 81 (1974), 754–757. MR 346647, DOI 10.2307/2319568
O. Bottema, Ein Schliessungssatz für zwei Kreise, Elem. Math. 20 (1965), 1–7.
Julian Lowell Coolidge, A treatise on the circle and the sphere, Chelsea Publishing Co., Bronx, N.Y., 1971. Reprint of the 1916 edition. MR 0389515
Vladimir Dragović and Milena Radnović, Poncelet porisms and beyond, Frontiers in Mathematics, Birkhäuser/Springer Basel AG, Basel, 2011. Integrable billiards, hyperelliptic Jacobians and pencils of quadrics. MR 2798784, DOI 10.1007/978-3-0348-0015-0
Arnold Emch, An application of elliptic functions to Peaucellier’s link-work (inversor), Ann. of Math. (2) 2 (1900/01), no. 1-4, 60–63. MR 1503478, DOI 10.2307/2007182
Leopold Flatto, Poncelet’s theorem, American Mathematical Society, Providence, RI, 2009. Chapter 15 by S. Tabachnikov. MR 2465164, DOI 10.1090/mbk/056
Lorenz Halbeisen and Norbert Hungerbühler, A simple proof of Poncelet’s theorem (on the occasion of its bicentennial), Amer. Math. Monthly 122 (2015), no. 6, 537–551. MR 3361732, DOI 10.4169/amer.math.monthly.122.6.537
András Hraskó, Poncelet-type problems, an elementary approach, Elem. Math. 55 (2000), no. 2, 45–62 (English, with German summary). MR 1761291, DOI 10.1007/s000170050071
V. V. Kozlov, Rationality conditions for the ratio of elliptic integrals and the great Poncelet theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4 (2003), 6–13, 71 (Russian, with Russian summary); English transl., Moscow Univ. Math. Bull. 58 (2003), no. 4, 1–7 (2004). MR 2054501
H. Lebesque, Les Coniques, Gauthier-Villars, Paris, 1942.
F. Nilov, Families of conics and circles with double tangencies, Sb. Math., submitted.
Jean-Victor Poncelet, Traité des propriétés projectives des figures. Tome I, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1995 (French). Reprint of the second (1865) edition. MR 1399774
V. Yu. Protasov, On a generalization of Poncelet’s theorem, Uspekhi Mat. Nauk 61 (2006), no. 6(372), 187–188 (Russian); English transl., Russian Math. Surveys 61 (2006), no. 6, 1180–1182. MR 2330019, DOI 10.1070/RM2006v061n06ABEH004375
V. Yu. Protasov, Generalized closing theorems, Elem. Math. 66 (2011), no. 3, 98–117 (English, with German summary). MR 2824426, DOI 10.4171/EM/177
I. J. Schoenberg, On Jacobi-Bertrand’s proof of a theorem of Poncelet, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 623–627. MR 820256