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Bicentennial of the Great Poncelet Theorem (1813-2013): Current advances

Authors: Vladimir Dragović and Milena Radnović
Journal: Bull. Amer. Math. Soc. 51 (2014), 373-445
MSC (2010): Primary 37J35, 14H70, 37A05
Published electronically: April 1, 2014
MathSciNet review: 3196793
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Abstract: We present very recent results related to the Poncelet Theorem on the occasion of its bicentennial. We are telling the story of one of the most beautiful theorems of geometry, recalling for general mathematical audiences the dramatic historic circumstances which led to its discovery, a glimpse of its intrinsic appeal, and the importance of its relationship to dynamics of billiards within confocal conics. We focus on the three main issues: A) The case of pseudo-Euclidean spaces, for which we present a recent notion of relativistic quadrics and apply it to the description of periodic trajectories of billiards within quadrics. B) The relationship between so-called billiard algebra and the foundations of modern discrete differential geometry which leads to double-reflection nets. C) We present a new class of dynamical systems--pseudo-integrable billiards generated by a boundary composed of several arcs of confocal conics having nonconvex angles. The dynamics of such billiards have several extraordinary properties, which are related to interval exchange transformations and which generate families of flows that are minimal but not uniquely ergodic. This type of dynamics provides a novel type of Poncelet porisms--the local ones.

References [Enhancements On Off] (What's this?)

  • [ABS2004] V. E. Adler, A. I. Bobenko, and Yu. B. Suris, Geometry of Yang-Baxter maps: pencils of conics and quadrirational mappings, Comm. Anal. Geom. 12 (2004), no. 5, 967-1007. MR 2103308 (2005i:14012)
  • [ABS2009] Vsevolod Eduardovich Adler, Alexander Ivanovich Bobenko, and Yuri Borisovich Suris, Integrable discrete nets in Grassmannians, Lett. Math. Phys. 89 (2009), no. 2, 131-139. MR 2534880 (2010f:37130),
  • [Arn1978] Vladimir Arnold, Mathematical Methods of Classical Mechanics, Springer Verlag, New York, 1978.
  • [Aud1994] Michèle Audin, Courbes algébriques et systèmes intégrables: géodésiques des quadriques, Exposition. Math. 12 (1994), no. 3, 193-226 (French, with English summary). MR 1295705 (95k:58068)
  • [BB1996] W. Barth and Th. Bauer, Poncelet theorems, Exposition. Math. 14 (1996), no. 2, 125-144. MR 1395253 (97f:14051)
  • [BM1993] W. Barth and J. Michel, Modular curves and Poncelet polygons, Math. Ann. 295 (1993), no. 1, 25-49. MR 1198840 (94c:14045),
  • [Ber1987a] Marcel Berger, Geometry. I, Universitext, Springer-Verlag, Berlin, 1987. Translated from the French by M. Cole and S. Levy. MR 882541 (88a:51001a)
  • [Ber1987b] Marcel Berger, Geometry. II, Universitext, Springer-Verlag, Berlin, 1987. Translated from the French by M. Cole and S. Levy. MR 882916 (88a:51001b)
  • [BM1962] Garrett Birkhoff and Robert Morris, Confocal conics in space-time, Amer. Math. Monthly 69 (1962), 1-4. MR 0133039 (24 #A2875)
  • [BS2008] Alexander I. Bobenko and Yuri B. Suris, Discrete differential geometry, Graduate Studies in Mathematics, vol. 98, American Mathematical Society, Providence, RI, 2008. Integrable structure. MR 2467378 (2010f:37125)
  • [BKOR1987] H. J. M. Bos, C. Kers, F. Oort, and D. W. Raven, Poncelet's closure theorem, Exposition. Math. 5 (1987), no. 4, 289-364. MR 917349 (88m:14041)
  • [Cay1853] Arthur Cayley, Note on the porism of the in-and-circumscribed polygon, Philosophical magazine 6 (1853), 99-102.
  • [Cay1854] Arthur Cayley, Developments on the porism of the in-and-circumscribed polygon, Philosophical magazine 7 (1854), 339-345.
  • [Cay1855] Arthur Cayley, On the porism of the in-and-circumscribed triangle, and on an irrational transformation of two ternary quadratic forms each into itself, Philosophical magazine 9 (1855), 513-517.
  • [Cay1857] Arthur Cayley, On the porism of the in-and-circumscribed triangle, Quarterly Mathematical Journal 1 (1857), 344-354.
  • [Cay1858] Arthur Cayley, On the a posteriori demonstration of the porism of the in-and-circumscribed triangle, Quarterly Mathematical Journal 2 (1858), 31-38.
  • [Cay1861] Arthur Cayley, On the porism of the in-and-circumscribed polygon, Philosophical Transactions of the Royal Society of London 51 (1861), 225-239.
  • [CCS1993] Shau-Jin Chang, Bruno Crespi, and Kang Jie Shi, Elliptical billiard systems and the full Poncelet's theorem in $ n$ dimensions, J. Math. Phys. 34 (1993), no. 6, 2242-2256. MR 1218986 (94g:58092),
  • [Cha1827] Chasles, Géométrie pure. Théorèmes sur les sections coniques confocales, Ann. Math. Pures Appl. [Ann. Gergonne] 18 (1827/28), 269-276 (French). MR 1556443
  • [CI2010] Sachiko Chino and Shyuichi Izumiya, Lightlike developables in Minkowski 3-space, Demonstratio Math. 43 (2010), no. 2, 387-399. MR 2668483 (2011f:58069)
  • [Dar1870] Gaston Darboux, Sur les polygones inscrits et circonscrits à l'ellipsoïde, Bulletin de la Société philomathique 7 (1870), 92-94.
  • [Dar1914] Gaston Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitesimal, Vol. 2 and 3, Gauthier-Villars, Paris, 1914.
  • [DR1998a] Vladimir Dragović and Milena Radnović, Conditions of Cayley's type for ellipsoidal billiard, J. Math. Phys. 39 (1998), no. 1, 355-362. MR 1489624 (98j:14041),
  • [DR1998b] Vladimir Dragović and Milena Radnović, On periodical trajectories of the billiard systems within an ellipsoid in $ \mathbf {R}^d$ and generalized Cayley's condition, J. Math. Phys. 39 (1998), no. 11, 5866-5869. MR 1653096 (99j:58158),
  • [DR2004] Vladimir Dragović and Milena Radnović, Cayley-type conditions for billiards within $ k$ quadrics in $ \mathbb{R}^d$, J. Phys. A 37 (2004), no. 4, 1269-1276. MR 2043219 (2005f:37123),
  • [DR2005] V. Dragović and M. Radnović, Corrigendum: ``Cayley-type conditions for billiards within $ k$ quadrics in $ {\mathbb{R}}^d$'' [J. Phys. A 37 (2004), no. 4, 1269-1276; MR2043219], J. Phys. A 38 (2005), no. 36, 7927. MR 2185422 (2006f:37086),
  • [DR2006a] V. Dragovich and M. Radnovich, A review of the analytic description of periodic billiard trajectories, Sovrem. Mat. Prilozh. 21, Geom. Zadachi Teor. Upr. (2004), 154-165 (Russian, with Russian summary); English transl., J. Math. Sci. (N. Y.) 135 (2006), no. 4, 3244-3255. MR 2157924 (2006f:37050),
  • [DR2006b] Vladimir Dragović and Milena Radnović, Geometry of integrable billiards and pencils of quadrics, J. Math. Pures Appl. (9) 85 (2006), no. 6, 758-790 (English, with English and French summaries). MR 2236243 (2007f:37083),
  • [DR2008] Vladimir Dragović and Milena Radnović, Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: beyond Poncelet porisms, Adv. Math. 219 (2008), no. 5, 1577-1607. MR 2458147 (2009k:14055),
  • [DR2010] V. Dragovich and M. Radnovich, Integrable billiards and quadrics, Uspekhi Mat. Nauk 65 (2010), no. 2(392), 133-194 (Russian, with Russian summary); English transl., Russian Math. Surveys 65 (2010), no. 2, 319-379. MR 2668802 (2011j:37107),
  • [DR2011] 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 (2012j:14059)
  • [DR2012a] Vladimir Dragović and Milena Radnović, Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics, Adv. Math. 231 (2012), no. 3-4, 1173-1201. MR 2964601,
  • [DR2012b] Vladimir Dragović and Milena Radnović, Billiard algebra, integrable line congruences, and double reflection nets, J. Nonlinear Math. Phys. 19 (2012), no. 3, 1250019, 18. MR 2978885,
  • [DR2012c] Vladimir Dragović and Milena Radnović, Pseudo-integrable billiards and arithmetic dynamics (2012), available at arXiv:1206.0163 [nlin.SI].
  • [Dui2010] Johannes J. Duistermaat, Discrete integrable systems, Springer Monographs in Mathematics, Springer, New York, 2010. QRT maps and elliptic surfaces. MR 2683025 (2012g:37178)
  • [Fla2009] Leopold Flatto, Poncelet's theorem, American Mathematical Society, Providence, RI, 2009. Chapter 15 by S. Tabachnikov. MR 2465164 (2011f:37001)
  • [GKT2007] Daniel Genin, Boris Khesin, and Serge Tabachnikov, Geodesics on an ellipsoid in Minkowski space, Enseign. Math. (2) 53 (2007), no. 3-4, 307-331. MR 2455947 (2009m:37163)
  • [GH1977] Phillip Griffiths and Joe Harris, A Poncelet theorem in space, Comment. Math. Helv. 52 (1977), no. 2, 145-160. MR 0498606 (58 #16695)
  • [GH1978] Phillip Griffiths and Joseph Harris, On Cayley's explicit solution to Poncelet's porism, Enseign. Math. (2) 24 (1978), no. 1-2, 31-40. MR 497281 (80g:51017)
  • [Har1967] Robin Hartshorne, Foundations of projective geometry, Lecture Notes, Harvard University, vol. 1966/67, W. A. Benjamin, Inc., New York, 1967. MR 0222751 (36 #5801)
  • [Jac1884] Carl Jacobi, Vorlesungen über Dynamic. Gesammelte Werke, Supplementband, Berlin, 1884.
  • [Jak1993] B. Jakob, Moduli of Poncelet polygons, J. Reine Angew. Math. 436 (1993), 33-44. MR 1207279 (94c:14010),
  • [Kea1975] Michael Keane, Interval exchange transformations, Math. Z. 141 (1975), 25-31. MR 0357739 (50 #10207)
  • [KT2009] Boris Khesin and Serge Tabachnikov, Pseudo-Riemannian geodesics and billiards, Adv. Math. 221 (2009), no. 4, 1364-1396. MR 2518642 (2010g:53163),
  • [Kin1994] Johnatan L. King, Three problems in search of a measure, The Americal Mathematical Monthly 101 (1994), no. 7, 609-628.
  • [KT1991] V. V. Kozlov and D. V. Treshchëv, Billiardy, Moskov. Gos. Univ., Moscow, 1991 (Russian). Geneticheskoe vvedenie v dinamiku sistem s udarami. [A genetic introduction to the dynamics of systems with impacts]. MR 1157370 (93k:58094b)
  • [Koz2003] 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 (2005d:33030)
  • [Leb1942] Henri Lebesgue, Les coniques, Gauthier-Villars, Paris, 1942.
  • [LT2007] Mark Levi and Serge Tabachnikov, The Poncelet grid and billiards in ellipses, Amer. Math. Monthly 114 (2007), no. 10, 895-908. MR 2363055 (2009b:37044)
  • [Mai1943] A. G. Maier, Trajectories on closable orientable surfaces, Sb. Math. 12 (1943), 71-84 (Russian).
  • [MT2002] Howard Masur and Serge Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015-1089. MR 1928530 (2003j:37002),
  • [MV1991] Jürgen Moser and Alexander P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys. 139 (1991), no. 2, 217-243. MR 1120138 (92g:58054)
  • [Mos1980] J. Moser, Geometry of quadrics and spectral theory, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York, 1980, pp. 147-188. MR 609560 (82j:58064)
  • [Pei1999] Donghe Pei, Singularities of $ {\bf R}{\rm P}^2$-valued Gauss maps of surfaces in Minkowski 3-space, Hokkaido Math. J. 28 (1999), no. 1, 97-115. MR 1673482 (2000a:58097)
  • [Pon1822] Jean Victor Poncelet, Traité des propriétés projectives des figures, Metz, Paris, 1822.
  • [Pon1862] Jean Victor Poncelet, Applications d'analyse et de géométrie, Mallet-Bachelier, Paris, 1862.
  • [1869] Obituary notices of fellows deceased. Jean Victor Poncelet, Proceedings of the Royal Society of London 18 (1869).
  • [Pre1999] Emma Previato, Poncelet's theorem in space, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2547-2556. MR 1662198 (2000d:14056),
  • [Pre2002] Emma Previato, Some integrable billiards, SPT 2002: Symmetry and perturbation theory (Cala Gonone), World Sci. Publ., River Edge, NJ, 2002, pp. 181-195. MR 1976669 (2004c:37134),
  • [QRT1988] G. R. W. Quispel, J. A. G. Roberts, and C. J. Thompson, Integrable mappings and soliton equations, Phys. Lett. A 126 (1988), no. 7, 419-421. MR 924318 (88m:58084),
  • [Sam1988] Pierre Samuel, Projective geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1988. Translated from the French by Silvio Levy; Readings in Mathematics. MR 960691 (89f:51003)
  • [Sch2003] Wolfgang Karl Schief, Lattice geometry of the discrete Darboux, KP, BKP and CKP equations. Menelaus' and Carnot's theorems, J. Nonlinear Math. Phys. 10 (2003), no. suppl. 2, 194-208. MR 2062279 (2005g:37136),
  • [Sch2007] Richard Evan Schwartz, The Poncelet grid, Adv. Geom. 7 (2007), no. 2, 157-175. MR 2314815 (2008f:37081),
  • [Smi] J. Smillie, The dynamics of billiard flows in rational polygons, Encyclopedia of Mathematical Sciences, Vol. 100, Springer-Verlag, 1999.
  • [Tab2005] Serge Tabachnikov, Geometry and billiards, Student Mathematical Library, vol. 30, American Mathematical Society, Providence, RI, 2005. MR 2168892 (2006h:51001)
  • [Tch1852] P. L. Tchebycheff, Report of the Extarordinary Professor of St Petersburg University Tchebycheff about the Trip Abroad, Complete Collected Works, Vol. 5, AN SSSR, Moscow-Leningrad, 1946-1951, 1852, pp. 246-255.
  • [Vee1969] William A. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem $ {\rm mod}\ 2$, Trans. Amer. Math. Soc. 140 (1969), 1-33. MR 0240056 (39 #1410)
  • [Vee1978] William A. Veech, Interval exchange transformations, J. Analyse Math. 33 (1978), 222-272. MR 516048 (80e:28034),
  • [Ves1988] A. P. Veselov, Integrable systems with discrete time, and difference operators, Funktsional. Anal. i Prilozhen. 22 (1988), no. 2, 1-13, 96 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 2, 83-93. MR 947601 (90a:58081),
  • [Via2008] Marcelo Viana, Dynamics of interval exchange maps and Teichmüller flows, 2008. lecture notes.
  • [WFS$^+$2009] Yao-Xiong Wang, Heng Fan, Kang-Jie Shi, Chun Wang, Kai Zhang, and Yu Zeng, Full Poncelet Theorem in Minkowski dS and AdS Spaces, Chinese Phys. Lett. 26 (2009), no. 1, 010201.
  • [Zor2006] Anton Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437-583. MR 2261104 (2007i:37070),

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Additional Information

Vladimir Dragović
Affiliation: Department of Mathematical Sciences, University of Texas at Dallas, Dallas, Texas; Mathematical Institute SANU, Kneza Mihaila 36, Belgrade, Serbia

Milena Radnović
Affiliation: Mathematical Institute SANU, Kneza Mihaila 36, Belgrade, Serbia

Keywords: Poncelet Theorem, periodic billiard trajectories, pencils of quadrics, relativistic quadrics, integrable line congruences, double reflection nets, pseudo-integrable billiards, interval exchange transformations
Received by editor(s): November 1, 2012
Received by editor(s) in revised form: March 25, 2013
Published electronically: April 1, 2014
Additional Notes: The research which led to this paper was partially supported by the Serbian Ministry of Education and Science (Project no. 174020: Geometry and Topology of Manifolds and Integrable Dynamical Systems).
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