Poncelet’s theorem in space

Previato, Emma

doi:10.1090/S0002-9939-99-05307-1

Poncelet’s theorem in space
HTML articles powered by AMS MathViewer

by Emma Previato

Proc. Amer. Math. Soc. 127 (1999), 2547-2556

DOI: https://doi.org/10.1090/S0002-9939-99-05307-1

Published electronically: May 4, 1999

PDF | Request permission

Abstract:

A plane polygon $\mathcal {P}$ inscribed in a conic $C$ and circumscribed to a conic $D$ can be continuously ‘rotated’, as it were. One of the many proofs consists in viewing each side of $\mathcal {P}$ as translation by a torsion point of an elliptic curve. In the $n$-space version, involving torsion points of hyperelliptic Jacobians, there is a $g=(n-1)$-dimensional family of rotations, where $g=\text {genus}$ of the hyperelliptic curve; the polygon is now inscribed in one and circumscribed to $n-1$ quadrics.

References

W. Barth and Th. Bauer, Poncelet theorems, Exposition. Math. 14 (1996), no. 2, 125–144. MR 1395253
W. Barth and J. Michel, Modular curves and Poncelet polygons, Math. Ann. 295 (1993), no. 1, 25–49. MR 1198840, DOI 10.1007/BF01444875
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
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, DOI 10.1063/1.530115
Bruno Crespi, Shau-Jin Chang, and Kang Jie Shi, Elliptical billiards and hyperelliptic functions, J. Math. Phys. 34 (1993), no. 6, 2257–2289. MR 1218987, DOI 10.1063/1.530116
David A. Cox, The arithmetic-geometric mean of Gauss, Enseign. Math. (2) 30 (1984), no. 3-4, 275–330. MR 767905
Shau-Jin Chang and Richard Friedberg, Elliptical billiards and Poncelet’s theorem, J. Math. Phys. 29 (1988), no. 7, 1537–1550. MR 946326, DOI 10.1063/1.527900
Percy Deift, Luen Chau Li, and Carlos Tomei, Loop groups, discrete versions of some classical integrable systems, and rank 2 extensions, Mem. Amer. Math. Soc. 100 (1992), no. 479, viii+101. MR 1124113, DOI 10.1090/memo/0479
U. V. Desale and S. Ramanan, Classification of vector bundles of rank $2$ on hyperelliptic curves, Invent. Math. 38 (1976/77), no. 2, 161–185. MR 429897, DOI 10.1007/BF01408570
Ron Donagi, Group law on the intersection of two quadrics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 2, 217–239. MR 581142
Bert van Geemen and Emma Previato, On the Hitchin system, Duke Math. J. 85 (1996), no. 3, 659–683. MR 1422361, DOI 10.1215/S0012-7094-96-08525-7
Phillip Griffiths and Joe Harris, A Poncelet theorem in space, Comment. Math. Helv. 52 (1977), no. 2, 145–160. MR 498606, DOI 10.1007/BF02567361
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
Joe Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), no. 4, 685–724. MR 552521
C.M. Jessop, A treatise on the line complex, Cambridge University Press, 1903.
Horst Knörrer, Geodesics on the ellipsoid, Invent. Math. 59 (1980), no. 2, 119–143. MR 577358, DOI 10.1007/BF01390041
Horst Knörrer, Geodesics on quadrics and a mechanical problem of C. Neumann, J. Reine Angew. Math. 334 (1982), 69–78. MR 667450, DOI 10.1515/crll.1982.334.69
Valeriĭ V. Kozlov and Dmitriĭ V. Treshchëv, Billiards, Translations of Mathematical Monographs, vol. 89, American Mathematical Society, Providence, RI, 1991. A genetic introduction to the dynamics of systems with impacts; Translated from the Russian by J. R. Schulenberger. MR 1118378, DOI 10.1090/mmono/089
H. P. McKean and P. van Moerbeke, Hill and Toda curves, Comm. Pure Appl. Math. 33 (1980), no. 1, 23–42. MR 544043, DOI 10.1002/cpa.3160330103
J. Moser, Geometry of quadrics and spectral theory, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York-Berlin, 1980, pp. 147–188. MR 609560
P. E. Newstead, Stable bundles of rank $2$ and odd degree over a curve of genus $2$, Topology 7 (1968), 205–215. MR 237500, DOI 10.1016/0040-9383(68)90001-3
S. Ramanan, Orthogonal and spin bundles over hyperelliptic curves, Proc. Indian Acad. Sci. Math. Sci. 90 (1981), no. 2, 151–166. MR 653952, DOI 10.1007/BF02837285
M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), no. 1, 207–233 (French). MR 688265, DOI 10.1007/BF01393342
M. Reid, The complete intersection of two or more quadrics, Thesis, Cambridge Univ. 1972.
Frank Sottile, Enumerative geometry for the real Grassmannian of lines in projective space, Duke Math. J. 87 (1997), no. 1, 59–85. MR 1440063, DOI 10.1215/S0012-7094-97-08703-2
G. Trautmann, Poncelet curves and associated theta characteristics, Exposition. Math. 6 (1988), no. 1, 29–64. MR 927588
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, DOI 10.1007/BF01077598