Poncelet's theorem in space
Author:
Emma Previato
Journal:
Proc. Amer. Math. Soc. 127 (1999), 25472556
MSC (1991):
Primary 14H40; Secondary 58F07
Published electronically:
May 4, 1999
MathSciNet review:
1662198
Fulltext PDF Free Access
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Abstract: A plane polygon inscribed in a conic and circumscribed to a conic can be continuously `rotated', as it were. One of the many proofs consists in viewing each side of as translation by a torsion point of an elliptic curve. In the space version, involving torsion points of hyperelliptic Jacobians, there is a dimensional family of rotations, where of the hyperelliptic curve; the polygon is now inscribed in one and circumscribed to quadrics.
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 H.J.M. Bos, C. Kers, F. Oort and D.W. Raven, Poncelet's closure theorem, Expositiones Math. 5 (1987), 238364. MR 88m:14041
 [CCS1]
 S.J. Chang, B. Crespi and K.J. Shi, Elliptical billiard systems and the full Poncelet's theorem in dimensions, J. Math. Phys. 34 (1993), 22422256. MR 94g:58092
 [CCS2]
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 [Co]
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 [CF]
 S.J. Chang and R. Friedberg, Elliptical billiards and Poncelet's theorem, J. Math. Phys. 29 (1988), 15371550. MR 89j:58043
 [DLT]
 P. Deift, L.C. Li and C. Tomei, Loop Groups, Discrete Versions of Some Classical Integrable Systems, and Rank 2 Extensions, Memoirs of the Amer. Math. Soc. 479 (1992). MR 93d:58065
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 U. Desale and S. Ramanan, Classification of vector bundles of rank 2 over hyperelliptic curves, Invent. Math. 38 (1977), 161186. MR 55:2906
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 R. Donagi, The group law on the intersection of two quadrics, Ann. Scuola Norm. Sup. Pisa 7 (1980), 217240. MR 82b:14025
 [vGP]
 B. van Geemen and E. Previato, On the Hitchin system, Duke Math. J. 85 (1996), 659683. MR 97k:14010
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 P. Griffiths and J. Harris, A Poncelet theorem in space, Comment. Math. Helvetici 52 (1977), 145160. MR 58:16695
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Additional Information
Emma Previato
Affiliation:
Department of Mathematics, Boston University, Boston, Massachusetts 02215
Email:
ep@math.bu.edu
DOI:
http://dx.doi.org/10.1090/S0002993999053071
PII:
S 00029939(99)053071
Received by editor(s):
May 20, 1997
Received by editor(s) in revised form:
September 28, 1997
Published electronically:
May 4, 1999
Additional Notes:
The author’s research was partly supported by NSA grant MDA90495H1031
Communicated by:
Ron Donagi
Article copyright:
© Copyright 1999 American Mathematical Society
