Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Poncelet's theorem in space

Author: Emma Previato
Journal: Proc. Amer. Math. Soc. 127 (1999), 2547-2556
MSC (1991): Primary 14H40; Secondary 58F07
Published electronically: May 4, 1999
MathSciNet review: 1662198
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • [BB] W. Barth and Th. Bauer, Poncelet theorems, Expositiones Math. 14 (1996), 125-144. MR 97f:14051
  • [BM] W. Barth and J. Michel, Modular curves and Poncelet polygons, Math. Ann. 295 (1993), 25-49. MR 94c:14045
  • [BKOR] H.J.M. Bos, C. Kers, F. Oort and D.W. Raven, Poncelet's closure theorem, Expositiones Math. 5 (1987), 238-364. MR 88m:14041
  • [CCS1] S.-J. Chang, B. Crespi and K.-J. Shi, Elliptical billiard systems and the full Poncelet's theorem in $n$ dimensions, J. Math. Phys. 34 (1993), 2242-2256. MR 94g:58092
  • [CCS2] B. Crespi, S.-J. Chang and K.-J. Shi, Elliptical billiards and hyperelliptic functions, J. Math. Phys. 34 (1993), 2257-2289. MR 94g:58093
  • [Co] D.A. Cox, The arithmetic-geometric mean of Gauss, Enseign. Math. 30 (1984), 275-330. MR 86a:01027
  • [CF] S.-J. Chang and R. Friedberg, Elliptical billiards and Poncelet's theorem, J. Math. Phys. 29 (1988), 1537-1550. 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
  • [DR] U. Desale and S. Ramanan, Classification of vector bundles of rank 2 over hyperelliptic curves, Invent. Math. 38 (1977), 161-186. MR 55:2906
  • [D] R. Donagi, The group law on the intersection of two quadrics, Ann. Scuola Norm. Sup. Pisa 7 (1980), 217-240. MR 82b:14025
  • [vGP] B. van Geemen and E. Previato, On the Hitchin system, Duke Math. J. 85 (1996), 659-683. MR 97k:14010
  • [GH1] P. Griffiths and J. Harris, A Poncelet theorem in space, Comment. Math. Helvetici 52 (1977), 145-160. MR 58:16695
  • [GH2] P. Griffiths and J. Harris, On Cayley's explicit solution to Poncelet's porism, Enseign. Math. 24 (1978), 31-40. MR 80g:51017
  • [H] J. Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), 685-724. MR 80m:14038
  • [J] C.M. Jessop, A treatise on the line complex, Cambridge University Press, 1903.
  • [K1] H. Knörrer, Geodesics on the ellipsoid, Invent. Math. 59 (1980), 119-143. MR 81h:58050
  • [K2] H. Knörrer, Geodesics on quadrics and a mechanical problem of C. Neumann, J. Reine Angew. Math. 334 (1982), 69-78. MR 84b:58089
  • [KT] V.V. Kozlov and D.V. Treshchëv, Billiards. A genetic introduction to the dynamics of systems with impacts, AMS Translations Math. Monographs 89 (1991). MR 93k:58094a
  • [McKvM] H.P. McKean and P. van Moerbeke, Hill and Toda curves, Comm. Pure Appl. Math. 33 (1980), 23-42. MR 81b:14016
  • [M] J. Moser, Geometry of quadrics and spectral theory, Chern Sympos., Springer-Verlag 1980, pp. 147-188. MR 82j:58064
  • [N] P. E. Newstead, Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology 7 (1968), 205-215. MR 38:5782
  • [R] S. Ramanan, Orthogonal and spin bundles over hyperelliptic curves, in Geometry and Analysis, Papers Dedicated to V.K. Patodi, Springer 1981, pp. 151-166. MR 83f:14014
  • [Ra] M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), 207-233. MR 84c:14021
  • [Re] M. Reid, The complete intersection of two or more quadrics, Thesis, Cambridge Univ. 1972.
  • [S] F. Sottile, Enumerative geometry for the real Grassmannian of lines in projective space, Duke Math. J. 87 (1997), 59-85. MR 99a:14079
  • [T] G. Trautmann, Poncelet curves and associated theta characteristics, Expositiones Math. 6 (1988), 29-64. MR 89c:14047
  • [V] A.P. Veselov, Integrable discrete-time systems and difference operators, Functional Anal. Appl. 22 (1988), 83-93. MR 90a:58081

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 14H40, 58F07

Retrieve articles in all journals with MSC (1991): 14H40, 58F07

Additional Information

Emma Previato
Affiliation: Department of Mathematics, Boston University, Boston, Massachusetts 02215

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 MDA904-95-H-1031
Communicated by: Ron Donagi
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society