Poncelet's theorem in space

Author:
Emma Previato

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

MSC (1991):
Primary 14H40; Secondary 58F07

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

Published electronically:
May 4, 1999

MathSciNet review:
1662198

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[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 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**

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

Email:
ep@math.bu.edu

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

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