Poncelet’s theorem in space

Previato, Emma

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

Poncelet’s theorem in space
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Proc. Amer. Math. Soc. 127 (1999), 2547-2556 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.

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