Poncelet Porisms and beyond

Abstract:

We start from the set $T$ of lines in $\mathbf R^d$ simultaneously tangent to $d-1$ quadrics from a given confocal family. We analyse its structure and derive a fundamental property of $T$: any two lines from this set can be obtained from each other by at most $d-1$ billiard reflections at some quadrics from the confocal family. We introduce two hierarchies of notions: {\em $s$-skew lines} in $T$ and {\em $s$-weak Poncelet trajectories}, $s=-1,0,...,d-2$. The interrelations between billiard dynamics, linear subspaces of intersections of quadrics and hyperelliptic Jacobians developed in our joint paper with M. Radnovic (arXiv 0710.3656) enabled us to obtain higher-dimensional and higher-genera generalizations of several classical genus $1$ results.

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