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A very famous theorem of Poncelet states that for an elliptic billiard all $n$-periodic trajectories are tangent to some ellipse. As far as I know, Poncelet proved this theorem while sitting in Russian jail so he didn't write it down. Could anybody give me a reference on a first book or article where the proof was actually given?

I want to make a correct reference in the article I am writing. Moreover, I am also thinking about mentioning a greek mathematician Proclus who lived in the 5th century BC and proved that once a line in an ellipse is tangent to some other smaller ellipse, its reflection is tangent to the smaller ellipse too.

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  • $\begingroup$ Here's an image illustrating the theorem, for the case of two circles dl.dropbox.com/u/9709624/Poncelet%20example.pdf $\endgroup$
    – Dror Atariah
    Commented Feb 8, 2013 at 9:00
  • $\begingroup$ @DrorAtariah The link in your comment no longer works (as far as I can tell). Still, I guess there are many images illustrating Poncelet's theorem available online - maybe some of them is close to what you had in mind. $\endgroup$
    – Martin Sleziak
    Commented Jan 25, 2018 at 6:01
  • $\begingroup$ Here's something related I'd like to know: given two non-singular conics $C, D \subset \mathbb{P^2}$, the curve $\{(p, L) \in C\times D^* | p \in L\}$ has genus one and so choosing a distinguished point we have an elliptic curve. What is a geometric construction of the group law? $\endgroup$
    – Oliver Nash
    Commented Jan 25, 2018 at 12:21

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Poncelet published his theorem ("Poncelet's porism) in 1822, after he returned to France following his captivity as war prisoner in Russia:

J.V. Poncelet, Traité des propriétés projectives des figures (Paris, 1822).

The book has been scanned and can be read here or on archive.org.

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There seems to be a misconception here: Poncelet Theorem (at least the great one, which I believe to remember is the one he proved while in jail) is a much deeper, and more difficult statement than what you state.

Consider an ellipse inside another ellipse, and play inner-outer billiard with them. This means that you start from a point on the outer ellipse, choose one of the two line from this point tangent to the inner ellipse, and take the second intersection point of this line with the outer ellipse. You continue, always taking the next line tangent to the inner ellipse from the current point, and the other intersection point with the outer ellipse from the current line.

Theorem (Poncelet) $-$ If one orbit of this dynamical system is periodic, then all orbits are periodic.

This, if I remember well, is in Berger's Geometry. There might be a reference there.

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  • $\begingroup$ @Benoît Kloeckner: the theorem which Poncelet proved in jail, and was published in the 1822 book I mentioned is stated here: komal.hu/lap/2002-ang/poncelet.e.shtml are we talking about different things? $\endgroup$
    – Carlo Beenakker
    Commented Feb 7, 2013 at 22:47
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    $\begingroup$ We are talking about the same result (although it has been extended to ellipses), but this is not what the OP asked about: "in elliptic billiard all n-periodic trajectories are tangent to some ellipse" which is a much more elementary result. That's why I think the OP has a misconception. $\endgroup$
    – Benoît Kloeckner
    Commented Feb 8, 2013 at 11:46
  • $\begingroup$ Yes, you are right, I wanted to ask about this theorem. $\endgroup$
    – Olga
    Commented Feb 15, 2013 at 18:34

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