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# Poncelet

[pawnsuh-le]
Poncelet, Jean Victor, 1788-1867, French mathematician and army engineer. He taught at the school of mechanics at Metz and at the Faculté des Sciences and the École Polytechnique, both in Paris. While a prisoner of war (1813-14) in Russia during Napoleon I's campaign, he evolved the foundations of the modern form of projective geometry and incorporated his results in Traité des propriétés projectives des figures (1822; 2d ed., 2 vol., 1865-66).

In geometry, Poncelet's porism, named after French engineer and mathematician Jean-Victor Poncelet, states the following: Let C and D be two plane conics. If it is possible to find, for a given n > 2, one n-sided polygon which is simultaneously inscribed in C and circumscribed around D, then it is possible to find infinitely many of them.

Poncelet's porism can be proved via elliptic curves; geometrically this depends on the representation of an elliptic curve as the double cover of C with four ramification points. (Note that C is isomorphic to the projective line.) The relevant ramification is over the four points of C where the conics intersect. (There are four such points by Bézout's theorem). One can also describe the elliptic curve as a double cover of D; in this case, the ramification is over the contact points of the four bitangents.

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