Poncelet's closure theorem

The closing sentence of Poncelet is a set of projective geometry and says: Can you use a corner () simultaneously circumscribe a conic section and enroll another conic, so there are infinitely many other corner with this property.

Alternative formulation: , are conic sections. lie within. It then starts following chain of constructions: From a point on the tangent line is drawn to that intersects another point, from this point the second tangent is drawn on and so the figure formed by the tangent sections Closes back in the point, stating the proposition that there are still infinitely many other such figures to the conics are. One can use any other point of start and gets a closed polygon again. The polygons obtained in this way are also called Poncelet polygons.

Jean -Victor Poncelet was in his Traité des propriétés projectives des figures of 1822 a ( "synthetic" ) geometric proof. Carl Gustav Jacobi (Journal of Pure and Applied Mathematics, Vol.3, 1828) gave a proof using elliptic functions. A modern proof of Phillip Griffiths makes clear that the group properties of elliptic curves behind this sentence. The sentence is by Griffiths equivalent to the addition law of elliptic integrals. Many other famous mathematicians have made ​​contributions for the set and its generalization, for example, Arthur Cayley gave explicit conditions for you when conics such Poncelet polygons have ( Philosophical Magazine Bd.6, 1852, 99, Phil.Trans.Royal Society Bd.151, 1861, pp. 225, in Henri Lebesgue: Les coniques 1942). . This is from the standpoint of the theory of elliptic curves also shown in Griffiths, Harris On Cayley 's explicit solution to Poncelet 's porism. L' enseignement Mathematique, 24 ( 1978).

The set is a prime example of a class of geometric problems, which are called the closure problems.