Poncelet's closure theorem

Illustration of Poncelet's porism for n = 3, a triangle that is inscribed in one circle and circumscribes another.

In geometry, Poncelet's porism (sometimes referred to as Poncelet's closure theorem), 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 that 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 complex 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.

Proof

Let p be a point of P² and ℓ a line of the dual projective plane. The key tool is the curve X given by the set of pairs (p, ℓ) where p is on the conic C and ℓ is tangent to the conic D. Then X is smooth; more specifically X is an elliptic curve. There is an involution $\sigma$ of X mapping (p, ℓ) to (p', ℓ) where p' is the other point of intersection of ℓ with C. There is another involution $\tau$ that sends (p, ℓ) to (p, ℓ') where ℓ' is the other tangent from p to D. With respect to the natural addition on X, it turns out that the composition $\tau \sigma$ is a translation. If it has one fixed point, it must be the identity. Translated back into the language of C and D, this means that if one point on C gives rise to an orbit that closes up (i.e. gives an n-gon), then every point does, as well.

See also

• Steiner's porism
• Tangent lines to circles

References

• Bos, H. J. M.; Kers, C.; Oort, F.; Raven, D. W. Poncelet's closure theorem. Expositiones Mathematicae 5 (1987), no. 4, 289–364.

External links

• David Speyer on Poncelet's Porism
• D. Fuchs, S. Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics
• Java applet by Michael Borcherds showing the cases n = 3, 4, 5, 6, 7, 8 (including the convex cases for n = 7, 8) made using GeoGebra.
• Java applet by Michael Borcherds showing Poncelet's Porism for a general Ellipse and a Parabola made using GeoGebra.
• Java applet by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 3) made using GeoGebra.
• Java applet by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 5) made using GeoGebra.
• Java applet by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 6) made using GeoGebra.
• Java applet showing the exterior case for n = 3 at National Tsing Hua University.
• Article on Poncelet's Porism at Mathworld.