Poncelet's closure theorem

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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 (i.e., a bicentric polygon), then it is possible to find infinitely many of them.

Poncelet's porism can be understood in terms of an elliptic curve.

Sketch of proof[edit]

View C and D as projective curves in P2. For simplicity, assume that C and D meet transversely. Then by Bézout's theorem, CD consists of 4 (complex) points. For d in D, let d be the tangent line to D at d. Let X be the subvariety of C × D consisting of (c,d) such that d passes through c. Given c, the number of d with (c,d) ∈ X is 1 if cCD and 2 otherwise. Thus the projection XCP1 presents X as a degree 2 cover ramified above 4 points, so X is an elliptic curve (once we fix a base point on X). Let \sigma be the involution of X sending a general (c,d) to the other point (c,d′) with the same first coordinate. Any involution of an elliptic curve with a fixed point, when expressed in the group law, has the form xpx for some p, so \sigma has this form. Similarly, the projection XD is a degree 2 morphism ramified over the contact points on D of the four lines tangent to both C and D, and the corresponding involution \tau has the form xqx for some q. Thus the composition \tau \sigma is a translation on X. If a power of \tau \sigma has a fixed point, that power must be the identity. Translated back into the language of C and D, this means that if one point cC (equipped with a corresponding d) gives rise to an orbit that closes up (i.e., gives an n-gon), then so does every point. The degenerate cases in which C and D are not transverse follow from a limit argument.

See also[edit]

References[edit]

  • 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[edit]

  • 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.