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) states that whenever a polygon is inscribed in one circle and circumscribes another circle, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two circles. It is named after French engineer and mathematician Jean-Victor Poncelet.

Poncelet's porism can be proved by an argument using an elliptic curve, whose points represent a combination of a line tangent to C and a crossing point of that line with D.

Statement[edit]

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 (meaning that all of its vertices lie on C) and circumscribed around D (meaning that all of its edges are tangent to D), then it is possible to find infinitely many of them. Each point of C or D is a vertex or tangency (respectively) of one such polygon.

The polygons that are inscribed in one circle and circumscribed about another are called bicentric polygons, so Poncelet's porism can be expressed more concisely by saying that every bicentric polygon is part of an infinite family of bicentric polygons with respect to the same two circles.

Proof sketch[edit]

View C and D as curves in the complex projective plane P2. For simplicity, assume that C and D meet transversely (meaning that each intersection point of the two is a simple crossing). Then by Bézout's theorem, the intersection CD of the two curves consists of four complex points. For an arbitrary point 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.