Kasner Meets Poncelet

Abstract

Given a planar pentagon P , construct two new pentagons, D(P ) and I(P ): the vertices of D(P ) are the intersection points of the diagonals of P , and the vertices of I(P ) are the tangency points of the conic inscribed in P (the vertices and the tangency points are taken in their cyclic order). The following result is due to E. Kasner [9] (published in 1928, but discovered much earlier, in 1896). Theorem 1 The two operations on pentagons, D and I, commute: ID(P ) = DI(P ), see Figure 1.Figure 1: Kasner's theorem.A polygon inscribed into a conic and circumscribed about a conic is called a Poncelet polygon. The celebrated Poncelet porism states that the existence of a Poncelet n-gon on a pair of nested ellipses implies that every point of the outer ellipse is a vertex of such a Poncelet n-gon, see, e.g., [2,4].Since a (generic) quintuple of points lies on a conic, a (generic) quintuple of lines is tangent to a conic, a (generic) pentagon is a Poncelet polygon. We generalize Kasner's theorem from pentagons to Poncelet polygons.Let P be a Poncelet n-gon, n ≥ 5. As before, the n-gon I(P ) is formed by the consecutive tangency points of the sides of P with the inscribed conic. Fix 2 ≤ k < n/2, and draw the k-diagonals of P , that is, connect ith vertex with (i + k)th vertex for i = 1, . . . , n. The consecutive intersection points of these diagonals form a new n-gon, denoted by D k (P ). Theorem 2 The two operations on Poncelet polygons commute: ID k (P ) = D k I(P ), see Figure 2.Figure 2: Theorem 2: n = 7, k = 3.We shall deduce Theorem 2 from the Poncelet grid theorem [15] which we now describe. The statements slightly differ for odd and even n, and we assume that n is odd.2 Let 1 , . . . , n be the lines containing the sides of a Poncelet n-gon, enumerated in such a way that their tangency points with the inscribed ellipse are in the cyclic order. The Poncelet grid is the collection of n(n + 1)/2 points i ∩ j , where i ∩ i is, by definition, the tangency point of the line i with the inscribed ellipse.Partition the Poncelet grid in two ways. Define the setswhere the indices are understood mod n. One has (n + 1)/2 sets Q k , each containing n points, and n sets R k , each containing (n + 1)/2 points. These sets are called concentric and radial, respectively, see Figure 3.Figure 3: Poncelet grid, n = 9: the concentric sets Q 0 , Q 2 , Q 3 , and Q 4 are shown.Theorem 3 (i) The concentric sets lie on nested ellipses, and the radial sets lie on disjoint hyperbolas.(ii) The complexifications of these conics have four common tangent lines (invisible in Figure 3). (iii) All the concentric sets are projectively equivalent to each other, and so are all the radial sets.The Poncelet grid theorem is a result in projective geometry. One can apply a projective transformation so that the initial two ellipses that support the Poncelet polygon become confocal (this is the case in Figure 2). Under 3 this assumption, an Euclidean version of the Poncelet grid theorem, proved in [11], asserts that all the concentric and the radial sets ...

Fun Problems in Geometry and Beyond

“…There are two operations on a pentagon: T , the pentagram map, and I, replacing the pentagon by the pentagon whose vertices are the tangency points of the conic that is tangent to its five sides. Kasner's theorem asserts that these two operations commute: T • I = I • T (see [31] for details and a generalization to Poncelet polygons).…”

Poncelet-Darboux, Kippenhahn, and Szegő: interactions between projective geometry, matrices and orthogonal polynomials

“…Poncelet's construction can be approached also from a point of view of the theory of elliptic billiards [16,17,40,53] and discrete dynamical systems such as the pentagram map [52,56,57,64].…”

Intersecting the sides of a polygon

“…Remark 1.3. S. Tabachnikov [7] proved that Kasner's theorem holds for all Poncelet polygons (i.e. polygons inscribed in conic and circumscribed about a conic).…”