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Abstract This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non‐focal point inside an elliptic billiard table, one considers the family of rays emanating from and the caustic of the reflected family after reflections off the ellipse, for each positive integer . It is known that has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of , though we do not prove that these are the only cusps. 

Abstract A bicycle path is a pair of trajectories in ${\mathbb{R}}^n$, the ‘front’ and ‘back’ tracks, traced out by the endpoints of a moving line segment of fixed length (the ‘bicycle frame’) and tangent to the back track. Bicycle geodesics are bicycle paths whose front track’s length is critical among all bicycle paths connecting two given placements of the line segment. We write down and study the associated variational equations, showing that for $n⩾3$each such geodesic is contained in a 3-dimensional affine subspace and that the front tracks of these geodesics form a certain subfamily ofKirchhoff rods, a class of curves introduced in 1859 by Kirchhoff, generalizing the planar elastic curves of Bernoulli and Euler. 

Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centroaffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centroaffine geometry. In particular, the Bäcklund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves. Our paper concerns self-Bäcklund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulam’s problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics. We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.