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Creators/Authors contains: "Tabachnikov, Serge"

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  1. (, Theoretical and Applied Mechanics)
    Given a projective Finsler metric in a convex domain in the projective plane, that is, a metric in which geodesics are straight lines, consider the respective Finsler billiard system. Choose a generic point inside the table and consider the billiard trajectories that start at this point and undergo N reflection off the boundary. The envelope of the resulting 1-parameter family of straight lines is the Nth caustic by reflection. We prove that, for every N, it has at least four cusps, generalizing a similar result for Euclidean metric, obtained recently jointly with G. Bor. 
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    Free, publicly-accessible full text available January 1, 2026
  2. ; ; ; ; (, Journal of the Association for Mathematical Research)
    In this article, we study polygonal symplectic billiards. We provide new results, some of which are inspired by numerical investigations. In particular, we present several polygons for which all orbits are periodic. We demonstrate their properties and derive various conjectures using two numerical implementations. 
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    Free, publicly-accessible full text available February 28, 2026
  3. ; ; (, Journal of the London Mathematical Society)
    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. 
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    Free, publicly-accessible full text available December 1, 2025
  4. (, Moscow mathematical journal)
    Accepted Manuscript Full Text Available
  5. ; (, Pacific Journal of Mathematics)
    Full Text Available 
  6. ; ; ; ; (, The Mathematical Intelligencer)
    Full Text Available 
  7. ; ; (, Annales de l'Institut Fourier)
    Full Text Available 
  8. ; (, The American Mathematical Monthly)
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  9. ; ; (, Nonlinearity)
    Abstract A bicycle path is a pair of trajectories in 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. 
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    Full Text Available 
  10. ; (, European Journal of Mathematics)
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