Bicycling geodesics are Kirchhoff rods

Gil Bor, Connor Jackman, Serge Tabachnikov

Research output: Contribution to journalArticlepeer-review


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 of Kirchhoff rods, a class of curves introduced in 1859 by Kirchhoff, generalizing the planar elastic curves of Bernoulli and Euler.

Original languageEnglish (US)
Pages (from-to)3572-3602
Number of pages31
Issue number7
StatePublished - Jul 1 2023

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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