Curve cuspless reconstruction via sub-Riemannian geometry

U. Boscain, R. Duits, F. Rossi, Y. Sachkov

Research output: Contribution to journalArticleAcademicpeer-review

20 Citations (Scopus)
175 Downloads (Pure)


We consider the problem of minimizing $\int_{0}^L \sqrt{\xi^2 +K^2(s)}\, ds $ for a planar curve having fixed initial and final positions and directions. The total length $L$ is free. Here $s$ is the variable of arclength parametrization, $K(s)$ is the curvature of the curve and $\xi>0$ a parameter. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem.
Original languageEnglish
Pages (from-to)748-770
JournalESAIM : Control, Optimisation and Calculus of Variations
Issue number3
Publication statusPublished - 2014


  • Mathematics - Optimization and Control
  • Mathematics - Differential Geometry


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