### Abstract

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 language | English |
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Pages (from-to) | 748-770 |

Journal | ESAIM : Control, Optimisation and Calculus of Variations |

Volume | 20 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 |

### Keywords

- Mathematics - Optimization and Control
- Mathematics - Differential Geometry

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## Cite this

Boscain, U., Duits, R., Rossi, F., & Sachkov, Y. (2014). Curve cuspless reconstruction via sub-Riemannian geometry.

*ESAIM : Control, Optimisation and Calculus of Variations*,*20*(3), 748-770. https://doi.org/10.1051/cocv/2013082