  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.
