We consider the problem of minimizing \hbox{$\int_{0}^\ell \sqrt{\xi^2 +K^2(s)}\, {\rm d}s $}${{}^{\mathrm{\int }}}_{\mathrm{0}}^{\mathit{\ell }}\sqrt{{\mathit{\xi }}^{\mathrm{2}}\mathrm{+}{\mathit{K}}^{\mathrm{2}}\mathrm{\left(}\mathit{s}\mathrm{\right)}}\hspace{0.17em}\mathrm{d}\mathit{s}$ for a planar curve having fixed initial and final positions and directions. The total length ℓ is free. Here s is the arclength parameter, K(s) is the curvature of the curve and ξ > 0 is a fixed constant. 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.