ESAIM: Control, Optimisation and Calculus of Variations

Research Article

Projective Reeds-Shepp car on S 2 with quadratic cost

Boscain, Ugoa1a2 and Rossi, Francescoa2

a1 LE2i, CNRS UMR5158, Université de Bourgogne, 9 avenue Alain Savary - BP 47870, 21078 Dijon Cedex, France.

a2 SISSA, via Beirut 2-4, 34014 Trieste, Italy.;


Fix two points $x,\bar{x}\in S^2$ and two directions (without orientation) $\eta,\bar\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost

$J[\gamma]=\int_0^T \left({\g}_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))+ 
 K^2_{\gamma(t)}{\g}_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) \right) ~{\rm d}t$

along all smooth curves starting from x with direction η and ending in $\bar{x}$ with direction $\bar\eta$ . Here g is the standard Riemannian metric on S 2 and $K_\gamma$ is the corresponding geodesic curvature. The interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1). We compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology.

(Received May 30 2008)

(Online publication December 19 2008)

Key Words:

  • Carnot-Caratheodory distance;
  • geometry of vision;
  • lens spaces;
  • global cut locus

Mathematics Subject Classification:

  • 49J15;
  • 53C17