Projective Reeds-Shepp car on S2 with quadratic cost
LE2i, CNRS UMR5158, Université de Bourgogne, 9 avenue Alain Savary - BP 47870,
21078 Dijon Cedex, France.
2 SISSA, via Beirut 2-4, 34014 Trieste, Italy. email@example.com; firstname.lastname@example.org
Fix two points and two directions (without orientation) of the velocities in these points. In this paper we are interested to the problem of minimizing the cost
along all smooth curves starting from x with direction η and ending in with direction . Here g is the standard Riemannian metric on S2 and 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.
Mathematics Subject Classification: 49J15 / 53C17
Key words: Carnot-Caratheodory distance / geometry of vision / lens spaces / global cut locus
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