%0 Journal Article
%J ESAIM COCV 16 (2010) 275-297
%D 2010
%T Projective Reeds-Shepp car on $S^2$ with quadratic cost
%A Ugo Boscain
%A Francesco Rossi
%X 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 $$ J[\\\\gamma]=\\\\int_0^T g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t))+\\nK^2_{\\\\gamma(t)}g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t)) ~dt$$ along all smooth curves starting from $x$ with direction $\\\\eta$ 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.\\nThe 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).\\nWe 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.
%B ESAIM COCV 16 (2010) 275-297
%G en_US
%U http://hdl.handle.net/1963/2668
%1 1429
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-06-11T11:22:54Z\\nNo. of bitstreams: 1\\n0805.4800v1.pdf: 610220 bytes, checksum: b0fa81a60fc43e6da6a4682e91b4d21e (MD5)
%R 10.1051/cocv:2008075