  A solution to the left-invariant sub-Riemannian problem on the group of
motions (rototranslations) of a plane SE(2) is obtained. Local and global
optimality of extremal trajectories is characterized.
  Lower and upper bounds on the first conjugate time are proved.
  The cut time is shown to be equal to the first Maxwell time corresponding to
the group of discrete symmetries of the exponential mapping. An explicit global
description of the cut locus is obtained.
  Optimal synthesis is described.
