The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal synthesis is constructed.

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. 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. Optimal synthesis on an open dense subset of the state space is described.

The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained.