We develop general methodologies for the minimal time routing problem of a vessel moving in stationary or time dependent environments, respectively. Local optimality considerations, combined with global boundary conditions, result in piecewise continuous optimal policies. In the stationary case, the velocity of the traveling vessel within each subregion depends only on the direction of motion. Variational calculus is used to derive the geometry of piecewise linear extremals. For the time dependent problem, the speed of the vessel within each subregion is assumed to be a known function of time and the direction of motion. Optimal control theory is used to reveal the nature of piecewise continuous optimal policies.