On Generalised Single-Heading Navigation

Introducing the notion of a pseudoloxodrome, we generalise a single-heading navigation to conformally flat Riemannian manifolds, under the action of a perturbing vector field (wind, current) of arbitrary force. The findings are applied to time-optimal navigation with the use of the Euler–Lagrange equations. We refer to the Zermelo navigation problem admitting space and time dependence of both a perturbation and a ship's speed. The necessary conditions for single-heading time-optimal navigation are obtained and the pseudoloxodromes of minimum and maximum time are discussed. Furthermore, we describe winds which yield the pseudoloxodromic and loxodromic time extremals. Our research is also illustrated with the examples in dimension two emphasising the single-heading solutions among the time-optimal trajectories in the presence of some space-dependent winds.