Abstract
This article aims at generalizing loxodromic (rhumb line) navigation to conformally flat Riemannian manifolds. We admit space-time dependence of both perturbing vector field and ship’s self-speed. Thereafter, the findings are applied to time-efficient navigation by a variational approach, referring to the local solutions of Zermelo’s navigation problem under arbitrary wind. This yields the corresponding conditions for loxodromic time-minimal and time-maximal navigation in relation to the navigation data. Our research is also illustrated by a two-dimensional example (an oblate ellipsoid), which distinguishes perturbations of different force: weak, critical and strong. It includes numerical simulations and discussion, emphasizing and comparing loxodromic solutions among the minimizing, maximizing and anomalous time extremals.
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