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.

本文旨在将loxodromic（大黄线）导航泛化为保形平坦的黎曼流形。我们接受扰动矢量场和船的自速度的时空相关性。此后，该发现通过变分方法应用于时效导航，这是指在任意风下策尔梅洛导航问题的局部解。这样就产生了关于导航数据的最低限度时间最小和最大时间导航的相应条件。我们的研究还通过一个二维示例（扁椭圆形）进行了说明，该示例区分了不同力（弱，临界和强）的扰动。它包括数值模拟和讨论，在最小化，最大化和异常时间极值之间强调和比较铁氧体解决方案。