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.

中文翻译：

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广义单航向导航

引入伪对等线的概念，我们将单航向导航推广到共形平坦黎曼流形，在任意力的扰动矢量场（风，流）的作用下。使用欧拉-拉格朗日方程将这些发现应用于时间最优导航。我们参考 Zermelo 导航问题，承认扰动和船速的空间和时间依赖性。得到了单航向时间最优导航的必要条件，讨论了最小时间和最大时间的伪对等差。此外，我们描述了产生伪对等和对等时间极值的风。