Generalized loxodromes with application to time-optimal navigation in arbitrary wind. Issue 1 (January 2021)

• Record Type: Journal Article
• Authors: Aldea, Nicoleta; Kopacz, Piotr
• Is Part Of: Journal of the Franklin Institute. Volume 358:Issue 1(2021)
• Page Start: 776
• Page End: 799
• Publication Date: 2021-01
• Subjects: Science -- Periodicals; Technology -- Periodicals; Patents -- United States -- Periodicals; 505
• DOI: 10.1016/j.jfranklin.2020.11.009
• Languages: English
• ISSNs: 0016-0032

Abstract

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.