Newton Solórzano, Víctor León, Alexandre Henrique, Marcelo Souza
{"title":"Navigation problem; $λ-$Funk metric; Finsler metric","authors":"Newton Solórzano, Víctor León, Alexandre Henrique, Marcelo Souza","doi":"arxiv-2409.12058","DOIUrl":null,"url":null,"abstract":"We investigate the travel time in a navigation problem from a geometric\nperspective. The setting involves an open subset of the Euclidean plane,\nrepresenting a lake perturbed by a symmetric wind flow proportional to the\ndistance from the origin. The Randers metric derived from this physical problem\ngeneralizes the well-known Euclidean metric on the Cartesian plane and the Funk\nmetric on the unit disk. We obtain formulas for distances, or travel times,\nfrom point to point, from point to line, and vice-versa","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Differential Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.12058","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate the travel time in a navigation problem from a geometric
perspective. The setting involves an open subset of the Euclidean plane,
representing a lake perturbed by a symmetric wind flow proportional to the
distance from the origin. The Randers metric derived from this physical problem
generalizes the well-known Euclidean metric on the Cartesian plane and the Funk
metric on the unit disk. We obtain formulas for distances, or travel times,
from point to point, from point to line, and vice-versa
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