Euclidean space perturbed by a constant vector field and its relation to a Zermelo navigation problem - Lujerio Garcia, Dik D. (2025)
In this work, the authors perturb the Euclidean plane with a constant vector field of the form W = (0, ε) with 0 ≤ ε < 1, which can be interpreted as wind currents affecting the movement of ships in a constant unidirectional way. It is observed that the resulting perturbed norm, called the ε-Euclidean metric, which is non-reversible, is a Finsler metric. In this way, a new non-Euclidean geometry is introduced. With this, the ε-Euclidean distance is induced and defined. This new way of measuring point-to-point distances can be interpreted, physically, as optimal travel time. Due to the non-reversibility of the ε-Euclidean metric, two types of circumferences are defined and characterized. Distance formulas (or optimal travel time) from point to line, from line to point, and from line to line are obtained, as well as a geometric construction technique for obtaining the distance from a point to a parabola, which can be adapted to other curves that simulate the Edge of a beach. Examples and graphs are presented for a better understanding of the work.