Loxodromes and geodesics on rotational surfaces in pseudo-isotropic space

TL;DR

This paper explores the properties of loxodromes and geodesics on rotational surfaces within pseudo-isotropic space, defining new angle concepts and deriving equations for these curves.

Contribution

It introduces the first study of loxodromes in pseudo-isotropic space I_p^3, defining angles and deriving equations for loxodromes and geodesics on rotational surfaces.

Findings

Derived equations for space-like loxodromes

Derived equations for time-like loxodromes

Provided examples illustrating these curves

Abstract

Special curves and surfaces have an important place in mathematics, engineering and other fields of science. Loxodromes are special curves which cut all meridians on the Earth's surface at a constant angle and they are very popular in engineering. Ships sailing and airplanes flying along a fixed magnetic compass course move along this curve. The Mercator projections of the loxodromes on the sphere correspond to the lines and their stereographic projections to the logarithmic spirals. In general, loxodromes are not great circle arcs (geodesics). Geodesics correspond to the shortest distance between two points on the Earth's surface. Since loxodromes do not need to change course, they are important in navigation. Up till now, there have been many important studies of loxodromes on different surfaces (sphere, ellipsoid, rotational, helicoidal, canal, twisted, etc.) and in different ambient spaces (Euclidean, Minkowski, simply isotropic, etc.). However, there is no study on loxodromes in pseudo-isotropic space I_p^3. In I_p^3, different pseudo-isotropic angles can be defined depending on whether the vectors are space-like or time-like as in Minkowski space. In this study, we will first define these angles in I_p^3. Then, the equations of space-like and time-like loxodromes and geodesics on rotational surfaces will be obtained. Some examples will also be provided.