Loxodromes on Twisted Surfaces in Lorentz-Minkowski 3-Space

TL;DR

This paper derives differential equations for loxodromes on twisted surfaces in Lorentz-Minkowski 3-space, generalizing previous formulas and providing visualizations to enhance understanding of their geometric properties.

Contribution

It introduces general formulas and differential equations for loxodromes on twisted surfaces in Lorentz-Minkowski space, extending existing theories and including visual examples.

Findings

Differential equations for loxodromes on three types of twisted surfaces.

Generalization of loxodrome equations for special cases.

Visualizations of loxodromes on twisted surfaces using Mathematica.

Abstract

In the present paper, firstly we give the general formulas according to first fundamental form of a surface for different types of loxodromes, meridians and surfaces in E^3_1. After that, we obtain the differential equations of loxodromes on Type-I, Type-II and Type-III twisted surfaces in E^3_1 and also, we state a theorem which generalizes the differential equations of different types of loxodromes on the twisted surfaces for a special case. Finally, we provide several examples for visualizing our obtained results and draw our loxodromes and meridians on twisted surfaces with the aid of Mathematica.