Timelike Loxodromes on Lorentzian Helicoidal Surfaces in Minkowski n--Space

TL;DR

This paper studies the properties and explicit parametrizations of timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski space, providing differential equations, solutions, and visual examples.

Contribution

It derives differential equations and explicit formulas for timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski space, a novel analysis in this geometric context.

Findings

Derived first order ODEs for timelike loxodromes

Obtained explicit parametrizations of these loxodromes

Provided visual examples of the results

Abstract

In this paper, we examine timelike loxodromes on three kinds of Lorentzian helicoidal surfaces in Minkowski $n$--space. First, we obtain the first order ordinary differential equations which determine timelike loxodromes on the Lorentzian helicoidal surfaces in $\mathbb{E}^n_1$ according to the causal characters of their meridian curves. Then, by finding general solutions, we get the explicit parametrizations of such timelike loxodromes. In particular, we investigate the timelike loxodromes on the three kinds of Lorentzian right helicoidal surfaces in $\mathbb{E}^n_1$. Finally, we give an example to visualize the results.