Characterizations of Loxodromes on Rotational Surfaces in Euclidean 3--Space

TL;DR

This paper characterizes loxodromes on various rotational surfaces in Euclidean 3-space, deriving parametrizations, curvatures, and special properties for surfaces with constant Gaussian curvature, flatness, or minimality.

Contribution

It provides explicit parametrizations and geometric properties of loxodromes on rotational surfaces with specific curvature conditions, including flat, constant Gaussian curvature, and minimal surfaces.

Findings

Loxodromes on flat rotational surfaces are general helices.

Loxodromes on minimal rotational surfaces intersect meridians at π/4 and are asymptotic curves.

Explicit formulas for loxodromes on surfaces with constant Gaussian curvature.

Abstract

In this paper, we study on the characterizations of loxodromes on the rotational surfaces satisfying some special geometric properties such as having constant Gaussian curvature, flat and minimality in Euclidean 3-space. First, we give the parametrizations of loxodromes parametrized by arc-length parameter on any rotational surfaces in $\mathbb{E}^{3}$ and then, we calculate the curvature and the torsion of such loxodromes. Then, we give the parametrizations of loxodromes on rotational surfaces with constant Gaussian curvature. In particular, we prove that the loxodrome on the flat rotational surface is a general helix. Also, we investigate the loxodromes on the rotational surfaces with a constant ratio of principal curvatures (CRPC rotational surfaces). Moreover, we give the parametrizations of loxodromes on the minimal rotational surface which is a special case of CRPC rotational surfaces. Then, we show that the loxodrome intersects the meridians of minimal rotational surface by the angle $\pi/{4}$ becomes an asymptotic curve. Finally, we give some visual examples to strengthen our main results via Wolfram Mathematica.