Characterizations of Loxodromes on Rotational Surfaces in Euclidean 3-Space

Year 2023, Volume: 16 Issue: 1, 147 - 159, 30.04.2023

Ferdağ Kahraman Aksoyak Burcu Bektaş Demirci Murat Babaarslan

https://doi.org/10.36890/iejg.1161830

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 and a constant ratio of principal curvatures (CRPC rotational surfaces).
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.
Also, we investigate the loxodromes on the CRPC rotational surfaces.
Moreover, we give the parametrizations of loxodromes on the minimal rotational surface which is a special case of CRPC rotational surfaces.
Finally, we give some visual examples to strengthen our main results via Wolfram Mathematica.

Keywords

Loxodrome, curvature, torsion, rotational surface, Gaussian curvature, mean curvature, minimal surfaces, helix, asymptotic curve, Euclidean 3--space

References

• [1] Alexander, J.: Loxodromes: a rhumb way to go. Mathematics Magazine. 77 (5), 349–356 (2004).
• [2] Babaarslan, M., Yayli, Y.: Differential equation of the loxodrome on a helicoidal surface. Journal of Navigation. 68 (5), 962–970 (2015).
• [3] Deshmukha, S., Alghanemib, A., Farouki, R. T.: Space curves defined by curvature–torsion relations and associated helices. Filomat. 33 (15), 4951–4966 (2019).
• [4] Ferreol, R.: Mathcurve. Rhumb line. http://www.mathcurve.com/courbes3d.gb/loxodromie/loxodromie.shtml, [Access Date: 14 August 2022].
• [5] Kendall, C. G.: A study of the loxodrome on a general surface of revolution with special application to the conical spiral. Master dissertation. University of California (1920).
• [6] Khalifa, Saad M., Abdel-Baky, R. A., Alharbi, F., Aloufi, A.: On minimal surfaces with the same asymptotic curve in Euclidean space. Applied Mathematical Sciences. 13 (21), 1021–1031 (2019).
• [7] Kos, S., Filjar, R., Hess, M.: Differential equation of the loxodrome on a rotational surface. In: Proceedings of the 2009 International Technical Meeting of The Institute of Navigation, Anaheim, CA (2009).
• [8] Kühnel, W.: Differential Geometry: curves–surfaces–manifolds (second ed.). American Mathematical Society. USA (2006).
• [9] Leitão, H., Gaspar, J. A.: Globes, rhumb tables, and the pre-history of the Mercator projection. Imago Mundi. 66 (2), 180–195 (2014).
• [10] Noble, C. A.: Note on loxodromes. Bulletin of the American Mathematical Society. 12 (3), 116–119 (1905).
• [11] O’Neill, B.: Elementary differential geometry (second ed.). Academic Press. USA (1997).
• [12] Petrovic, M.: Differential equation of a loxodrome on the spheroid. Nase More 54 (3-4), 87–89 (2007).
• [13] Pérez, J.: A new golden age of minimal surfaces. Notices of the AMS. 64 (4), 347–358 (2017).
• [14] Pressley, A.: Elementary differential geometry. Springer (2001).
• [15] Xu, G., Wang, G. Z.: Quintic parametric polynomial minimal surfaces and their properties. Differential Geometry and its Applications. 28 (6), 697–704 (2010).
• [16] Wang, H., Pottmann, H.: Characteristic parameterizations of surfaces with a constant ratio of principal curvatures. Computer Aided Geometric Design 93 102074 (2022).