Loxodromes On Twisted Surfaces in Euclidean 3-Space

Abstract

In the present paper, loxodromes, which cut all meridians and parallels of twisted surfaces (that can be considered as a generalization of rotational surfaces) at a constant angle, have been studied in Euclidean 3-space and also some examples have been constructed to visualize and support our theory.

Keywords

• Loxodromes
• Meridian
• Twisted surface.

References

1. J. Alexander; Loxodromes: a rhumb way to go, Math. Mag., 77(5), (2004), 349–356.
2. M. Babaarslan and Y. Yaylı; Differential Equation of the Loxodrome on a Helicoidal Surface, The Journal of Navigation, 68, (2015), 962–970.
3. M. Babaarslan, Loxodromes on Canal Surfaces in Euclidean 3-Space, Ann. Sofia Univ. Fac. Math and Inf., 103, (2016), 97–103.
4. M. Babaarslan and Y. Yaylı; Space-like loxodromes on rotational surfaces in Minkowski 3-space, J. Math. Anal. Appl., 409, (2014), 288–298.
5. M. Babaarslan and M.I. Munteanu; Timelike loxodromes on rotational surfaces in Minkowski 3–space, Annals of the Alexandru Ioan Cuza University-Mathematics, (2015), DOI: 10.2478/aicu-2013-0021.
6. M. Babaarslan and M. Kayacik; Differential Equations of the Space-Like Loxodromes on the Helicoidal Surfaces, Differ Equ Dyn Syst, 28(2), (2020), 495–512.
7. M. Dede, C. Ekici, W. Goemans and Y ¨Unl¨ut¨urk; Twisted surfaces with vanishing curvature in Galilean 3-space, International Journal of Geometric Methods in Modern Physics, 15(1), (2018).
8. W. Goemans and I. Van de Woestyne, Twisted surfaces in Euclidean and Minkowski 3-space, Pure and Applied Differential Geometry, Joeri Van der Veken, Ignace Van de Woestyne, Leopold Verstraelen and Luc Vrancken (Editors), Shaker Verlag (Aachen, Germany), (2013), 143-151.

9. W. Goemans and I. Van de Woestyne; Twisted Surfaces with Null Rotation Axis in Minkowski 3-Space, Results in Mathematics, 70, (2016), 81–93.
10. A. Kazan and H.B. Karada˘g; Twisted Surfaces in the Pseudo-Galilean Space, NTMSCI, 5(4), (2017), 72-79.
11. S. Kos, D. Vranic and D. Zec; Differential Equation of a Loxodrome on a Sphere, The Journal of Navigation, 52, (1999), 418–420.
12. S. Kos, R. Filjar and M. Hess; Differential equation of the loxodrome on a rotational surface, ION ITM Conference, (2009), Anaheim, California, USA.
13. M. Petrovic; Differential Equation of a Loxodrome on the Spheroid, ”Naˇse more”, 54(3-4), (2007), 87-89.