Timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski n-space

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 $E_1^n$ 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 $E_1^n$. Finally, we give an example to visualize the results.

Keywords

• Loxodromes (rhumb lines)
• navigation
• rotational surfaces
• helicoidal surfaces
• Minkowski space

References

1. Alexander, J., Loxodromes: a rhumb way to go, Math. Mag., 77(5) (2004), 349–356. https://doi.org/10.1080/0025570X.2004.11953279
2. ldea, N., Kopacz, P., Generalized loxodromes with application to timeoptimal navigation in arbitrary wind, J. Franklin Inst., 358(1) (2021), 776–799. https://doi.org/10.1016/j.jfranklin.2020.11.009
3. Babaarslan, M., Yaylı, Y., Space–like loxodromes on rotational surfaces in Minkowski 3–space, J. Math. Anal. Appl., 409(1) (2014), 288–298. https://doi.org/10.1016/j.jmaa.2013.06.035
4. Babaarslan, M., Munteanu, M. I., Time–like loxodromes on rotational surfaces in Minkowski 3–space, An. S¸tiint. Univ. Al. I. Cuza Ia¸si, Ser. Nou˘a, Mat., 61(2) (2015), 471–484.
5. Babaarslan, M., Yaylı, Y., Differential equation of the loxodrome on a helicoidal surface, J. Navig., 68(5) (2015), 962–970. https://doi.org/10.1017/S0373463315000181
6. Babaarslan, M., Kayacık, M., Time–like loxodromes on helicoidal surfaces in Minkowski 3–space, Filomat, 31(14) (2017), 4405–4414. https://doi.org/10.2298/fil1714405b
7. Babaarslan, M., Loxodromes on helicoidal surfaces and tubes with variable radius in $E^4$, Commun. Fac. Sci. Univ. Ank. Ser. A1, Math. Stat., 68(2) (2019), 1950–1958. https://doi.org/10.31801/cfsuasmas.455372
8. Babaarslan, M., Kayacık, M. Differential equations of the space–like loxodromes on the helicoidal surfaces in Minkowski 3–space, Differ. Equ. Dyn. Syst., 28(2) (2020), 495–512. https://doi.org/10.1007/s12591-016-0343-5

9. Babaarslan, M., A note on loxodromes on helicoidal surfaces in Euclidean n–space, Appl. Math. E-Notes, 20 (2020), 458–461.
10. Babaarslan, M., Gümüş, M., On the parametrizations of loxodromes on time–like rotational surfaces in Minkowski space–time, Asian-Eur. J. Math., 14(5) (2021), 2150080. https://doi.org/10.1142/S1793557121500807
11. Babaarslan, M., Sönmez, N., Loxodromes on non–degenerate helicoidal surfaces in Minkowksi space–time, Indian J. Pure Appl. Math., 52(4) (2021), 1212–1228. https://doi.org/10.1007/s13226-021-00030-x
12. Babaarslan, M., Demirci, B.B, Gen¸c, R., Spacelike loxodromes on helicoidal surfaces in Lorentzian n–space, Differ. Geom. Dyn. Syst., 24 (2022), 18-33.
13. Byrd, P.F., Friedman, M.D., Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag Berlin Heidelberg, 1954.
14. Blackwood, J., Dukehart, A., Javaheri, M., Loxodromes on hypersurfaces of revolution, Involve, 10(3) (2016), 465–472. https://doi.org/10.2140/involve.2017.10.465
15. Carlton–Wippern, K. C., On loxodromic navigation, J. Navig., 45(2) (1992), 292–297. https://doi.org/10.1017/s0373463300010791
16. Caddeo, R., Onnis, I. I, Piu, P., Loxodromes on invariant surfaces in three–manifolds, Mediterr. J. Math., 17(1) (2020), 1–24. https://doi.org/10.1007/s00009-019-1439-2
17. Hitt, L. R., Roussos, I. M., Computer graphics of helicoidal surfaces with constant mean curvature, An. Acad. Bras. Ci, 63(3) (1991), 211–228.
18. Jensen, B., Electronic states on the helicoidal surface, Phys. Rev. A, 80(2) (2009), 022101. https://doi.org/10.1103/PhysRevA.80.022101
19. Kos, S., Filjar, R., Hess, M., Differential equation of the loxodrome on a rotational surface, Proceedings of the 2009 International Technical Meeting of the Institute of Navigation, (2009), 958–960.
20. Manfio, F., Tojeiro R., Van der Veken J., Geometry of submanifolds with respect to ambient vector fields, Ann. Mat. Pura Appl., 199(6) (2020), 2197–2225. https://doi.org/10.1007/s10231-020-00964-9
21. Noble, C. A., Note on loxodromes, Amer. M. S. Bull., 12(2) (1905), 116–119. https://doi.org/10.1090/S0002-9904-1905-01296-9
22. Pollard, D. D., Fletcher, R. C., Fundamentals of Structural Geology, Cambridge University Press, 2005.
23. Petrovic M, Differential equation of a loxodrome on the sphereoid, Int. J. Mar. Sci. Technol. ”Our Sea”, 54(3–4) (2007), 87–89. https://hrcak.srce.hr/16504
24. Ratcliffe, J.G., Foundations of Hyperbolic Manifolds, Springer Graduate Texts in Mathematics, 149, Second Edition, 2006.
25. Tseng, W.K, Earle, M. A., Guo, J.L., Direct and inverse solutions with geodetic latitude in terms of longitude for rhumb line sailing, J. Navig., 65(3) (2012), 549–559. https://doi.org/10.1017/S0373463312000148
26. Tseng, W.K, Chang, W.J., Analogues between 2D linear equations and great circle sailing, J. Navig., 67(1) (2014), 101–112. https://doi.org/10.1017/S0373463313000532
27. Weintrit, A., Kopacz, P., A novel approach to loxodrome (rhumb line), orthodrome (great circle) and geodesic line in ECDIS and navigation in general, Int. J. Mar. Navig. Saf. Sea Transp., 5(4) (2011), 507–517.
28. Yoon, D.W., Loxodromes and geodesics on rotational surfaces in a simply isotropic space, J. Geom., 108(2) (2017), 429–435. https://doi.org/10.1007/s00022-016-0349-8