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Timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski n-space

Yıl 2022, Cilt: 71 Sayı: 3, 856 - 869, 30.09.2022

Burcu Bektaş Demirci , Murat Babaarslan , Zehra Öge

https://doi.org/10.31801/cfsuasmas.1007599

Öz

In this paper, we examine timelike loxodromes on three kinds of Lorentzian helicoidal surfaces
in Minkowski nn-space.
First, we obtain the first order ordinary differential equations which determine timelike loxodromes
on the Lorentzian helicoidal surfaces in En1E1n
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 En1E1n. Finally, we give an example to visualize the results.

Anahtar Kelimeler

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

Kaynakça

  • Alexander, J., Loxodromes: a rhumb way to go, Math. Mag., 77(5) (2004), 349–356. https://doi.org/10.1080/0025570X.2004.11953279
  • 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
  • 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
  • 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.
  • 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
  • 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
  • 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
  • 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
  • Babaarslan, M., A note on loxodromes on helicoidal surfaces in Euclidean n–space, Appl. Math. E-Notes, 20 (2020), 458–461.
  • 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
  • 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
  • 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.
  • Byrd, P.F., Friedman, M.D., Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag Berlin Heidelberg, 1954.
  • 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
  • Carlton–Wippern, K. C., On loxodromic navigation, J. Navig., 45(2) (1992), 292–297. https://doi.org/10.1017/s0373463300010791
  • 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
  • Hitt, L. R., Roussos, I. M., Computer graphics of helicoidal surfaces with constant mean curvature, An. Acad. Bras. Ci, 63(3) (1991), 211–228.
  • Jensen, B., Electronic states on the helicoidal surface, Phys. Rev. A, 80(2) (2009), 022101. https://doi.org/10.1103/PhysRevA.80.022101
  • 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.
  • 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
  • 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
  • Pollard, D. D., Fletcher, R. C., Fundamentals of Structural Geology, Cambridge University Press, 2005.
  • 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
  • Ratcliffe, J.G., Foundations of Hyperbolic Manifolds, Springer Graduate Texts in Mathematics, 149, Second Edition, 2006.
  • 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
  • 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
  • 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.
  • 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

Yıl 2022, Cilt: 71 Sayı: 3, 856 - 869, 30.09.2022

Burcu Bektaş Demirci , Murat Babaarslan , Zehra Öge

https://doi.org/10.31801/cfsuasmas.1007599

Öz

Kaynakça

  • Alexander, J., Loxodromes: a rhumb way to go, Math. Mag., 77(5) (2004), 349–356. https://doi.org/10.1080/0025570X.2004.11953279
  • 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
  • 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
  • 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.
  • 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
  • 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
  • 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
  • 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
  • Babaarslan, M., A note on loxodromes on helicoidal surfaces in Euclidean n–space, Appl. Math. E-Notes, 20 (2020), 458–461.
  • 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
  • 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
  • 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.
  • Byrd, P.F., Friedman, M.D., Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag Berlin Heidelberg, 1954.
  • 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
  • Carlton–Wippern, K. C., On loxodromic navigation, J. Navig., 45(2) (1992), 292–297. https://doi.org/10.1017/s0373463300010791
  • 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
  • Hitt, L. R., Roussos, I. M., Computer graphics of helicoidal surfaces with constant mean curvature, An. Acad. Bras. Ci, 63(3) (1991), 211–228.
  • Jensen, B., Electronic states on the helicoidal surface, Phys. Rev. A, 80(2) (2009), 022101. https://doi.org/10.1103/PhysRevA.80.022101
  • 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.
  • 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
  • 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
  • Pollard, D. D., Fletcher, R. C., Fundamentals of Structural Geology, Cambridge University Press, 2005.
  • 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
  • Ratcliffe, J.G., Foundations of Hyperbolic Manifolds, Springer Graduate Texts in Mathematics, 149, Second Edition, 2006.
  • 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
  • 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
  • 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.
  • 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
Toplam 28 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Research Article
Yazarlar

Burcu Bektaş Demirci 0000-0002-5611-5478

Murat Babaarslan 0000-0002-2770-4126

Zehra Öge 0000-0001-7416-5236

Yayımlanma Tarihi 30 Eylül 2022
Gönderilme Tarihi 9 Ekim 2021
Kabul Tarihi 13 Mayıs 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 71 Sayı: 3

Kaynak Göster

APA Bektaş Demirci, B., Babaarslan, M., & Öge, Z. (2022). Timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski n-space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(3), 856-869. https://doi.org/10.31801/cfsuasmas.1007599
AMA Bektaş Demirci B, Babaarslan M, Öge Z. Timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski n-space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Eylül 2022;71(3):856-869. doi:10.31801/cfsuasmas.1007599
Chicago Bektaş Demirci, Burcu, Murat Babaarslan, ve Zehra Öge. “Timelike Loxodromes on Lorentzian Helicoidal Surfaces in Minkowski N-Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, sy. 3 (Eylül 2022): 856-69. https://doi.org/10.31801/cfsuasmas.1007599.
EndNote Bektaş Demirci B, Babaarslan M, Öge Z (01 Eylül 2022) Timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski n-space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 3 856–869.
IEEE B. Bektaş Demirci, M. Babaarslan, ve Z. Öge, “Timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski n-space”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 71, sy. 3, ss. 856–869, 2022, doi: 10.31801/cfsuasmas.1007599.
ISNAD Bektaş Demirci, Burcu vd. “Timelike Loxodromes on Lorentzian Helicoidal Surfaces in Minkowski N-Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/3 (Eylül 2022), 856-869. https://doi.org/10.31801/cfsuasmas.1007599.
JAMA Bektaş Demirci B, Babaarslan M, Öge Z. Timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski n-space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:856–869.
MLA Bektaş Demirci, Burcu vd. “Timelike Loxodromes on Lorentzian Helicoidal Surfaces in Minkowski N-Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 71, sy. 3, 2022, ss. 856-69, doi:10.31801/cfsuasmas.1007599.
Vancouver Bektaş Demirci B, Babaarslan M, Öge Z. Timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski n-space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(3):856-69.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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