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On the pole indicatrix curve of the spacelike Salkowski curve with timelike principal normal in Lorentzian 3-space

Year 2022, Volume: 12 Issue: 4, 1168 - 1179, 15.10.2022
https://doi.org/10.17714/gumusfenbil.1176243

Abstract

In this study, we compute the Frenet frame, the curvatures, the Frenet derivative formulas, the Darboux vector, the arc length, the geodesic curvatures according to and (Lorentzian sphere) of the pole indicatrix curve of the spacelike Salkowski curve with the timelike principal normal in Lorentzian 3-space and show the graph of the indicatrix curve on the Lorentzian sphere.

References

  • Akutagava, K., & Nishikawa, S. (1990) The Gauss map and space-like surfaces with prescribed mean curvature in Minkowski 3-space. Tohoku Mathematical Journal, Second Series 42(1), 67–82.
  • Ali, A. T. (2011) Spacelike Salkowski and anti-Salkowski curves with timelike principal normal in Minkowski 3-space. Mathematica Aeterna, 1(4), 201–210.
  • Ali, A. T. (2012) Position vectors of slant helices in Euclidean 3-space, Journal of the Egyptian Mathematical Society, 20(1), 1–6. https://doi.org/10.1016/j.joems.2011.12.005
  • Alluhaibi, N., & Abdel-Baky, R. A. (2022) Kinematic geometry of timelike ruled surfaces in Minkowski 3-space , Symmetry, 14(4), 749. https://doi.org/10.3390/sym14040749
  • Beem, J. K., Paul, E. E., & Kevin, L. E. (2017) Global Lorentzian geometry. Routledge. https://doi.org/10.1201/9780203753125
  • Birman, G. S., & Nomizu, K. (1984) Trigonometry in Lorentzian geometry. The American Mathematical Monthly, 91(9), 534–549. https://doi.org/10.1080/00029890.1984.11971490
  • Bulut, F., & Bektaş, M. (2020) Special helices on equiform differential geometry of spacelike curves in Minkowski space-time. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(2), 1045–1056. https://doi.org/10.31801/cfsuasmas.686311
  • Bükcü, B., & Karacan, M. K. (2007) On the involute and evolute curves of the spacelike curve with a spacelike binormal in Minkowski 3-space. International Journal of Contemporary Mathematical Sciences, 2(5), 221–232.
  • Gür, S., & Şenyurt, S. (2010) Frenet vectors and geodesic curvatures of spheric indicators of Salkowski curve in, Hadronic Journal, 33(5), 485–512.
  • Gür, S., & Şenyurt, S. (2013) Spacelike–timelike involute–evolute curve couple on dual Lorentzian space. Journal of Mathematical and Computational Science, 3(4), 1054–1075.
  • Gür Mazlum, S., Şenyurt, S. & Bektaş, M. (2022) Salkowski curves and their modified orthogonal frames in, Journal of New theory, 40, 12–26.
  • Hacısalihoğlu, H. H. (1983) Differential geometry. İnönü University, Publication of Faculty of Sciences and Arts: Malatya, Turkiye.
  • Izumiya, S., & Tkeuchi, N. (2004) New special curves and developable surfaces, Turkish Journal of Mathematics, 28(2), 153–163.
  • Kılıçoğlu, Ş, Hacısalihoğlu, H. (2008) On the b-scrolls with time-like generating vector in 3-dimensional Minkowski space . Beykent University Journal of Science and Technology, 3(2), 55–67.
  • Külahcı, M., Bektaş, M., & Ergüt, M. (2009) On harmonic curvatures of a Frenet curve in Lorentzian space. Chaos, Solitons and Fractals, 41(4), 1668–1675. https://doi.org/10.1016/j.chaos.2008.07.013
  • Li, Y., Wang, Z., & Zhao, T. (2020) Slant helix of order n and sequence of Darboux developables of principal-directional curves. Mathematical Methods in the Applied Sciences, 43(17), 9888–9903. https://doi.org/10.1002/mma.6663
  • Li, Y., Pei, D., Takahashi, M., & Yu, H. (2018) Envelopes of Legendre curves in the unit spherical bundle over the unit sphere, The Quarterly Journal of Mathematics, 69(2), 631-653. https://doi.org/10.1093/qmath/hax056
  • Li, Y., & Pei, D. (2016) Evolutes of dual spherical curves for ruled surfaces. Mathematical Methods in the Applied Sciences, 39(11), 3005-3015. https://doi.org/10.1002/mma.3748
  • Li, Y., Gür Mazlum, S., & Şenyurt, S. (2022) The Darboux trihedrons of timelike surfaces in the Lorentzian 3-space, International Journal of Geometric Methods in Modern Physics, Accepted, https://doi.org/10.1142/S0219887823500305
  • Lopez, R. (2014) Differential geometry of curves and surfaces in Lorentz-Minkowski space. International Electronic Journal of Geometry, 7(1), 44–107.
  • Monterde, J. (2009) Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion. Computer Aided Geometric Design, 26(3), 271–278. https://doi.org/10.1016/j.cagd.2008.10.002
  • O’Neill, B. (1983) Semi-Riemannian geometry with applications to relativity. Academic Press: London, England.
  • Önder, M., & Uğurlu, H. H. (2013). Frenet frames and invariants of timelike ruled surfaces. Ain Shams Engineering Journal, 4(3), 507-513. https://doi.org/10.1016/j.asej.2012.10.003
  • Özdemir, M. (2020) Diferansiyel geometri, Altın Nokta Printing and Distribution, İzmir, Turkiye.
  • Ratcliffe, J. G. (1994) Foundations of hyperbolic manifolds. Springer-Verlag: Tokyo, Japanese.
  • Salkowski, E. (1909) Zur transformation von raumkurven. Mathematische Annalen, 66(4), 517–557.
  • Şenatalar, M. (1978) Differential geometry (curves and surfaces theory), Istanbul State Engineering and Architecture Academy Publications: Istanbul, Turkiye.
  • Şenyurt, S., & Gür, S. (2017) Spacelike surface geometry. International Journal of Geometric Methods in Modern Physics, 14(09), 689–700. https://doi.org/10.1142/S0219887817501183
  • Şenyurt, S., & Gür, S. (2012) Timelike–spacelike involute–evolute curve couple on dual Lorentzian space, Journal of Mathematical and Computational Science, 2(6), 1808–1823.
  • Şenyurt, S., Gür, S., & Özyılmaz, E. (2015) The Frenet vectors and the geodesic curvatures of spherical indicatrix of the timelike Salkowski curve in Minkowski 3-space. Journal of Advanced Research in Dynamical and Control Systems, 7(4), 20–42.
  • Şenyurt, S., & Uzun, M. (2020) Salkowski eğrisinin birim Darboux vektörünün Sabban çatısından elde edilen Smarandache eğrileri, Journal of the Institute of Science and Technology, 10(3), 1966–1974. https://doi.org/10.21597/jist.703495
  • Şenyurt, S., & Kemal, E. (2020) Smarandache curves of spacelike anti-Salkowski curve with a spacelike principal normal according to Frenet frame. Gümüşhane University, Journal of Science and Technology, 10, 251–260. https://doi.org/10.17714/gumusfenbil.621363
  • Şenyurt, S., & Öztürk, B. (2018) Smarandache curves of Salkowski curve according to Frenet frame. Turkish Journal of Mathematics and Computer Science, 10, 190–201.
  • Struik, D. J. (1961) Lectures on classical differential geometry. Courier Corporation.
  • Uğurlu, H. H. (1997) On the geometry of time-like surfaces, Communications, Faculty of Sciences, University of Ankara, A1 Series, 46, 211–223.
  • Uğurlu, H. H., & Çalışkan, A. (2012) Darboux ani dönme vektörleri ile spacelike ve timelike yüzeyler geometrisi. Celal Bayar University Press: Manisa, Turkiye.
  • Uğurlu, H. H., & Topal, A. (1996) Relation between Darboux instantaneous rotation vectors of curves on time-like surface. Mathematical and Computational Applications, 1(2), 149–157. https://doi.org/10.3390/mca1020149
  • Uğurlu, H. H., & Kocayiğit, H. (1996) The Frenet and Darboux instantaneous rotation vectors of curves on time-like surface. Mathematical and Computational Applications, 1(2), 133–141.
  • Walrave, J. (1995) Curves and surfaces in Minkowski space, [Ph.D. Thesis, Katholieke Universiteit, Leuven, Belgium.]
  • Woestijne, I. V. (1990) Minimal surfaces of the 3-dimensional Minkowski space, Geometry and Topology of Submanifolds: II, Word Scientific, Singapore, 344–369. https://doi.org/10.1142/9789814540339
  • Yüksel, N., Saltık, B., & Damar, E. (2014) Parallel curves in Minkowski 3-space, Gümüşhane Univerrsity Journal of Science and Technology, 12(2), 480-486. https://doi.org/10.17714/gumusfenbil.855265
  • Yüksel, N., Vanlı, A. T., & Damar, E. (2015). A new approach for geometric properties of DNA structure in , Life Science Journal, 12(2), 71-79.

On the pole indicatrix curve of the spacelike Salkowski curve with timelike principal normal in Lorentzian 3-space

Year 2022, Volume: 12 Issue: 4, 1168 - 1179, 15.10.2022
https://doi.org/10.17714/gumusfenbil.1176243

Abstract

Bu çalışmada 3-boyutlu Lorentz uzayında timelike asli normalli spacelike Salkowski eğrisinin pol gösterge eğrisinin Frenet çatısı, eğrilikleri, Frenet türev formülleri, Darboux vektörü, yay uzunluğu, e ve (Lorentzian küre) ye göre geodezik eğrilikleri hesaplanmıştır ve bu pol gösterge eğrisi Lorentzian küre üzerinde gösterilmiştir.

References

  • Akutagava, K., & Nishikawa, S. (1990) The Gauss map and space-like surfaces with prescribed mean curvature in Minkowski 3-space. Tohoku Mathematical Journal, Second Series 42(1), 67–82.
  • Ali, A. T. (2011) Spacelike Salkowski and anti-Salkowski curves with timelike principal normal in Minkowski 3-space. Mathematica Aeterna, 1(4), 201–210.
  • Ali, A. T. (2012) Position vectors of slant helices in Euclidean 3-space, Journal of the Egyptian Mathematical Society, 20(1), 1–6. https://doi.org/10.1016/j.joems.2011.12.005
  • Alluhaibi, N., & Abdel-Baky, R. A. (2022) Kinematic geometry of timelike ruled surfaces in Minkowski 3-space , Symmetry, 14(4), 749. https://doi.org/10.3390/sym14040749
  • Beem, J. K., Paul, E. E., & Kevin, L. E. (2017) Global Lorentzian geometry. Routledge. https://doi.org/10.1201/9780203753125
  • Birman, G. S., & Nomizu, K. (1984) Trigonometry in Lorentzian geometry. The American Mathematical Monthly, 91(9), 534–549. https://doi.org/10.1080/00029890.1984.11971490
  • Bulut, F., & Bektaş, M. (2020) Special helices on equiform differential geometry of spacelike curves in Minkowski space-time. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(2), 1045–1056. https://doi.org/10.31801/cfsuasmas.686311
  • Bükcü, B., & Karacan, M. K. (2007) On the involute and evolute curves of the spacelike curve with a spacelike binormal in Minkowski 3-space. International Journal of Contemporary Mathematical Sciences, 2(5), 221–232.
  • Gür, S., & Şenyurt, S. (2010) Frenet vectors and geodesic curvatures of spheric indicators of Salkowski curve in, Hadronic Journal, 33(5), 485–512.
  • Gür, S., & Şenyurt, S. (2013) Spacelike–timelike involute–evolute curve couple on dual Lorentzian space. Journal of Mathematical and Computational Science, 3(4), 1054–1075.
  • Gür Mazlum, S., Şenyurt, S. & Bektaş, M. (2022) Salkowski curves and their modified orthogonal frames in, Journal of New theory, 40, 12–26.
  • Hacısalihoğlu, H. H. (1983) Differential geometry. İnönü University, Publication of Faculty of Sciences and Arts: Malatya, Turkiye.
  • Izumiya, S., & Tkeuchi, N. (2004) New special curves and developable surfaces, Turkish Journal of Mathematics, 28(2), 153–163.
  • Kılıçoğlu, Ş, Hacısalihoğlu, H. (2008) On the b-scrolls with time-like generating vector in 3-dimensional Minkowski space . Beykent University Journal of Science and Technology, 3(2), 55–67.
  • Külahcı, M., Bektaş, M., & Ergüt, M. (2009) On harmonic curvatures of a Frenet curve in Lorentzian space. Chaos, Solitons and Fractals, 41(4), 1668–1675. https://doi.org/10.1016/j.chaos.2008.07.013
  • Li, Y., Wang, Z., & Zhao, T. (2020) Slant helix of order n and sequence of Darboux developables of principal-directional curves. Mathematical Methods in the Applied Sciences, 43(17), 9888–9903. https://doi.org/10.1002/mma.6663
  • Li, Y., Pei, D., Takahashi, M., & Yu, H. (2018) Envelopes of Legendre curves in the unit spherical bundle over the unit sphere, The Quarterly Journal of Mathematics, 69(2), 631-653. https://doi.org/10.1093/qmath/hax056
  • Li, Y., & Pei, D. (2016) Evolutes of dual spherical curves for ruled surfaces. Mathematical Methods in the Applied Sciences, 39(11), 3005-3015. https://doi.org/10.1002/mma.3748
  • Li, Y., Gür Mazlum, S., & Şenyurt, S. (2022) The Darboux trihedrons of timelike surfaces in the Lorentzian 3-space, International Journal of Geometric Methods in Modern Physics, Accepted, https://doi.org/10.1142/S0219887823500305
  • Lopez, R. (2014) Differential geometry of curves and surfaces in Lorentz-Minkowski space. International Electronic Journal of Geometry, 7(1), 44–107.
  • Monterde, J. (2009) Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion. Computer Aided Geometric Design, 26(3), 271–278. https://doi.org/10.1016/j.cagd.2008.10.002
  • O’Neill, B. (1983) Semi-Riemannian geometry with applications to relativity. Academic Press: London, England.
  • Önder, M., & Uğurlu, H. H. (2013). Frenet frames and invariants of timelike ruled surfaces. Ain Shams Engineering Journal, 4(3), 507-513. https://doi.org/10.1016/j.asej.2012.10.003
  • Özdemir, M. (2020) Diferansiyel geometri, Altın Nokta Printing and Distribution, İzmir, Turkiye.
  • Ratcliffe, J. G. (1994) Foundations of hyperbolic manifolds. Springer-Verlag: Tokyo, Japanese.
  • Salkowski, E. (1909) Zur transformation von raumkurven. Mathematische Annalen, 66(4), 517–557.
  • Şenatalar, M. (1978) Differential geometry (curves and surfaces theory), Istanbul State Engineering and Architecture Academy Publications: Istanbul, Turkiye.
  • Şenyurt, S., & Gür, S. (2017) Spacelike surface geometry. International Journal of Geometric Methods in Modern Physics, 14(09), 689–700. https://doi.org/10.1142/S0219887817501183
  • Şenyurt, S., & Gür, S. (2012) Timelike–spacelike involute–evolute curve couple on dual Lorentzian space, Journal of Mathematical and Computational Science, 2(6), 1808–1823.
  • Şenyurt, S., Gür, S., & Özyılmaz, E. (2015) The Frenet vectors and the geodesic curvatures of spherical indicatrix of the timelike Salkowski curve in Minkowski 3-space. Journal of Advanced Research in Dynamical and Control Systems, 7(4), 20–42.
  • Şenyurt, S., & Uzun, M. (2020) Salkowski eğrisinin birim Darboux vektörünün Sabban çatısından elde edilen Smarandache eğrileri, Journal of the Institute of Science and Technology, 10(3), 1966–1974. https://doi.org/10.21597/jist.703495
  • Şenyurt, S., & Kemal, E. (2020) Smarandache curves of spacelike anti-Salkowski curve with a spacelike principal normal according to Frenet frame. Gümüşhane University, Journal of Science and Technology, 10, 251–260. https://doi.org/10.17714/gumusfenbil.621363
  • Şenyurt, S., & Öztürk, B. (2018) Smarandache curves of Salkowski curve according to Frenet frame. Turkish Journal of Mathematics and Computer Science, 10, 190–201.
  • Struik, D. J. (1961) Lectures on classical differential geometry. Courier Corporation.
  • Uğurlu, H. H. (1997) On the geometry of time-like surfaces, Communications, Faculty of Sciences, University of Ankara, A1 Series, 46, 211–223.
  • Uğurlu, H. H., & Çalışkan, A. (2012) Darboux ani dönme vektörleri ile spacelike ve timelike yüzeyler geometrisi. Celal Bayar University Press: Manisa, Turkiye.
  • Uğurlu, H. H., & Topal, A. (1996) Relation between Darboux instantaneous rotation vectors of curves on time-like surface. Mathematical and Computational Applications, 1(2), 149–157. https://doi.org/10.3390/mca1020149
  • Uğurlu, H. H., & Kocayiğit, H. (1996) The Frenet and Darboux instantaneous rotation vectors of curves on time-like surface. Mathematical and Computational Applications, 1(2), 133–141.
  • Walrave, J. (1995) Curves and surfaces in Minkowski space, [Ph.D. Thesis, Katholieke Universiteit, Leuven, Belgium.]
  • Woestijne, I. V. (1990) Minimal surfaces of the 3-dimensional Minkowski space, Geometry and Topology of Submanifolds: II, Word Scientific, Singapore, 344–369. https://doi.org/10.1142/9789814540339
  • Yüksel, N., Saltık, B., & Damar, E. (2014) Parallel curves in Minkowski 3-space, Gümüşhane Univerrsity Journal of Science and Technology, 12(2), 480-486. https://doi.org/10.17714/gumusfenbil.855265
  • Yüksel, N., Vanlı, A. T., & Damar, E. (2015). A new approach for geometric properties of DNA structure in , Life Science Journal, 12(2), 71-79.
There are 42 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Birkan Aksan 0000-0002-1533-6557

Sümeyye Gür Mazlum 0000-0003-2471-1627

Publication Date October 15, 2022
Submission Date September 16, 2022
Acceptance Date October 6, 2022
Published in Issue Year 2022 Volume: 12 Issue: 4

Cite

APA Aksan, B., & Gür Mazlum, S. (2022). On the pole indicatrix curve of the spacelike Salkowski curve with timelike principal normal in Lorentzian 3-space. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 12(4), 1168-1179. https://doi.org/10.17714/gumusfenbil.1176243