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On Characterization of Smarandache Curves Constructed by Modified Orthogonal Frame

Year 2024, Volume: 12 Issue: 3, 101 - 112

Kemal Eren Soley Ersoy

https://doi.org/10.36753/mathenot.1409228

Abstract

In this study, we investigate Smarandache curves constructed by a space curve with a modified orthogonal frame. Firstly, the relations between the Frenet frame and the modified orthogonal frame are summarized. Later, the Smarandache curves based on the modified orthogonal frame are obtained. Finally, the tangent, normal, binormal vectors and the curvatures of the Smarandache curves are determined. A special curve known as the Gerono lemniscate curve whose curvature is not differentiable, the principal normal and binormal vectors are discontinuous at zero point is considered as an example, and the Smarandache curves of this curve are obtained by the aid of its modified orthogonal frame, and their graphics are given.

Keywords

Gerono lemniscate curve Modified orthogonal frame Smarandache curves

References

  • [1] Ali, A. T.: Special Smarandache curves in the Euclidean space. International Journal of Mathematical Combinatorics. 2, 30-36 (2010).
  • [2] Okuyucu, O. Z., Değirmen, Ç, Yıldız, Ö. G.: Smarandache curves in three dimensional Lie groups. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 68(1), 1175-1185 (2019).
  • [3] Taşköprü, K., Tosun, M.: Smarandache curves on S2. Boletim da Sociedade Paranaense de Matematica. 32(1), 51-59 (2014).
  • [4] Gürses, B. N., Bektaş, Ö, Yüce, S.: Special Smarandache curves in R^3_1. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 65(2), 143-160 (2016).
  • [5] Turgut, M., Yılmaz, S.: Smarandache curves in Minkowski space-time. International Journal of Mathematical Combinatorics. 3, 51-55 (2008).
  • [6] Ergut, M., Yılmaz, S., Ünlütürk, Y.: Isotropic Smarandache curves in the complex 4-space. Honam Mathematical Journal. 40(1), 47–59 (2018).
  • [7] Solouma, E. M., Mahmoud,W. M.: On spacelike equiform-Bishop Smarandache curves on S2 1 . Journal of the Egyptian Mathematical Society. 27(1), 7-17 (2019).
  • [8] Solouma, E. M.: Characterization of Smarandache trajectory curves of constant mass point particles as they move along the trajectory curve via PAF. Bulletin of Mathematical Analysis and Applications. 13(4), 14-30 (2021).
  • [9] Şenyurt, S., Ayvacı, K. H., Canlı, D.: Smarandache curves according to Flc-frame in Euclidean 3-space. Fundamentals of Contemporary Mathematical Sciences. 4(1), 16-30 (2023).
  • [10] Şenyurt, S., Eren, K.: Smarandache curves of spacelike anti-Salkowski curve with a spacelike principal normal according to Frenet frame. Gümü¸shane University journal of Science. 10(1), 251-260 (2020).
  • [11] Şenyurt, S., Eren, K.: Smarandache curves of spacelike Salkowski curve with a spacelike principal normal according to Frenet frame. Erzincan University Journal of Science and Technology. 13(Special Issue -I), 7-17 (2020).
  • [12] Şenyurt, S., Eren, K.: Some Smarandache curves constructed by a spacelike Salkowski curve with timelike principal normal. Punjab University Journal of Mathematics. 53(9), 679-690 (2021).
  • [13] Şenyurt, S., Eren, K.: Smarandache curves of spacelike anti-Salkowski curve with a timelike principal normal according to Frenet frame. Erzincan University Journal of Science and Technology. 13(2), 404-416 (2020).
  • [14] Özen K. E., Tosun, M.: Trajectories generated by special Smarandache curves according to positional adapted frame. KMU Journal of Engineering and Natural Sciences. 3(1), 15-23 (2021).
  • [15] Özen K. E., Tosun, M., Avcı, K.: Type 2-positional adapted frame and its application to Tzitzeica and Smarandache curves. Karatekin University Journal of Science. 1(1), 42-53 (2022).
  • [16] Sasai, T.: The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations. Tohoku Mathematical Journal. 36, 17–24 (1984).
  • [17] Bükcü, B., Karacan, M. K.: On the modified orthogonal frame with curvature and torsion in 3-space. Math. Sci. Appl. E-Notes. 4, 184–188 (2016).
  • [18] Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC, New York, 2010.

Year 2024, Volume: 12 Issue: 3, 101 - 112

Kemal Eren Soley Ersoy

https://doi.org/10.36753/mathenot.1409228

Abstract

References

  • [1] Ali, A. T.: Special Smarandache curves in the Euclidean space. International Journal of Mathematical Combinatorics. 2, 30-36 (2010).
  • [2] Okuyucu, O. Z., Değirmen, Ç, Yıldız, Ö. G.: Smarandache curves in three dimensional Lie groups. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 68(1), 1175-1185 (2019).
  • [3] Taşköprü, K., Tosun, M.: Smarandache curves on S2. Boletim da Sociedade Paranaense de Matematica. 32(1), 51-59 (2014).
  • [4] Gürses, B. N., Bektaş, Ö, Yüce, S.: Special Smarandache curves in R^3_1. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 65(2), 143-160 (2016).
  • [5] Turgut, M., Yılmaz, S.: Smarandache curves in Minkowski space-time. International Journal of Mathematical Combinatorics. 3, 51-55 (2008).
  • [6] Ergut, M., Yılmaz, S., Ünlütürk, Y.: Isotropic Smarandache curves in the complex 4-space. Honam Mathematical Journal. 40(1), 47–59 (2018).
  • [7] Solouma, E. M., Mahmoud,W. M.: On spacelike equiform-Bishop Smarandache curves on S2 1 . Journal of the Egyptian Mathematical Society. 27(1), 7-17 (2019).
  • [8] Solouma, E. M.: Characterization of Smarandache trajectory curves of constant mass point particles as they move along the trajectory curve via PAF. Bulletin of Mathematical Analysis and Applications. 13(4), 14-30 (2021).
  • [9] Şenyurt, S., Ayvacı, K. H., Canlı, D.: Smarandache curves according to Flc-frame in Euclidean 3-space. Fundamentals of Contemporary Mathematical Sciences. 4(1), 16-30 (2023).
  • [10] Şenyurt, S., Eren, K.: Smarandache curves of spacelike anti-Salkowski curve with a spacelike principal normal according to Frenet frame. Gümü¸shane University journal of Science. 10(1), 251-260 (2020).
  • [11] Şenyurt, S., Eren, K.: Smarandache curves of spacelike Salkowski curve with a spacelike principal normal according to Frenet frame. Erzincan University Journal of Science and Technology. 13(Special Issue -I), 7-17 (2020).
  • [12] Şenyurt, S., Eren, K.: Some Smarandache curves constructed by a spacelike Salkowski curve with timelike principal normal. Punjab University Journal of Mathematics. 53(9), 679-690 (2021).
  • [13] Şenyurt, S., Eren, K.: Smarandache curves of spacelike anti-Salkowski curve with a timelike principal normal according to Frenet frame. Erzincan University Journal of Science and Technology. 13(2), 404-416 (2020).
  • [14] Özen K. E., Tosun, M.: Trajectories generated by special Smarandache curves according to positional adapted frame. KMU Journal of Engineering and Natural Sciences. 3(1), 15-23 (2021).
  • [15] Özen K. E., Tosun, M., Avcı, K.: Type 2-positional adapted frame and its application to Tzitzeica and Smarandache curves. Karatekin University Journal of Science. 1(1), 42-53 (2022).
  • [16] Sasai, T.: The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations. Tohoku Mathematical Journal. 36, 17–24 (1984).
  • [17] Bükcü, B., Karacan, M. K.: On the modified orthogonal frame with curvature and torsion in 3-space. Math. Sci. Appl. E-Notes. 4, 184–188 (2016).
  • [18] Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC, New York, 2010.
There are 18 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Articles
Authors

Kemal Eren 0000-0001-5273-7897

Soley Ersoy 0000-0002-7183-7081

Early Pub Date April 14, 2024
Publication Date
Submission Date December 24, 2023
Acceptance Date March 13, 2024
Published in Issue Year 2024 Volume: 12 Issue: 3

Cite

APA Eren, K., & Ersoy, S. (2024). On Characterization of Smarandache Curves Constructed by Modified Orthogonal Frame. Mathematical Sciences and Applications E-Notes, 12(3), 101-112. https://doi.org/10.36753/mathenot.1409228
AMA Eren K, Ersoy S. On Characterization of Smarandache Curves Constructed by Modified Orthogonal Frame. Math. Sci. Appl. E-Notes. April 2024;12(3):101-112. doi:10.36753/mathenot.1409228
Chicago Eren, Kemal, and Soley Ersoy. “On Characterization of Smarandache Curves Constructed by Modified Orthogonal Frame”. Mathematical Sciences and Applications E-Notes 12, no. 3 (April 2024): 101-12. https://doi.org/10.36753/mathenot.1409228.
EndNote Eren K, Ersoy S (April 1, 2024) On Characterization of Smarandache Curves Constructed by Modified Orthogonal Frame. Mathematical Sciences and Applications E-Notes 12 3 101–112.
IEEE K. Eren and S. Ersoy, “On Characterization of Smarandache Curves Constructed by Modified Orthogonal Frame”, Math. Sci. Appl. E-Notes, vol. 12, no. 3, pp. 101–112, 2024, doi: 10.36753/mathenot.1409228.
ISNAD Eren, Kemal - Ersoy, Soley. “On Characterization of Smarandache Curves Constructed by Modified Orthogonal Frame”. Mathematical Sciences and Applications E-Notes 12/3 (April 2024), 101-112. https://doi.org/10.36753/mathenot.1409228.
JAMA Eren K, Ersoy S. On Characterization of Smarandache Curves Constructed by Modified Orthogonal Frame. Math. Sci. Appl. E-Notes. 2024;12:101–112.
MLA Eren, Kemal and Soley Ersoy. “On Characterization of Smarandache Curves Constructed by Modified Orthogonal Frame”. Mathematical Sciences and Applications E-Notes, vol. 12, no. 3, 2024, pp. 101-12, doi:10.36753/mathenot.1409228.
Vancouver Eren K, Ersoy S. On Characterization of Smarandache Curves Constructed by Modified Orthogonal Frame. Math. Sci. Appl. E-Notes. 2024;12(3):101-12.
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