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Year 2025, Volume: 13 Issue: 2, 92 - 105, 26.06.2025

Zülal Derin Yaqub , Mehmet Ali Güngör

https://doi.org/10.36753/mathenot.1658156

Abstract

References

  • [1] Bishop, R.L.: There is more than one way to frame a curve. The American Mathematical Monthly. 82(3), 246-251 (1975).
  • [2] Choi, J.H., Kim, Y.H.: Associated curves of a Frenet curve and their applications. Applied Mathematics and Computation. 218(18), 9116-9124 (2012).
  • [3] Arslan, K., Hacısalihoğlu, H.H.: On harmonic curvatures of a Frenet curve. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Journal. 49, 015-023 (2000).
  • [4] Romero-Fuster, M.C., Sanabria-Codesal, E.: Generalized helices, twistings and flattenings of curves in n-space. Matemática Contemporânea. 17, 267-280 (1999).
  • [5] Deshmukh, S., Chen, B.Y., Alshamari, S.: On rectifying curves in Euclidean 3-space. Turkish Journal of Mathematics. 42, 609-620 (2018).
  • [6] Fukunaga, T., Takahashi, M.: Existence conditions of framed curves for smooth curves. Journal of Geometry. 108, 763-774 (2017).
  • [7] Fukunaga, T., Takahashi, M.: Framed curves in the Euclidean space. Advances in Geometry. 16, 265-276 (2016).
  • [8] Doğan Yazıcı, B., İşbilir, Z., Tosun, M.: Spinor representation of framed Mannheim curves . Turkish Journal of Mathematics. 46, 2690-2700 (2022).
  • [9] İşbilir, Z., Doğan Yazıcı, B., Tosun, M.: The Spinor representations of framed Bertrand curves. Filomat. 37(9), 2831-2843 (2023).
  • [10] Doğan Yazıcı, B., Okuyucu, O. Z., Tosun, M.: On special singular curve couples of framed curves in 3D Lie groups. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 72(3), 710–720 (2023).
  • [11] Yıldız, Ö. G., Akyiğit, M., Tosun, M.: On the trajectory ruled surfaces of framed base curves in the Euclidean space. Mathematical Methods in the Applied Sciences. 44(9), 7463–7470 (2020).
  • [12] Doğan Yazıcı, B., Özkaldı Karakuş, S., Tosun, M.: Framed normal curves in Euclidean space. Tbilisimathematics, Sciendo. 27-37 (2020).
  • [13] Okuyucu, O. Z., Canbirdi, C.: Framed slant helices in Euclidean 3-space. Advances in Difference Equations. 2021 (504), 1-14 (2021).
  • [14] Doğan Yazıcı, B., Özkaldı Karaku¸s, S., Tosun, M.: Characterizations of framed curves in four-dimensional Euclidean space. Universal Journal of Mathematics and Applications. 4(4), 125-131 (2021).
  • [15] Akyiğit M., Yıldız Ö. G.: On the framed normal curves in Euclidean 4-space. Fundamental Journal of Mathematics and Applications. 4(4), 258-263 (2021).
  • [16] Bharathi, K., Nagaraj, M.: Quaternion valued function of a real variable Serret-Frenet formula. Indian Journal of Pure and Applied Mathematics. 18(6), 507-511 (1987).
  • [17] Tuna, A.: Yarı Öklid uzayındaki kuaterniyonik eğriler için Serret-Frenet formülleri [Ph D Thesis]. Süleyman Demirel Üniversitesi, Isparta, 2002.
  • [18] Güngör, M.A., Tosun, M.: Some characterizations of quaternionic rectifying curves. Differential Geometry-Dynamical Systems. 13, 89-100 (2011).
  • [19] Soyfidan, T., Güngör, M. A.: On the quaternionic curves in the semi-Euclidean space E_2^4 . Caspian Journal of Mathematical Sciences. 7(1), 36-45 (2018).
  • [20] Hacısalihoğlu, H. H.: Differential Geometry I. Ankara University, Ankara, 2000.
  • [21] Kahraman Aksoyak, F.: A new type quaternionic frame in R4. International Journal of Geometric Methods in Modern Physics. 16(6), 1950084 (2019).
  • [22] Çöken Ceylan, A., Tuna, A.: On the quaternionic inclined curves in the semi-Euclidean space E_2^4 . Applied Mathematics and Computation. 155(2), 373–389 (2004).
  • [23] Cansu, G., Yaylı, Y., Gök, .I.: A new quaternion valued frame of curves with an Application. Filomat. 35(1), 315–330 (2021).
  • [24] Gök, İ., Okuyucu, O. Z., Kahraman, F., Hacısalihoğlu, H. H.: On the quaternionic B2 slant helices in Euclidean 4-space E4. Advances in Applied Clifford Algebras. 21, 707–719 (2011).
  • [25] Kahraman, F., Gök, İ., Hacısalihoğlu, H.H.: On the quaternionic B2 slant helices in the semi-Euclidean space E_2^4. Applied Mathematics and Computation. 218(11), 6391-6400 (2012).

Serret-Frenet Formula of Quaternionic Framed Curves

Year 2025, Volume: 13 Issue: 2, 92 - 105, 26.06.2025

Zülal Derin Yaqub , Mehmet Ali Güngör

https://doi.org/10.36753/mathenot.1658156

Abstract

In this study, we present an approach by introducing the quaternionic structure of framed curves. Furthermore, we derive Serret-Frenet formulas and give specific results for quaternionic framed curves. Initially, we focus on the moving frame and its curvatures corresponding to the frame $ T,N,B $ along the quaternionic framed base curve in three-dimensional Euclidean space ${\mathbb{R}^3}$. Then, we establish the Serret-Frenet type formulas of quaternionic framed curves. We then generalize these formulas and the definition of quaternionic framed curves to four-dimensional Euclidean space, highlighting the relationship between the curvatures in both 3-dimensional and four-dimensional Euclidean spaces. In addition, the theorems are supported by examples, demonstrating the applicability of the proposed results.

Keywords

Framed curves Serret-Frenet formulas Singular point Quaternion

Thanks

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

References

  • [1] Bishop, R.L.: There is more than one way to frame a curve. The American Mathematical Monthly. 82(3), 246-251 (1975).
  • [2] Choi, J.H., Kim, Y.H.: Associated curves of a Frenet curve and their applications. Applied Mathematics and Computation. 218(18), 9116-9124 (2012).
  • [3] Arslan, K., Hacısalihoğlu, H.H.: On harmonic curvatures of a Frenet curve. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Journal. 49, 015-023 (2000).
  • [4] Romero-Fuster, M.C., Sanabria-Codesal, E.: Generalized helices, twistings and flattenings of curves in n-space. Matemática Contemporânea. 17, 267-280 (1999).
  • [5] Deshmukh, S., Chen, B.Y., Alshamari, S.: On rectifying curves in Euclidean 3-space. Turkish Journal of Mathematics. 42, 609-620 (2018).
  • [6] Fukunaga, T., Takahashi, M.: Existence conditions of framed curves for smooth curves. Journal of Geometry. 108, 763-774 (2017).
  • [7] Fukunaga, T., Takahashi, M.: Framed curves in the Euclidean space. Advances in Geometry. 16, 265-276 (2016).
  • [8] Doğan Yazıcı, B., İşbilir, Z., Tosun, M.: Spinor representation of framed Mannheim curves . Turkish Journal of Mathematics. 46, 2690-2700 (2022).
  • [9] İşbilir, Z., Doğan Yazıcı, B., Tosun, M.: The Spinor representations of framed Bertrand curves. Filomat. 37(9), 2831-2843 (2023).
  • [10] Doğan Yazıcı, B., Okuyucu, O. Z., Tosun, M.: On special singular curve couples of framed curves in 3D Lie groups. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 72(3), 710–720 (2023).
  • [11] Yıldız, Ö. G., Akyiğit, M., Tosun, M.: On the trajectory ruled surfaces of framed base curves in the Euclidean space. Mathematical Methods in the Applied Sciences. 44(9), 7463–7470 (2020).
  • [12] Doğan Yazıcı, B., Özkaldı Karakuş, S., Tosun, M.: Framed normal curves in Euclidean space. Tbilisimathematics, Sciendo. 27-37 (2020).
  • [13] Okuyucu, O. Z., Canbirdi, C.: Framed slant helices in Euclidean 3-space. Advances in Difference Equations. 2021 (504), 1-14 (2021).
  • [14] Doğan Yazıcı, B., Özkaldı Karaku¸s, S., Tosun, M.: Characterizations of framed curves in four-dimensional Euclidean space. Universal Journal of Mathematics and Applications. 4(4), 125-131 (2021).
  • [15] Akyiğit M., Yıldız Ö. G.: On the framed normal curves in Euclidean 4-space. Fundamental Journal of Mathematics and Applications. 4(4), 258-263 (2021).
  • [16] Bharathi, K., Nagaraj, M.: Quaternion valued function of a real variable Serret-Frenet formula. Indian Journal of Pure and Applied Mathematics. 18(6), 507-511 (1987).
  • [17] Tuna, A.: Yarı Öklid uzayındaki kuaterniyonik eğriler için Serret-Frenet formülleri [Ph D Thesis]. Süleyman Demirel Üniversitesi, Isparta, 2002.
  • [18] Güngör, M.A., Tosun, M.: Some characterizations of quaternionic rectifying curves. Differential Geometry-Dynamical Systems. 13, 89-100 (2011).
  • [19] Soyfidan, T., Güngör, M. A.: On the quaternionic curves in the semi-Euclidean space E_2^4 . Caspian Journal of Mathematical Sciences. 7(1), 36-45 (2018).
  • [20] Hacısalihoğlu, H. H.: Differential Geometry I. Ankara University, Ankara, 2000.
  • [21] Kahraman Aksoyak, F.: A new type quaternionic frame in R4. International Journal of Geometric Methods in Modern Physics. 16(6), 1950084 (2019).
  • [22] Çöken Ceylan, A., Tuna, A.: On the quaternionic inclined curves in the semi-Euclidean space E_2^4 . Applied Mathematics and Computation. 155(2), 373–389 (2004).
  • [23] Cansu, G., Yaylı, Y., Gök, .I.: A new quaternion valued frame of curves with an Application. Filomat. 35(1), 315–330 (2021).
  • [24] Gök, İ., Okuyucu, O. Z., Kahraman, F., Hacısalihoğlu, H. H.: On the quaternionic B2 slant helices in Euclidean 4-space E4. Advances in Applied Clifford Algebras. 21, 707–719 (2011).
  • [25] Kahraman, F., Gök, İ., Hacısalihoğlu, H.H.: On the quaternionic B2 slant helices in the semi-Euclidean space E_2^4. Applied Mathematics and Computation. 218(11), 6391-6400 (2012).
There are 25 citations in total.

Details

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

Zülal Derin Yaqub 0000-0001-9712-0428

Mehmet Ali Güngör 0000-0003-1863-3183

Early Pub Date June 20, 2025
Publication Date June 26, 2025
Submission Date March 14, 2025
Acceptance Date June 14, 2025
Published in Issue Year 2025 Volume: 13 Issue: 2

Cite

APA Derin Yaqub, Z., & Güngör, M. A. (2025). Serret-Frenet Formula of Quaternionic Framed Curves. Mathematical Sciences and Applications E-Notes, 13(2), 92-105. https://doi.org/10.36753/mathenot.1658156
AMA Derin Yaqub Z, Güngör MA. Serret-Frenet Formula of Quaternionic Framed Curves. Math. Sci. Appl. E-Notes. June 2025;13(2):92-105. doi:10.36753/mathenot.1658156
Chicago Derin Yaqub, Zülal, and Mehmet Ali Güngör. “Serret-Frenet Formula of Quaternionic Framed Curves”. Mathematical Sciences and Applications E-Notes 13, no. 2 (June 2025): 92-105. https://doi.org/10.36753/mathenot.1658156.
EndNote Derin Yaqub Z, Güngör MA (June 1, 2025) Serret-Frenet Formula of Quaternionic Framed Curves. Mathematical Sciences and Applications E-Notes 13 2 92–105.
IEEE Z. Derin Yaqub and M. A. Güngör, “Serret-Frenet Formula of Quaternionic Framed Curves”, Math. Sci. Appl. E-Notes, vol. 13, no. 2, pp. 92–105, 2025, doi: 10.36753/mathenot.1658156.
ISNAD Derin Yaqub, Zülal - Güngör, Mehmet Ali. “Serret-Frenet Formula of Quaternionic Framed Curves”. Mathematical Sciences and Applications E-Notes 13/2 (June 2025), 92-105. https://doi.org/10.36753/mathenot.1658156.
JAMA Derin Yaqub Z, Güngör MA. Serret-Frenet Formula of Quaternionic Framed Curves. Math. Sci. Appl. E-Notes. 2025;13:92–105.
MLA Derin Yaqub, Zülal and Mehmet Ali Güngör. “Serret-Frenet Formula of Quaternionic Framed Curves”. Mathematical Sciences and Applications E-Notes, vol. 13, no. 2, 2025, pp. 92-105, doi:10.36753/mathenot.1658156.
Vancouver Derin Yaqub Z, Güngör MA. Serret-Frenet Formula of Quaternionic Framed Curves. Math. Sci. Appl. E-Notes. 2025;13(2):92-105.
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