Quaternionic Sabban frame in $\mathbb{R}^{3}$ and $\mathbb{R}^{4}$

Year 2025, Volume: 74 Issue: 3, 513 - 522, 23.09.2025

Ferdağ Kahraman Aksoyak

https://doi.org/10.31801/cfsuasmas.1650280

Abstract

In this paper, we first reconsider Sabban frame of curves on $2-$sphere $S^{2}$ in $\mathbb{R}^{3}$ with the help of the properties of quaternion algebra and then, we define two different types of quaternionic Sabban frame of the curves on $3$-sphere $S^{3}$ in $\mathbb{R}^{4}$. Also, we support the theory in the paper with some examples.

Keywords

Quaternions , quaternionic frame , Sabban frame

References

• Acar, N., Kahraman Aksoyak, F., Generalized quaternionic Mannheim curves according to the type 2-quaternionic frame in Euclidean 4-space, Uzbek Math. J., 66(1) (2022), 14-22. https://doi.org/10.29229/uzmj.2022-1-2.
• Bektaş, Ö., Gürses, N., Yüce, S., Quaternionic osculating curves in Euclidean and semi-Euclidean space, J. Dyn. Syst. Geom.Theor., 14(1) (2016), 65–84. https://doi.org/10.1080/1726037x.2016.1177935.
• Bharathi, K., Nagaraj, M., Quaternion valued function of a real Serret-Frenet formulae, Indian J. Pure Appl. Math., 18(6) (1987), 507-511.
• Cansu, G., Yaylı, Y., Gök, İ., A new quaternion valued frame of curves with an application, Filomat, 35(1) (2021), 315-330. https://doi.org/10.2298/FIL2101315C.
• Çetin, M., Kocayiğit, H., On the quaternionic Smarandache curves in Euclidean 3-space, Int. J. Contemp. Math. Sci., 8(3) (2013), 139-150.
• Çöken, A. C., Ekici, C., Kocayusufoğlu, İ., Görgülü, A., Formulas for dual-split quaternionic curves, Kuwait J. Sci. Engrg., 36(1A) (2009), 1–14.
• Çöken, A. C., Tuna, A., On the quaternionic inclined curves in the semi-Euclidean space, Appl. Math. Comput., 155(2) (2004), 373-389. https://doi.org/doi:10.1016/S0096-3003(03)00783-5.
• Erdem, E., Some new characterizations of F-rectifying curves respect to type-2 quaternionic frame in R4, Prespacetime Journal, 14(3) (2023), 344-353.
• Giardino, S., A primer on the differential geometry of quaternionic curves, Math. Methods Appl. Sci., 44(18) (2021), 14428-14436. https://doi.org/10.1002/mma.7709.
• Giardino, S., Differential geometry using quaternions, Int. Electron. J. Geom., 17(2) (2024), 700-711. https://doi. org/10.36890/IEJG.1362006.
• Giardino, S., Winding number and homotopy for quaternionic curves, Int. J. Geom. Methods Mod. Phys., 19(6) (2022), 2250087, 13 pp. https://doi.org/10.1142/S0219887822500876.
• Gök, İ., Okuyucu, O. Z., Kahraman, F., Hacsalihoğlu, H. H., On the quaternionic $B_{2}$-slant helices in the Euclidean space $\mathbb{E}^{4}$, Adv. Appl. Clifford Algebr., 21(4) (2011), 707-719. https://doi.org/10.1007/s00006-011-0284-6.
• Güngör, M. A., Tosun, M., Some characterizations of quaternionic rectifying curves, Differ. Geom. Dyn. Syst., 13 (2011), 89-100.
• Izumiya, S., Nagai T., Generalized Sabban curves in the Euclidean n-sphere and spherical duality, Results Math., 72 (2017), 401-417. https://doi.org/10.1007/s00025-017-0685-5.
• Kahraman Aksoyak F., A new type of quaternionic frame in $\mathbb{R}^{4}$, Int. J. Geom. Methods Mod. Phys., 16(6) (2019), 1950084, 11 pp. https://doi.org/10.1142/S0219887819500841.
• Kahraman Aksoyak F., Quaternionic $\left (1, 3\right)$-Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$, Konuralp J. Math., 9(2) (2021), 346-355.
• Kahraman Aksoyak F., Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 71(2) (2022), 395-406. https://doi.org/10.31801/cfsuasmas.991631.
• Karadağ, M., Sivridağ, A. İ., Quaternion valued functions of a single real variable and inclined curves, Erciyes Univ. J. Inst. Sci. Technol., 13 (1997), 23-36.
• Keçilioğlu, O., İlarslan, K., Quaternionic Bertrand curves in Euclidean 4-space, Bull. Math. Anal. Appl., 5(3) (2013), 27–38.
• Koenderink, J. J., Solid Shape, MIT Press, Cambridge, 1990.
• Önder, M., Quaternionic Salkowski curves and quaternionic similar curves, Proc. Natl. Acad. Sci. India, Sect. A Phys. Sci., 90 (3) (2020), 447-456. https://doi.org/10.1007/s40010-019-00601-y.
• Öztürk, G., Kişi, İ., Büyükkütük, S., Constant ratio quaternionic curves in Euclidean spaces, Adv. Appl. Clifford Algebr., 27 (2017), no. 2, 1659–1673. https://doi.org/10.1007/s00006-016-0716-4.
• Şenyurt, S., Cevahir, C., Altun, Y., On spatial quaternionic involute curve a new view, Adv. Appl. Clifford Algebr., 27(2) (2017), 1815–1824. https://doi.org/10.1007/s00006-016-0669-7.
• Yıldız, Ö. G., İçer, Ö., A note on evolution of quaternionic curves in the Euclidean space $\mathbb{R}^{4}$, Konuralp J. Math., 7(2) (2019), 462-469.
• Yıldız, Ö. G., Usta, H., Inextensible flow of quaternionic curves according to type 2-quaternionic frame in the Euclidean space, Math. Sci. Appl. E-Notes, 12(4) (2024), 169-177. https://doi.org/10.36753/mathenot.1481075.
• Yoon, D. W., Tunçer,Y., Karacan, M.K., Generalized Mannheim quaternionic curves in Euclidean 4-space, Appl. Math. Sci. (Ruse), 7(132), (2013), 6583–6592. https://doi.org/10.12988/ams.2013.310560.
• Yoon, D. W., On the quaternionic general helices in Euclidean 4-space, Honam Math. J., 34(3) (2012), 381-390. https://doi.org/10.5831/HMJ.2012.34.3.381.