Research Article
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Year 2022, Volume: 71 Issue: 2, 395 - 406, 30.06.2022
https://doi.org/10.31801/cfsuasmas.991631

Abstract

References

  • Bertrand, J. M., Memoire sur la theorie des courbes a double courbure, Comptes Rendus, 15 (1850), 332-350. https://doi.org/10.24033/bsmf.387
  • Bharathi, K., Nagaraj, M., Quaternion valued function of a real Serret-Frenet formulae, Indian J. Pure Appl. Math., 18(6) (1987), 507-511.
  • Ekmekci, N., İlarslan, K., On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst., 3 (2001), 17-24.
  • Gök, İ., Okuyucu, O. Z., Kahraman, F., Hacısalihoglu H. H., On the quaternionic $B_{2}$−slant helices in the Euclidean space $\mathbb{E}^{4}$, Adv. Appl. Clifford Algebr., 21 (2011), 707-719. https://doi.org/ 10.1007/s00006-011-0284-6
  • Görgülü, A., Özdamar, E., A Generalization of the Bertrand curves as general inclined curves in $\mathbb{E}^{n}$, Commun. Fac. Sci. Univ. Ankara, Ser. A1, 35 (1986), 53-60. https://doi.org/10.1501/Commua1-0000000254.
  • Güngör, M. A., Tosun, M., Some characterizations of quaternionic rectifying curves, Differ. Geom. Dyn. Syst., 13 (2011), 89-100.
  • Irmak, Y., Bertrand Curves and Geometric Applications in Four Dimensional Euclidean Space, MSc thesis, Ankara University, Institute of Science, 2018.
  • Kahraman Aksoyak, F., A new type of quaternionic Frame in $\mathbb{R}^{4}$, Int. J. Geom. Methods Mod. Phys., 16(6) (2019), 1950084 (11 pages). https://doi.org/10.1142/S0219887819500841.
  • Karadag, M., Sivridag, A. İ., Quaternion valued functions of a single real variable and inclined curves, Erciyes Univ. J. Inst. Sci. Technol., 13 (1997), 23-36.
  • Keçilioglu, O., İlarslan, K., Quaternionic Bertrand curves in Euclidean 4-space, Bull. Math. Anal. Appl., 5(3) (2013), 27-38.
  • 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(2) (2017), 1659-1673. https://doi.org/10.1007/s00006-016-0716-4
  • Pears, L. R., Bertrand curves in Riemannian space, J. London Math. Soc. 1-10(2) ( 1935), 180-183. https://doi.org/10.1112/jlms/s1-10.2.180
  • 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
  • Tanrıöver, N., Bertrand curves in n−dimensional Euclidean space, Journal of Karadeniz University, Faculty of Arts and Sciences, Series of Mathematics-Physics, 9 (1986), 61-62.
  • 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.
  • Yoon, D. W., On the quaternionic general helices in Euclidean 4-space, Honam Mathematical J., 34(3) (2012), 381-390. https://doi.org/10.5831/HMJ.2012.34.3.381
  • Yoon, D. W., Tuncer Y., Karacan, M. K., Generalized Mannheim quaternionic curves in Euclidean 4-space, Appl. Math. Sci. (Ruse), 7 (2013), 6583-6592. https://doi.org/6583-6592.10.12988/ams.2013.310560

Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$

Year 2022, Volume: 71 Issue: 2, 395 - 406, 30.06.2022
https://doi.org/10.31801/cfsuasmas.991631

Abstract

In this paper, we give some characterization of quaternionic Bertrand curves whose the torsion is non-zero but bitorsion is zero in $\mathbb{R}^{4}$ according to Type 2-Quaternionic Frame. One of the most important points in working on quaternionic curves is that given a curve in $\mathbb{R}^{4}$, the curve in $\mathbb{R}^{3}$ associated with this curve is determined individually. So, we obtain some relationships between quaternionic Bertrand curve $\alpha^{(4)}$ in $\mathbb{R}^{4}$ and its associated spatial quaternionic curve $\alpha$ in $\mathbb{R}^{3}$. Also, we support some theorems in the paper by means of an example.

References

  • Bertrand, J. M., Memoire sur la theorie des courbes a double courbure, Comptes Rendus, 15 (1850), 332-350. https://doi.org/10.24033/bsmf.387
  • Bharathi, K., Nagaraj, M., Quaternion valued function of a real Serret-Frenet formulae, Indian J. Pure Appl. Math., 18(6) (1987), 507-511.
  • Ekmekci, N., İlarslan, K., On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst., 3 (2001), 17-24.
  • Gök, İ., Okuyucu, O. Z., Kahraman, F., Hacısalihoglu H. H., On the quaternionic $B_{2}$−slant helices in the Euclidean space $\mathbb{E}^{4}$, Adv. Appl. Clifford Algebr., 21 (2011), 707-719. https://doi.org/ 10.1007/s00006-011-0284-6
  • Görgülü, A., Özdamar, E., A Generalization of the Bertrand curves as general inclined curves in $\mathbb{E}^{n}$, Commun. Fac. Sci. Univ. Ankara, Ser. A1, 35 (1986), 53-60. https://doi.org/10.1501/Commua1-0000000254.
  • Güngör, M. A., Tosun, M., Some characterizations of quaternionic rectifying curves, Differ. Geom. Dyn. Syst., 13 (2011), 89-100.
  • Irmak, Y., Bertrand Curves and Geometric Applications in Four Dimensional Euclidean Space, MSc thesis, Ankara University, Institute of Science, 2018.
  • Kahraman Aksoyak, F., A new type of quaternionic Frame in $\mathbb{R}^{4}$, Int. J. Geom. Methods Mod. Phys., 16(6) (2019), 1950084 (11 pages). https://doi.org/10.1142/S0219887819500841.
  • Karadag, M., Sivridag, A. İ., Quaternion valued functions of a single real variable and inclined curves, Erciyes Univ. J. Inst. Sci. Technol., 13 (1997), 23-36.
  • Keçilioglu, O., İlarslan, K., Quaternionic Bertrand curves in Euclidean 4-space, Bull. Math. Anal. Appl., 5(3) (2013), 27-38.
  • 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(2) (2017), 1659-1673. https://doi.org/10.1007/s00006-016-0716-4
  • Pears, L. R., Bertrand curves in Riemannian space, J. London Math. Soc. 1-10(2) ( 1935), 180-183. https://doi.org/10.1112/jlms/s1-10.2.180
  • 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
  • Tanrıöver, N., Bertrand curves in n−dimensional Euclidean space, Journal of Karadeniz University, Faculty of Arts and Sciences, Series of Mathematics-Physics, 9 (1986), 61-62.
  • 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.
  • Yoon, D. W., On the quaternionic general helices in Euclidean 4-space, Honam Mathematical J., 34(3) (2012), 381-390. https://doi.org/10.5831/HMJ.2012.34.3.381
  • Yoon, D. W., Tuncer Y., Karacan, M. K., Generalized Mannheim quaternionic curves in Euclidean 4-space, Appl. Math. Sci. (Ruse), 7 (2013), 6583-6592. https://doi.org/6583-6592.10.12988/ams.2013.310560
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Ferdağ Kahraman Aksoyak 0000-0003-4633-034X

Publication Date June 30, 2022
Submission Date September 6, 2021
Acceptance Date October 12, 2021
Published in Issue Year 2022 Volume: 71 Issue: 2

Cite

APA Kahraman Aksoyak, F. (2022). Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(2), 395-406. https://doi.org/10.31801/cfsuasmas.991631
AMA 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. June 2022;71(2):395-406. doi:10.31801/cfsuasmas.991631
Chicago Kahraman Aksoyak, Ferdağ. “Quaternionic Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 2 (June 2022): 395-406. https://doi.org/10.31801/cfsuasmas.991631.
EndNote Kahraman Aksoyak F (June 1, 2022) Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 2 395–406.
IEEE F. Kahraman Aksoyak, “Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 2, pp. 395–406, 2022, doi: 10.31801/cfsuasmas.991631.
ISNAD Kahraman Aksoyak, Ferdağ. “Quaternionic Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/2 (June 2022), 395-406. https://doi.org/10.31801/cfsuasmas.991631.
JAMA 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. 2022;71:395–406.
MLA Kahraman Aksoyak, Ferdağ. “Quaternionic Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 2, 2022, pp. 395-06, doi:10.31801/cfsuasmas.991631.
Vancouver 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. 2022;71(2):395-406.

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

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