Research Article
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Year 2025, Volume: 18 Issue: 2, 185 - 195, 19.10.2025

Ayşe Zeynep Azak , Arzu Sürekçi , Mehmet Ali Güngör

Abstract

References

  • Aksoyak, F.K.: A new type of quaternionic frame in R4, Int. J. Geom. Methods Mod. Phys. 16 (6), 1959984 (2019). https://doi.org/10.1142/S0219887819500841
  • Aksoyak, F.K.: Quaternionic Bertrand curves according to type-2-quaternionic frame in R4, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 71 (2), 395-406 (2022). https://doi.org/10.31801/cfsuasmas.991631
  • Aksoyak, F.K.: Quaternionic (1, 3)-Bertrand curves according to type-2-quaternionic frame in R4, Konuralp J. Math. 9 (2), 346-355 (2021).
  • Balgetir, H., Bektaş, M., Ergüt, M.: Bertrand curves for nonnull curves in 3-dimensional Lorentzian space, Hadronic J. 27, 229-236 (2004).
  • Balgetir, H., Bekta¸s, M., Inoguchi, J.: Null Bertrand curves in Minkowski 3-space and their characterizations, Not. Mat. 23 (1), 7–13 (2004). https://doi.org/10.1285/i15900932v23n1p7
  • Bertrand, J.M.: Mémoire sur la théorie des courbes á double courbure. J. Math. Pures Appl. 15, 332–350 (1850).
  • Bharathi, K., Nagaraj, M.: Quaternion valued function of a real Serret-Frenet formulae, Indian J. Pure Appl. Math. 18 (6), 507-511 (1987).
  • Camci, Ç., Uçum, A., İIlarslan, K.: A new approach to Bertrand curves in Euclidean 3-space, J. Geom. 111 (49), 1–15 (2020). https://doi.org/10.1007/s00022-020-00560-5
  • Dede, M., Ekici, C.: Directional Bertrand curves, Gazi Univ. J. Sci. 31 (1), 202–211 (2018).
  • Ekmekci, N., İlarslan, K.: On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst. 3 (2), 17–24 (2001).
  • Elsayied, H.K., Altaha, A.A., Elsharkawy, A.: Bertrand curves with the modified orthogonal frame in Minkowski 3-space $E_1^3$, Rev. Educ. 392 (6), 43-55 (2022).
  • Elsharkawy, A., Tashkandy, Y., Emam W., Cesarano, C., Elsharkawy, N.: On some quasi-curves in Galilean three-space, Axioms 12 (9), 823 (2023). https://doi.org/10.3390/axioms12090823
  • Elzawya, M., Mosab, S.: Quaternionic Bertrand curves in the Galilean space, Filomat 34 (1), 59–66 (2020). https://doi.org/10.2298/FIL2001059E
  • Ergüt, M., Öztekin, H., Gün Bozok, H.: Representation formulae for Bertrand curves in Galilean and pseudo-Galilean 3-space, Adv. Model. Optim. 16 (2), 371-385 (2014).
  • Ersoy, S., Tosun, M.: Timelike Bertrand curves in semi-Euclidean space, Int. J. Math. Stat. 14 (2), 78-89 (2013). [16] Fukunaga, T., Takahashi, M.: Existence conditions of framed curves for smooth curves, J. Geom. 108 (2), 763-774 (2017). https://doi.org/10.1007/s00022-017-0371-5
  • Güner, G., Ekmekçi, N.: On the spherical curves and Bertrand curves in Minkowski 3-space, J. Math. Comput. Sci. 2 (4), 898-906 (2012).
  • Gök, İ., Nurkan, S.K., İlarslan, K.: On pseudo null Bertrand curves in Minkowski space-time, Kyungpook Math. J. 54 (4), 685-697 (2014). 108 (2), 763-774 (2017). https://doi.org/10.5666/KMJ.2014.54.4.685
  • Honda, S., Takahashi, M.: Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space, Turk. J. Math. 44 (3), 883-899 (2020). https://doi.org/10.3906/mat-1906-71
  • Honda, S., Takahashi, M., Yu, H.: Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space, J. Geom. 114 (2), 12 (2023). https://doi.org/10.1007/s00022-023-00673-7
  • Honda, S., Takahashi, M.: Framed curves in the Euclidean space, Adv. Geom. 16 (3), 265–276 (2016). https://doi.org/10.1515/advgeom-2015- 0035
  • Izumiya, S., Takeuchi, N.: Generic properties of helices and Bertrand curves, J. Geom. 74, 97–109 (2002). https://doi.org/10.1007/PL00012543
  • İlarslan, K., Aslan, N.K.: On spacelike Bertrand curve in Minkowski 3-space, Konuralp J. Math. 5 (1), 214-222 (2017).
  • İşbilir, Z., Özen, K.E., Tosun, M.: Bertrand partner p-trajectories in the Euclidean 3-space E3, Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat. 72 (1), 216–228 (2023).
  • Kazaz, M., Uğurlu, H.H., Önder, M., Oral, S.: Bertrand partner D-curves in the Euclidean 3-space E3, AKU J. Sci. Eng. 16 (1), 76-83 (2016).
  • Keçilioğlu, O., İlarslan, K.: Quaternionic Bertrand curves in Euclidean 4-space, Bull. Math. Anal. Appl. 5 (3), 27-38 (2013).
  • Lone, M.S., Es, H., Karacan, M.K., Bukcu, B.: On some curves with modified orthogonal frame in Euclidean 3-space, Iran. J. Sci. Technol. Trans. A Sci. 43, 1905-1916 (2009).
  • Lucas, P., Ortega-Yagües, J.A.: Bertrand curves in the three-dimensional sphere, J. Geom. Phys. 62 (9), 1903–1914 (2012). https://doi.org/10.1016/j.geomphys.2012.05.013
  • Masal, M., Azak, A.Z.: Bertrand curves and Bishop frame in the 3-dimensional Euclidean space, Sakarya Univ. J. Sci. 21 (6), 1140-1145 (2017).
  • Matsuda, H., Yorozu, S.: Notes on Bertrand curves, Yokohama Math. J. 50 (1-2), 41–58 (2003).
  • Öğrenmiş, A.O., Öztekin, H., Ergüt, M.: Bertrand curves in Galilean space and their characterizations, Kragujevac J. Math. 32 (32), 139–147 (2009).
  • Pears, L.R.: Bertrand curves in Riemannian space, J. Lond. Math. Soc. 1 (3), 180–183 (1935). https://doi.org/10.1112/jlms/s1-10.2.180
  • Takahashi, M., Yu, H.: Bertrand and Mannheim curves of spherical framed curves in a three-dimensional sphere, Mathematics 10 (8), 1292 (2022). https://doi.org/10.3390/math10081292
  • Tunçer, Y., Ünal, S.: New representations of Bertrand pairs in Euclidean 3-space, ppl. Math. Comput. 219 (4), 1833-1842 (2012). https://doi.org/10.1016/j.amc.2012.06.021
  • Yakub, Z.D., Güngör, M.A.: Quaternionic framed curves, Math. Sci. Appl. E-Notes (accepted, in press) (2025).
  • Zhang, C., Pei, D.: Generalized Bertrand curves in Minkowski 3-space, Mathematics 8 (12), 2199 (2020). https://doi.org/10.3390/math8122199
  • Yerlikaya, F., Karaahmetoglu, S., Aydemir, I.: On the Bertrand B-pair curve in 3-dimensional Euclidean space, J. Sci. Arts 3 (36), 215–224 (2016). https://doi.org/10.5269/bspm.v35i2.24309

Quaternionic Framed Bertrand Curves

Year 2025, Volume: 18 Issue: 2, 185 - 195, 19.10.2025

Ayşe Zeynep Azak , Arzu Sürekçi , Mehmet Ali Güngör

Abstract

There is no pair of Bertrand quaternionic curves in the classical sense in quaternionic space. Therefore, realizing that investigating singular points and analyzing their properties and behavior is a rising topic in differential geometry, framed Bertrand mates, which are singular have been defined in the quaternionic space. The characteristic properties of these curves have been given. Thus, we have proven that the distance between these pairs of curves remains constant. The conditions for framed quaternionic curves to be Bertrand mates have been shown. Then, we demonstrate that Bertrand’s partner of any framed quaternionic curve is a framed quaternionic curve. Furthermore, the relationship between the curvatures of quaternionic framed Bertrand curves has been given. Finally, an example has been given that supports the proven theorem in our study.

Keywords

Singular curve Bertrand curves quaternionic curve framed curve

References

  • Aksoyak, F.K.: A new type of quaternionic frame in R4, Int. J. Geom. Methods Mod. Phys. 16 (6), 1959984 (2019). https://doi.org/10.1142/S0219887819500841
  • Aksoyak, F.K.: Quaternionic Bertrand curves according to type-2-quaternionic frame in R4, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 71 (2), 395-406 (2022). https://doi.org/10.31801/cfsuasmas.991631
  • Aksoyak, F.K.: Quaternionic (1, 3)-Bertrand curves according to type-2-quaternionic frame in R4, Konuralp J. Math. 9 (2), 346-355 (2021).
  • Balgetir, H., Bektaş, M., Ergüt, M.: Bertrand curves for nonnull curves in 3-dimensional Lorentzian space, Hadronic J. 27, 229-236 (2004).
  • Balgetir, H., Bekta¸s, M., Inoguchi, J.: Null Bertrand curves in Minkowski 3-space and their characterizations, Not. Mat. 23 (1), 7–13 (2004). https://doi.org/10.1285/i15900932v23n1p7
  • Bertrand, J.M.: Mémoire sur la théorie des courbes á double courbure. J. Math. Pures Appl. 15, 332–350 (1850).
  • Bharathi, K., Nagaraj, M.: Quaternion valued function of a real Serret-Frenet formulae, Indian J. Pure Appl. Math. 18 (6), 507-511 (1987).
  • Camci, Ç., Uçum, A., İIlarslan, K.: A new approach to Bertrand curves in Euclidean 3-space, J. Geom. 111 (49), 1–15 (2020). https://doi.org/10.1007/s00022-020-00560-5
  • Dede, M., Ekici, C.: Directional Bertrand curves, Gazi Univ. J. Sci. 31 (1), 202–211 (2018).
  • Ekmekci, N., İlarslan, K.: On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst. 3 (2), 17–24 (2001).
  • Elsayied, H.K., Altaha, A.A., Elsharkawy, A.: Bertrand curves with the modified orthogonal frame in Minkowski 3-space $E_1^3$, Rev. Educ. 392 (6), 43-55 (2022).
  • Elsharkawy, A., Tashkandy, Y., Emam W., Cesarano, C., Elsharkawy, N.: On some quasi-curves in Galilean three-space, Axioms 12 (9), 823 (2023). https://doi.org/10.3390/axioms12090823
  • Elzawya, M., Mosab, S.: Quaternionic Bertrand curves in the Galilean space, Filomat 34 (1), 59–66 (2020). https://doi.org/10.2298/FIL2001059E
  • Ergüt, M., Öztekin, H., Gün Bozok, H.: Representation formulae for Bertrand curves in Galilean and pseudo-Galilean 3-space, Adv. Model. Optim. 16 (2), 371-385 (2014).
  • Ersoy, S., Tosun, M.: Timelike Bertrand curves in semi-Euclidean space, Int. J. Math. Stat. 14 (2), 78-89 (2013). [16] Fukunaga, T., Takahashi, M.: Existence conditions of framed curves for smooth curves, J. Geom. 108 (2), 763-774 (2017). https://doi.org/10.1007/s00022-017-0371-5
  • Güner, G., Ekmekçi, N.: On the spherical curves and Bertrand curves in Minkowski 3-space, J. Math. Comput. Sci. 2 (4), 898-906 (2012).
  • Gök, İ., Nurkan, S.K., İlarslan, K.: On pseudo null Bertrand curves in Minkowski space-time, Kyungpook Math. J. 54 (4), 685-697 (2014). 108 (2), 763-774 (2017). https://doi.org/10.5666/KMJ.2014.54.4.685
  • Honda, S., Takahashi, M.: Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space, Turk. J. Math. 44 (3), 883-899 (2020). https://doi.org/10.3906/mat-1906-71
  • Honda, S., Takahashi, M., Yu, H.: Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space, J. Geom. 114 (2), 12 (2023). https://doi.org/10.1007/s00022-023-00673-7
  • Honda, S., Takahashi, M.: Framed curves in the Euclidean space, Adv. Geom. 16 (3), 265–276 (2016). https://doi.org/10.1515/advgeom-2015- 0035
  • Izumiya, S., Takeuchi, N.: Generic properties of helices and Bertrand curves, J. Geom. 74, 97–109 (2002). https://doi.org/10.1007/PL00012543
  • İlarslan, K., Aslan, N.K.: On spacelike Bertrand curve in Minkowski 3-space, Konuralp J. Math. 5 (1), 214-222 (2017).
  • İşbilir, Z., Özen, K.E., Tosun, M.: Bertrand partner p-trajectories in the Euclidean 3-space E3, Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat. 72 (1), 216–228 (2023).
  • Kazaz, M., Uğurlu, H.H., Önder, M., Oral, S.: Bertrand partner D-curves in the Euclidean 3-space E3, AKU J. Sci. Eng. 16 (1), 76-83 (2016).
  • Keçilioğlu, O., İlarslan, K.: Quaternionic Bertrand curves in Euclidean 4-space, Bull. Math. Anal. Appl. 5 (3), 27-38 (2013).
  • Lone, M.S., Es, H., Karacan, M.K., Bukcu, B.: On some curves with modified orthogonal frame in Euclidean 3-space, Iran. J. Sci. Technol. Trans. A Sci. 43, 1905-1916 (2009).
  • Lucas, P., Ortega-Yagües, J.A.: Bertrand curves in the three-dimensional sphere, J. Geom. Phys. 62 (9), 1903–1914 (2012). https://doi.org/10.1016/j.geomphys.2012.05.013
  • Masal, M., Azak, A.Z.: Bertrand curves and Bishop frame in the 3-dimensional Euclidean space, Sakarya Univ. J. Sci. 21 (6), 1140-1145 (2017).
  • Matsuda, H., Yorozu, S.: Notes on Bertrand curves, Yokohama Math. J. 50 (1-2), 41–58 (2003).
  • Öğrenmiş, A.O., Öztekin, H., Ergüt, M.: Bertrand curves in Galilean space and their characterizations, Kragujevac J. Math. 32 (32), 139–147 (2009).
  • Pears, L.R.: Bertrand curves in Riemannian space, J. Lond. Math. Soc. 1 (3), 180–183 (1935). https://doi.org/10.1112/jlms/s1-10.2.180
  • Takahashi, M., Yu, H.: Bertrand and Mannheim curves of spherical framed curves in a three-dimensional sphere, Mathematics 10 (8), 1292 (2022). https://doi.org/10.3390/math10081292
  • Tunçer, Y., Ünal, S.: New representations of Bertrand pairs in Euclidean 3-space, ppl. Math. Comput. 219 (4), 1833-1842 (2012). https://doi.org/10.1016/j.amc.2012.06.021
  • Yakub, Z.D., Güngör, M.A.: Quaternionic framed curves, Math. Sci. Appl. E-Notes (accepted, in press) (2025).
  • Zhang, C., Pei, D.: Generalized Bertrand curves in Minkowski 3-space, Mathematics 8 (12), 2199 (2020). https://doi.org/10.3390/math8122199
  • Yerlikaya, F., Karaahmetoglu, S., Aydemir, I.: On the Bertrand B-pair curve in 3-dimensional Euclidean space, J. Sci. Arts 3 (36), 215–224 (2016). https://doi.org/10.5269/bspm.v35i2.24309
There are 36 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Ayşe Zeynep Azak 0000-0002-2686-6043

Arzu Sürekçi 0000-0003-2003-3507

Mehmet Ali Güngör 0000-0003-1863-3183

Early Pub Date October 13, 2025
Publication Date October 19, 2025
Submission Date February 7, 2025
Acceptance Date May 31, 2025
Published in Issue Year 2025 Volume: 18 Issue: 2

Cite

APA Azak, A. Z., Sürekçi, A., & Güngör, M. A. (2025). Quaternionic Framed Bertrand Curves. International Electronic Journal of Geometry, 18(2), 185-195.
AMA Azak AZ, Sürekçi A, Güngör MA. Quaternionic Framed Bertrand Curves. Int. Electron. J. Geom. October 2025;18(2):185-195.
Chicago Azak, Ayşe Zeynep, Arzu Sürekçi, and Mehmet Ali Güngör. “Quaternionic Framed Bertrand Curves”. International Electronic Journal of Geometry 18, no. 2 (October 2025): 185-95.
EndNote Azak AZ, Sürekçi A, Güngör MA (October 1, 2025) Quaternionic Framed Bertrand Curves. International Electronic Journal of Geometry 18 2 185–195.
IEEE A. Z. Azak, A. Sürekçi, and M. A. Güngör, “Quaternionic Framed Bertrand Curves”, Int. Electron. J. Geom., vol. 18, no. 2, pp. 185–195, 2025.
ISNAD Azak, Ayşe Zeynep et al. “Quaternionic Framed Bertrand Curves”. International Electronic Journal of Geometry 18/2 (October2025), 185-195.
JAMA Azak AZ, Sürekçi A, Güngör MA. Quaternionic Framed Bertrand Curves. Int. Electron. J. Geom. 2025;18:185–195.
MLA Azak, Ayşe Zeynep et al. “Quaternionic Framed Bertrand Curves”. International Electronic Journal of Geometry, vol. 18, no. 2, 2025, pp. 185-9.
Vancouver Azak AZ, Sürekçi A, Güngör MA. Quaternionic Framed Bertrand Curves. Int. Electron. J. Geom. 2025;18(2):185-9.
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