A New Generalization of Some Curve Pairs

Year 2022, Volume: 15 Issue: 2, 214 - 224, 31.10.2022

Oğuzhan Çelik , Mustafa Ozdemır

https://doi.org/10.36890/iejg.1110327

Cited By: 1

Abstract

In this study, we give a new curve pair that generalizes some of the famous pairs of curves as Bertrand and constant torsion curves. This curve pair is defined with the help of a vector obtained by the intersection of the osculating planes such that this vector makes the same angle $\gamma$ with the tangents of the curves. We examine the relations between torsions and
curvatures of these curve mates. Also, We have seen that the unit quaternion corresponding to the rotation matrix between the Frenet vectors of the curves is $q=\cos (\theta/2)-\mathbf{i}\sin (\theta/2)\cos \gamma -\mathbf{j}\sin (\theta/2)\sin \gamma$, where $\theta$ is the angle between the reciprocal binormals of the curves. Finally, we show in which specific case which well-known pairs of curves will be obtained.

Keywords

Bertrand Mate, Backlund Transformation, constant torsion curves, curve mates

References

• [1] Babaarslan, M., & Yayli, Y. On helices and Bertrand curves in Euclidean 3-space Mathematical and Computational Applications, 18(1), 1-11 (2013).
• [2] Balgetir, H., Bektas, M., & Inoguchi, J. I. Null Bertrand curves in Minkowski 3-space and their characterizations. Note di matematica, 23(1), 7-13 (2004).
• [3] Bukcu, B., Karacan, M. K., & Yuksel, N. New Characterizations for Bertrand Curves in Minkowski 3-Space. Mathematical Combinatorics, 2, 98-103 (2011).
• [4] Burke, J. F. Bertrand curves associated with a pair of curves. Mathematics Magazine, 34(1), 60-62 (1960).
• [5] Calini, A., & Ivey, T. Bäcklund transformations and knots of constant torsion. Journal of Knot Theory and Its Ramifications, 7(06), 719-746 (1998).
• [6] Cheng, Y. M., & Lini, C. C. On the Generalized Bertrand Curves in Euclidean-spaces. Note di Matematica, 29(2), 33-39 (2010).
• [7] Choi, J. H., & Kim, Y. H. Associated curves of a Frenet curve and their applications. Applied Mathematics and Computation, 218(18), 9116-9124 (2012).
• [8] Dede, M., & Ekici, C. Directional Bertrand curves. Gazi University Journal of Science. 31(1), 202-211 (2018).
• [9] Deshmukh, S., Chen, B. Y., & Alghanemi, A. Natural mates of Frenet curves in Euclidean 3-space. Turkish Journal of Mathematics, 42(5), 2826-2840 (2018).
• [10] Ekmekci, N., Ilarslan, K., (2001) On Bertrand curves and their characterization. Differ. Geom. Dyn. Syst 3 (2), 17-24.
• [11] Gok, I., Nurkan, S. K., & Ilarslan, K. On pseudo null Bertrand curves in Minkowski space-time. Kyungpook Mathematical Journal, 54(4), 685-697 (2014).
• [12] Görgülü, A., & Özdamar, E. A generalization of the Bertrand curves as general inclined curves in En, Commun. Fac. Sci. Univ. Ank., Series A 1 , V.35, 00. 53-60 (1986).
• [13] Güner, G., & Ekmekci, N. On the spherical curves and Bertrand curves in Minkowski-3 space. J. Math. Comput. Sci., 2(4), 898-906 (2012).
• [14] Hanif M. and Hou, Z. H., Generalized involute and evolute curve-couple In Euclidean space. Int. J. Open Problems Compt. Math., 11(2) (2018).
• [15] Honda, S. I., & Takahashi, M. Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space. Turkish Journal of Mathematics, 44(3), 883-899 (2020).
• [16] Ilarslan, K., & Aslan, N. K. On Spacelike Bertrand Curves in Minkowski 3-Space. Konuralp Journal of Mathematics, 5(1), 214-222 (2016).
• [17] Ilarslan, K., & Nešovi´c, E. Some characterizations of osculating curves in the Euclidean spaces. Demonstratio Mathematica, 41(4), 931-939 (2008).
• [18] Izumiya, S., & Takeuchi, N. Generic properties of helices and Bertrand curves. Journal of Geometry, 74(1-2), 97-109 (2002).
• [19] Karacan, M. K., & Tunçer, Y. Bäcklund transformations according to bishop frame in Euclidean 3-space. In Siauliai Mathematical Seminar (Vol. 7, No. 15), (2012).
• [20] Liu, H., & Wang, F. Mannheim partner curves in 3-space. Journal of Geometry, 88(1-2), 120-126 (2008).
• [21] Matsuda, H., & Yorozu, S. Notes on Bertrand Curves. Yokohama Mathematical Journal, 50(1), 41-58 (2003).
• [22] Matsuda, H., & Yorozu, S. On generalized Mannheim curves in Euclidean 4-space. Nihonkai Mathematical Journal, 20(1), 33-56 (2009).
• [23] Nemeth, S. Z. Bäcklund transformations of constant torsion curves in 3 dimensional constant curvature spaces. Italian Journal of Pure and Applied Mathematics, (7), 125-138 (2000).
• [24] Nemeth, S. Z. Bäcklund transformations of n-dimensional constant torsion curves. Publicationes Mathematicae Debrecen, 53(3-4), 271-279 (1998).
• [25] Orbay, K., & Kasap, E. (2009). On Mannheim partner curves in E3. International Journal of Physical Sciences, 4(5), 261-264.
• [26] Önder, M. Construction of curve pairs and their applications. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 1-8 (2019).
• [27] Özdemir, M., & Çöken, A. C. Bäcklund transformation for non-lightlike curves in Minkowski 3-space. Chaos, Solitons & Fractals, 42(4), 2540-2545 (2009).
• [28] Öztekin, H. B., & Bektas, M. Representation formulae for Bertrand curves in the Minkowski 3-space. Scientia Magna, 6(1), 89 (2010).
• [29] Ozturk, G., Arslan, K., & Bulca, B. A Characterization of Involutes and Evolutes of a Given Curve in E^n. Kyungpook Mathematical Journal, 58(1), 117-135 (2018).
• [30] Rutter, J. W. (2018). Geometry of curves. CRC press.
• [31] Tunçer, Y., & Ünal, S. New representations of Bertrand pairs in Euclidean 3-space. Applied Mathematics and Computation, 219(4), 1833-1842 (2012).
• [32] Ucum, A., and Ilarslan K., On timelike Bertrand Curves in Minkowski 3-space. Honam Mathematical Journal, Vol.38, No.3„ pp.467-477 (2016).
• [33] Ucum, A., Keçilio˘glu, O., & Ilarslan, K. Generalized Bertrand Curves with Spacelike (1 , 3)-Normal Plane in Minkowski Space-Time. Turkish Journal of Mathematics, 40(3), 487-505 (2016).
• [34] Wang F, Liu H Mannheim Partner Curve in 3-Euclidean Space. Mathematics in Practice and Theory 37: 141-143 (2007).
• [35] Zhang, C., & Pei, D. Generalized Bertrand Curves in Minkowski 3-Space. Mathematics, 8(12), 2199 (2020).
• [36] Rogers, C, and William F. S. Bäcklund transformations and their applications. Vol. 161. New York: Academic press, 1982.