A Generalization of Curve Mates: Normal Mate of a Curve

Yıl 2024, Cilt: 17 Sayı: 2, 338 - 352, 31.08.2024

Oğuzhan Çelik

https://doi.org/10.18185/erzifbed.1396745

Öz

This a paper, a new curve pair is defined that generalizes some pairs of curves well known as Mannheim and Bertrand curve pairs. A normal curve pair is defined in such a way that a vector u obtained by overlapping the normal planes of the G and G* curves makes the same angle as the binormals of these curves. The relationship between torsions and curvatures of curve pairs was analyzed. Moreover, The unit quaternion q corresponding to the rotation matrix between the Frenet vectors of the curves was defined. In the conclusion, it is expressed express which famous pairs of curves will be obtained in which particular case.

Anahtar Kelimeler

Normal mate, Curve mates, Mannheim mate, Quaternion.

Teşekkür

Kolaylıklar ve iyi çalışmalar diliyorum.

Kaynakça

• [1] Babaarslan, M., Yayli, Y., (2013) On helices and Bertrand curves in Euclidean 3-space, Mathematical and Computational Applications, 18 (1) 1 - 11.
• [2] Balgetir, H., Bektas, M., Inoguchi, J. I., (2004) Null Bertrand curves in Minkowski 3-space and their characterizations, Note di matematica, 23 (1) 7 - 13.
• [3] Bektaş, Ö., (2018) Normal curves in n-dimensional Euclidean Space, Advances in Difference Equations, 2018 456.
• [4] Bukcu, B., Karacan, M. K., Yuksel, N., (2011) New Characterizations for Bertrand Curves in Minkowski 3-Space, Mathematical Combinatorics, 2 98-103.
• [5] Burke, J. F., (1960) Bertrand curves associated with a pair of curves, Mathematics Magazine, 34 (1) 60 - 62.
• [6] Calini, A., Ivey, T., (1998) Backlund transformations and knots of constant torsion, Journal of Knot Theory and Its Ramifications, 7 (06) 719 - 746.
• [7] Çelik, O., Özdemir, M., (2022) A New Generalization of Some Curve Pairs, International Electronic Journal of Geometry, 15 (2) 214 - 224.
• [8] Cheng, Y. M., Lini, C. C., (2010) On the Generalized Bertrand Curves in Euclidean-spaces, Note di Matematica, 29 (2) 33 - 39.
• [9] Choi, J. H., Kim, Y. H., (2012) Associated curves of a Frenet curve and their applications, Applied Mathematics and Computation, 218 (18) 9116 - 9124.
• [10] Dede, M., Ekici, C., (2018) Directional Bertrand curves, Gazi University Journal of Science, 31 (1) 202 - 211.
• [11] Deshmukh S, Chen B. Y., (2018) Alghanemi A, Natural mates of Frenet curves in Euclidean 3-space. Turkish Journal of Mathematics, 42 (5) 2826 - 2840.
• [12] Ekmekci, N., Ilarslan, K., (2001) On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst., 3 (2) 17 - 24.
• [13] Gok, I., Nurkan, S. K., Ilarslan, K., (2014) On pseudo null Bertrand curves in Minkowski space-time, Kyungpook Mathematical Journal, 54 (4) 685-697.
• [14] Görgülü, A., Özdamar, E., (1986)“A generalization of the Bertrand curves as general inclined curves in E^n, Commun. Fac. Sci. Univ. Ank., Series A1 (35) 53 - 60.
• [15] Güner G., Ekmekci, N., (2012) On the spherical curves and Bertrand curves in Minkowski-3 space, J. Math. Comput. Sci., 2 (4) 898 - 906.
• [16] Hanif, M., Hou, Z. H., (2018) Generalized involute and evolute curve-couple In Euclidean space, Int. J. Open Problems Compt. Math., 11 (2).
• [17] Honda, S. I., Takahashi, M., (2020) Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space, Turkish Journal of Mathematics, 44 (3) 883 – 899.
• [18] Ilarslan, K,, Aslan, N. K., (2016) On Spacelike Bertrand Curves in Minkowski 3-Space, Konuralp Journal of Mathematics, 5 (1) 214 - 222.
• [19] Ilarslan, K., (2005) Timelike and null normal curves in Minkowski space E (1,3). Turk. J.Math., 29 (1) 53 - 63.
• [20] Ilarslan, K., Nesovic, E., (2009) Spacelike and timelike normal curves in Minkowski space-time, Publ. Inst. Math., 85 (99) 111 - 118.
• [21] Izumiya, S., Takeuchi, N., (2002) Generic properties of helices and Bertrand curves, Journal of Geometry, 74 (1-2) 97 - 109.
• [22] Karacan, M. K., Tunçer, Y., (2012) Backlund transformations according to bishop frame in Euclidean 3-space, In Siauliai Mathematical Seminar, 7 (15).
• [23] Liu, H., Wang, F., (2008) Mannheim partner curves in 3-space, Journal of Geometry, 88 (1-2) 120 - 126.
• [24] Matsuda, H., Yorozu, S., (2003) Notes on Bertrand Curves, Yokohama Mathematical Journal, 50 (1) 41 - 58.
• [25] Matsuda, H., Yorozu, S., (2009) On generalized Mannheim curves in Euclidean 4-space, Nihonkai Mathematical Journal, 20 (1) 33 - 56.
• [26] Nemeth, S. Z., (2000) Backlund transformations of constant torsion curves in 3 dimensional constant curvature spaces, Italian Journal of Pure and Applied Mathematics, (7) 125 - 138.
• [27] Nemeth, S. Z., (1998) Backlund transformations of n-dimensional constant torsion curves, Publicationes Mathematicae Debrecen, 53 (3-4) 271 - 279.
• [28] Orbay, K., Kasap, E., (2009) On Mannheim partner curves in E^3, International Journal of Physical Sciences, 4 (5) 261 - 264.
• [29] Önder, M., (2019) Construction of curve pairs and their application, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences:1-8.
• [30] Özdemir, M., Çöken, A. C., (2009) Backlund transformation for non-lightlike curves in Minkowski 3-space, Chaos, Solitons & Fractals, 42 (4) 2540 - 2545.
• [31] Özdemir, M., (2020) Kuaterniyonlar ve Geometri, (72). Altın Nokta Yayınları, İzmir.
• [32] Öztekin, H. B., Bektas, M., (2020) Representation formulae for Bertrand curves in the Minkowski 3-space, Scientia Magna, 6 (1) 89.
• [33] Ozturk, G., Arslan, K., Bulca, B., (2018) A Characterization of Involutes and Evolutes of a Given Curve in E^n, Kyungpook Mathematical Journal, 58 (1) 117 - 135.
• [34] Tunçer, Y., Ünal, S., (2012) New representations of Bertrand pairs in Euclidean 3-space, Applied Mathematics and Computation, 219 (4) 1833 - 1842.
• [35] Uçum, A., İlarslan, K., (2016) On timelike Bertrand Curves in Minkowski 3-space, Honam Mathematical Journal, 38 (3) 467 - 477.
• [36]Uçum, A., Keçilioğlu, O., İlarslan, K., (2016) Generalized Bertrand Curves with Spacelike ( 1,3) Normal Plane in Minkowski Space-Time, Turkish Journal of Mathematics, 40 (3) 487 – 505.
• [37]Wang, F., Liu, H., (2007) Mannheim Partner Curve in 3-Euclidean Space, Mathematics in Practice and Theory, 37 141 - 143.
• [38]Zhang, C., Pei, D., (2020) Generalized Bertrand Curves in Minkowski 3-Space, Mathematics, 8 (12) 21 - 99.