3-Boyutlu öklid uzayında bertrand eğriler ve bishop çatısı

Year 2017, , 1140 - 1145, 01.12.2017

Ayşe Zeynep Azak , Melek Masal

https://doi.org/10.16984/saufenbilder.267557

Cited By: 2

Abstract

Bu çalışmada 1975 yılında L. R.
Bishop tarafından tanımlanan Bishop çatısına ait eğrilikliklerin geometrik
anlamları verildi. Daha sonra 1850 yılında Bertrand’ın tanımladığı Bertrand
eğri çiftlerinin Bishop vektörleri arasındaki bağıntılar elde edildi. Ayrıca bu
Bertrand eğri çiftlerinin paralel eğri olması durumunda bazı ilginç sonuçlar
elde edildi.

Keywords

Bertrand eğrileri, Bishop çatısı, Frenet çatısı, Öklid uzayı, paralel eğriler

Bertrand curves and bishop frame in the 3-dimensional euclidean space

Abstract

In this paper, the geometric
meanings of the curvatures belong to Bishop frame, which was defined by L.R.
Bishop in 1975, has been given. Afterwards, the relations between the Bishop
vectors of Bertrand curve couple, which Bertrand defined in 1850, has been
obtained. Furthermore, some interesting results have been found when these
curves become parallel curves.

Keywords

Bertrand curves, Bishop frame, Frenet frame, Euclidean space, parallel curves

References

• [1] E. As, A. Sarıoğlugil (2014) , “On the Bishop curvatures of involute-evolute curve couple in ”, Int. J. Phys. Sci., Cilt 9, No 7, pp. 140-145.
• [2] H. Balgetir, M. Bektaş, J. Inoguchi (2004), “Null Bertrand curves and their characterizations”, Note Mat., Cilt 23, No 1, pp. 7-13.
• [3] H. Balgetir, M. Bektaş, M. Ergüt (2004), “Bertrand curves for nonnull curves in three dimensional Lorentzian space”, Hadronic J., Cilt 27, pp. 229-236.
• [4] J. Bertrand (1850), “La theories de courbes a double courbure”, J. Math. Pures et Appl., Cilt 15, pp. 332-350.
• [5] L.R. Bishop (1975), “There is more than one way to frame a curve”, Amer. Math. Monthly, Cilt 82, No 3, pp. 246–251.
• [6] B. Bükcü, M.K. Karacan (2008), “Special Bishop motion and Bishop darboux rotation axis of the space curve”, J. Dyn. Syst. Geom. Theor., Cilt 6, No 1, pp. 27-34.
• [7] B. Bükcü, M.K. Karacan (2009), “The slant helices according to Bishop frame”, World Academy of Science, Engineering and Technology, Cilt 59, pp. 1039-1042.
• [8] J.H. Choi, T.H. Kang, Y.H. Kim (2012), “Bertrand curves in 3-dimensional space forms”, Appl. Math. Comput., Cilt 219, No 3, pp. 1040-1046.
• [9] M. Çetin, Y. Tunçer, M.K. Karacan (2014), “Smarandache curves according to Bishop frame in Euclidean 3-space”, Gen. Math. Notes, Cilt 20, No 2, pp. 50-66.
• [10] M. P. Do Carmo, Differential Geometry of Curves and Surfaces, New Jersey, Prentice-Hall, Englewood Cliffs, 1976.
• [11] R. Ghadami, Y. Yaylı (2012), “A new characterization for inclined curves by the help of spherical representations according to Bishop frame”, Int. J. Pure Appl. Math., Cilt 74, No 4, pp. 455-463.
• [12] S. Izumiya, N. Takeuchi (2002), “Generic properties of helices and Bertrand curves”, J. Geom., Cilt 74, pp. 97-109.
• [13] T. Körpınar, V. Asil, S. Baş (2011), “On characterization inextensible flows of curves according to Bishop frame”, Revista Notas de Matematica, Cilt 7(1), No 302, pp. 37-45.
• [14] P. Lucas, J.A. Ortega-Yagües (2012), “Bertrand curves in the three-dimensional sphere”, J. Geom. Phys., Cilt 62, No 9, pp. 1903-1914.
• [15] A.W. Nutbourne, R.R. Martin, Differential geometry applied to the design of curves and surfaces, UK, Ellis Horwood, Chichester, 1988.
• [16] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, New York, Academic Press, 1983.
• [17] A.O. Öğrenmiş, H. Öztekin, M. Ergüt (2009), “Bertrand curves in Galilean space and their characterizations”, Kragujevac J. Math., Cilt 32, pp. 139-147.
• [18] S.G. Papaioannou, D. Kiritsis (1985), “An application of Bertrand curves and surface to CAD/CAM”, Comput. Aided Geom. Design, Cilt 17, No 8, pp. 348-352.
• [19] D. Ünal, İ. Kişi, M. Tosun (2013), “Spinor Bishop equations of the curves in Euclidean 3-space”, Adv. Appl. Clifford Algebr., Cilt 23, No 3, pp. 757–765.
• [20] M.Y. Yılmaz, M. Bektaş (2008), “General properties of Bertrand curves in Riemann–Otsuki space”, Nonlinear Anal., Cilt 69, No 10, pp. 3225–3231.
• [21] S. Yılmaz, E. Özyılmaz, M. Turgut (2010), “New spherical indicatrices and their characterizations”, An. Şt. Univ. Ovidius Constanta, Cilt 18, No 2, pp. 337-354.