New moving frames for the curves lying on a surface

Yıl 2024, Cilt: 42 Sayı: 4, 1023 - 1029, 01.08.2024

Akın Alkan , Hüseyin Kocayiğit Tuba Ağırman Aydın

Öz

In this article, three new orthogonal frames are defined for the curves lying on a surface. These moving frames, obtained based on the Darboux frame, are called “Osculator Darboux Frame”, “Normal Darboux Frame” and “Rectifying Darboux Frame”, respectively. Also, the Osculator Darboux Frame components and curvatures are calculated for a presented example.

Anahtar Kelimeler

Darboux Frame, Moving Frame, Normal Darboux Vector, Osculator Darboux Vector, Rectifying Darboux Vector

Kaynakça

• REFERENCES
• [1] Liu H, Wang F. Mannheim partner curves in 3-space. J Geom 2008;88:120126.
• [2] J Burke. Bertrand curves associated with a pair of curves. Math Mag 1960;34:60–62.
• [3] Gluck H. Higher curvatures of curves in euclidean space. Am Math Mon 1966;73:699704.
• [4] Sabuncuoğlu A, Hacısalihoğlu HH. On higher curvatures of a curve. Commun Fac Sci Univ Ankara Ser A1 Math Stat 1975;24:3346.
• [5] Özdamar E, Hacısalihoğlu HH. A Characterization of Inclined Curves in Euclidean n-Space, Communications, Commun Fac Sci Univ Ankara Ser A1 Math 1975;24:1523.
• [6] Özdamar E, Hacısalihoğlu HH. Characterizations of spherical curves in euclidean n-space. Commun Fac Sci Univ Ankara Ser A1 Math Stat 1974:109125.
• [7] Izumiya S, Takeuchi N. New special curves and developable surfaces. Turk J Math 2004;28:153163.
• [8] Kula L, Ekmekçi N, Yaylı Y, İlarslan K. Characterizations of slant helices in euclidean 3-space. Turk J Math 2010;34:261273.
• [9] Öztürk U, Hacısalihoğlu HH. Helices on a surface in Euclidean 3-space.Celal Bayar Univ J Sci 2017;13:113123.
• [10] Zıplar E, Şenol A, Yaylı Y. On darboux helices in euclidean 3-space. Glob J Science Front Res Math Decis Sci 2012;12:7380.
• [11] Uzunoğlu B, Gök ·I, Yayli Y. A new approach on curves of constant precession. Appl Math Comput 2016;275:317323.
• [12] Yılmaz S, Turgut M. 2010. A New version of bishop frame and application to spherical images. J Mathe Anal Appl 2010;371:764776.
• [13] Bishop LR. There is more than one way to frame a curve. Am Math Mon 1975;82:246251.
• [14] Düldül M, Uyar Düldül B. Characterizations of helices by using their Darboux vectors. Sigma J Eng Nat Sci 2020;38:12991306.
• [15] Hananoi S, Ito N, Izumiya S. Spherical Darboux images of curves on surfaces. Beitr Algebra Geom 2015;56:575585.
• [16] Macit N, Düldül M. Relatively normal-slant helices lying on a surface and their characterizations. Hacettepe J Math Stat 2017;46:397408.
• [17] Doğan F, Yaylı Y. On isophote curves and their characterizations. Turk J Math 2015;39:650664.
• [18] Önder M. Helices associated to helical curves, relatively normal-slant helices and isophote curves. Available at: https://arxiv.org/abs/2201.09684. Accessed on Jul 2, 2024.
• [19] O'Neill B. Elementary differential geometry. Cambridge: Academic Press; 1966.