On the modified orthogonal frames of the non-unit speed curves in Euclidean Space E3

Abstract

In this study, the modified frames with both the non-zero curvature and the torsion of the non-unit speed curves in Euclidean space E3 are examined. The relationships between the derivative vectors of the modified frames and the Frenet vectors or the vectors of the modified frames of the curve are given. Besides, the Darboux vectors obtained from the modified orthogonal frames with both the curvature and torsion of the curve and the unit vectors in the direction of these Darboux vectors are investigated. Finally, all these results are shown on the example curves.

Keywords

• Modified orthogonal frame
• Frenet frame
• Darboux vector

References

1. \bibitem{Arikan} Arikan M, Nurkan SK. Adjoint Curve According to Modified Orthogonal Frame with Torsion in 3-Space. Usak University Journal of Science and Natural Sciences. 2, 2020, 54 -- 64.
2. \bibitem{Azak} Azak AZ. Involute-Evolute Curves According to Modified Orthogonal Frame. Journal of Science and Arts. 21(2), 2021, 385 -- 394.
3. \bibitem{Bukcu} Bukcu B, Karacan MK. On the modified orthogonal frame with curvature and torsion in 3-space. Mathematical Sciences and Applications E-Notes. 4(1), 2016, 184 -- 188.
4. \bibitem{Clelland} Clelland JN. From Frenet to Cartan: the method of moving frames. American Mathematical Soc. 178, 2017.
5. \bibitem{Eren} Eren K, Kosal HH. Evolution of space curves and the special ruled surfaces with modified orthogonal frame. AIMS Mathematics. 5(3), 2020, 2027 -- 2039.
6. \bibitem{Gray} Gray A. Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC New York. 1998.
7. \bibitem{Hacisalihoglu} Hacisalihoglu HH. Differantial Geometry. Ankara University Faculty of Science Press, 2000.
8. \bibitem{Hord} Hord AM. Torsion at an inflection point of a space curve. Am. Math. Mon. 79, 1972, 371 -- 374.

9. \bibitem{Lone} Lone MS, Es H, Karacan MK, Bukcu B. On some curves with modified orthogonal frame in Euclidean 3-Space, Iranian Journal of Science and Technology, Transactions A: Science. 43(4), 2019, 1905 -- 1916.
10. \bibitem{Lone2} Lone MS, Es H, Karacan MK, Bukcu B. Mannheim curves with modified orthogonal frame in Euclidean 3-space. arXiv e-print. 1809, 2018.
11. \bibitem{Lone3} Lone MS, Karacan MK, Tuncer Y, Es H. Spherical indicatrices with the modified orthogonal frame. arXiv preprint. arXiv: 1810.01082, 2018.
12. \bibitem{Carmo} Do Carmo MP. Differential geometry of curves and surfaces: revised and updated second edition. Courier Dover Publications, 2016.
13. \bibitem{Ozdemir} Ozdemir M. Diferansiyel Geometri. Altin Nokta Yayinevi, 2020.
14. \bibitem{Sasai} Sasai T. The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations. Tohoku Math Journal. 36, 1984, 17 -- 24.
15. \bibitem{Sasai2} Sasai T. Geometry of analytic space curves with singularities and regular singularities of differential equations. Funkcial. Ekvac. 30, 1987, 283 -- 303.
16. \bibitem{Yayli} Yayli Y, Gok I, Hacisalihoglu HH. Extended rectifying curves as new kind of modified Darboux vectors. TWMS J Pure Appl Math. 9, 2018, 18 -- 31.