3 Boyutlu Uzaylarda Torsiyon İle Oluşturulan Modifiye dik çatıya göre Adjoint Eğriler

Year 2020, , 54 - 64, 30.12.2020

Murat Arıkan , Semra Kaya Nurkan

https://doi.org/10.47137/usufedbid.798189

Cited By: 2

Abstract

Bu çalışmanın esas amacı, 3 Boyutlu Öklid ve Minkowski uzaylarında bir eğrinin adjoint eğrisini torsiyon ile oluşturulan modifiye dik çatıya göre elde etmektir. Bunun yanı sıra, diğer bir amaç ise bahsedilen eğrilerin modifiye dik çatıları arasındaki ilişkilere ulaşmaktır. Bu makalede ilk olarak 3 Boyutlu Öklid uzayda bir uzay eğrisinin torsiyon ile oluşturulan modifiye dik çatıya göre adjoint eğrisi tanımlandı ve adjoint eğrisinin modifiye dik çatısı esas eğrinin modifiye dik çatısı cinsinden bir teoremle verildi. Bu teorem yardımı ile esas eğri ve adjoint eğrisi arasında genel helis olma, Salkowski eğrisi olma gibi özellikler ile ilgili sonuçlar verildi. Benzer yaklaşımlar 3 Boyutlu Minkowski uzayında da null olmayan eğriler için tartışıldı ve bir uzay eğrisinin adjoint eğrisi tanımlandı.

Keywords

adjoint curve, Modified orthogonal frame, euclidean space, minkowski space

Adjoint Curve According to Modified Orthogonal Frame with Torsion in 3-Space

Abstract

The main purpose in this study is to create the adjoint curve of a curve according to the modified orthogonal frame created by torsion in Euclidean and Minkowski 3-space. Besides, it is another purpose to reveal the relationships between the modified orthogonal frame of these curves.In this paper, firstly, the adjoint curve of a space curve in 3D Euclidean space with respect to the modified orthogonal frame formed by torsion is defined and the modified orthogonal frame of the adjoint curve is given by a theorem in terms of the modified orthogonal frame of the main curve.With the help of this theorem, results about properties such as being a general helix and a Salkowski curve between main curve and adjoint curve were given. Similar approaches are discussed for non-null curves in 3D Minkowski space and the adjoint curve of a space curve is defined.

Keywords

adjoint curve, modified orthogonal Frame, euclidean 3-space, minkowski 3-space

References

• 1. Alev Kelleci; (2019), Conjugate mates for non-null Frenet curves. Sakarya University Journal of Science, 23(4), 600-604, DOI: 10.16984/saufenbilder.494471.
• 2. Bukcu B, Karacan MK. On the modified orthogonal frame with curvature and torsion in 3-space. Mathematical Sciences and Applications E-Notes 2016; 4
• 3. Bukcu B, Karacan MK. Spherical curves with modified orthogonal frame. Journal of New Results in Science 2016;10: 60-68.
• 4. J. Choi, Y. H. Kim, Associated curves of a Frenet curve and their applications, Appl. Math.Comput, 2012; 218: 9116-9124.
• 5. S. Desmukh, BY. Chen and A. Alghanemi, Natural mates of Frenet curves in Euclidean 3-space, Turk. J. Math, (2018) 42: 2826-2840.
• 6. Do Carmo, Manfredo, P., Differential geometry of curves and surfaces. Prentice-Hall, 1976.
• 7. 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 (in press). doi.org/10.1007/s40995-018-0661-2.
• 8. Lone MS, Es H, Karacan MK, Bukcu B. Mannheim curves with modified orthogonal frame in Euclidean 3-space. Turkish Journal of Mathematics. 2019; 43(2): 648-663.
• 9. S. K. Nurkan, I. A. Güven, M. K. Karacan, Characterizations of adjoint curves in Euclidean 3-space. Proc Natl Acad Sci. India Sect A Phys Sci. https://doi.org/10.1007/s40010-017-0425-y.
• 10. Sasai T. The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations. Tohoku Math Journal 1984; 36: 17-24.