E3'te Adjoint Eğriler ile İlişkili Normal Yüzeyler

Yıl 2022, , 779 - 783, 30.09.2022

Talat Körpınar , Ahmet Sazak

https://doi.org/10.17798/bitlisfen.1086972

Öz

Bu çalışmada, ilişkili eğrilere bağlı olarak belirlenen yüzeyler üzerinde çalışıyoruz. Özel bir ilişkili eğri türü olan adjoint eğrilerin yardımıyla tanımlanan normal yüzeyleri inceliyoruz. Bunun için öncelikle çalıştığımız uzay olan 3 boyutlu Öklid uzayının temel denklemlerini ve adjoint eğri konusunu hatırlıyoruz. Daha sonra, bir adjoint eğrinin normal yüzeyinin birinci ve ikinci temel formlarını, asal eğriliklerini ve Gauss ve ortalama eğriliklerini hesaplayarak, bu yüzeyin karakterizasyonlarını ve ilgili bazı sonuçlar elde ediyoruz.

Anahtar Kelimeler

Weingarten yüzey, Normal yüzey, Serret-Frenet çatı, Adjoint eğri

Normal Surfaces Associated with Adjoint Curves in E3

Öz

In this study, we work on the surfaces determined in relation to associated curves. We study normal surfaces defined with the help of adjoint curves, a special type of associated curve. For this, we first remember the basic equations of the 3-dimensional Euclidean space, which is the space we work with, and the adjoint curve issue. Then, by computing the first and second fundamental forms, principal curvatures and Gaussian and mean curvatures of the normal surface of an adjoint curve, we obtain the characterizations of this surface and related some results.

Anahtar Kelimeler

Weingarten surface, Normal surface, Serret-Frenet frame, Adjoint curve

Kaynakça

• M. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, 1976.
• S. K. Nurkan, I. A. Güven and M. K. Karacan, "Characterizations of adjoint curves in Euclidean 3-space", Proc Natl Acad Sci. India Sect A Phys Sci., vol. 89, pp. 155-161, 2019.
• G. U. Kaymanli, S. Okur and C. Ekici, "The Ruled Surfaces Generated By Quasi Vectors", IV. International Scientific and Vocational Studies Congress - Science and Health, Ankara, Turkey, November, 2019.
• R. Lopez, "Differential geometry of curves and surfaces in Lorentz-Minkowski space", MiniCourse taught at IME-USP, Sao Paulo, Brasil, October, 2008.
• F. Sakai, "Weil divisors on normal surfaces", Duke Mathematical Journal, vol. 51, no. 4, pp. 877-887, 1984.
• R. Lopez, Z. M. Sipus, L.P. Gajcic and I. Protrka, "Harmonic evolutes of B-scrolls with constant mean curvature in Lorentz-Minkowski space" , International Journal of Geometric Methods in Modern Physics, vol. 16, no. 5, 2019.
• I. Aydemir and K. Orbay, " The Ruled Surfaces Generated By Frenet Vectors of Timelike Ruled Surface in the Minkowski Space R_1^3", World Applied Science Journal, vol. 6, no. 5, pp. 692-696, 2009.
• I. Protrka, "Harmonic Evolutes of Timelike Ruled Surfaces in Minkowski Space", 18th Scientific-Professional Colloquium on Geometry and Graphics, Beli Manastir, Croatia, September 6-10, 2015.
• I. Protrka, "The harmonic evolute of a helicoidal surfaces in Minkowski 3-space", Proceedings of the Young Researcher Workshop on Differential Geometry in Minkowski Space, Granada, Spain, pp. 133-142, 2017.
• A. Sarıoglugil and A. Tutar, "On Ruled Surface in Euclidean Space E3", Int. J. Contemp. Math. Sci., vol. 2, no. 1, pp. 1-11, 2007.
• G. Y. Senturk and S. Yuce, "Characteristic properties of the ruled surface with Darboux frame in E3", Kuwait J. Sci., vol. 42, no. 2, pp. 14-33, 2015.
• W. Kühnel, “Ruled W-surfaces”, Arch. Math. (Basel), vol. 62, pp. 475-480, 1994.
• Y. Ünlütürk and C. Ekici, "Parallel Surfaces of Spacelike Ruled Weingarten Surfaces in Minkowski 3-space", New Trends in Mathematical Sciences, vol.1, no. 1, pp. 85-92, 2013.
• J. H. Choi and Y. H. Kim, "Associated curves of a Frenet curve and their applications", Appl. Math. Comput., vol. 218, no. 18, pp. 9116-9124, 2012.
• T. Körpınar, M. T. Sarıaydın, and E. Turhan, "Associated curves according to Bishop frame in Euclidean 3-space" , Adv. Model. Optim., vol. 15, no. 3, pp. 713-717, 2013.
• N. Macit and M. Düldül, "Some new associated curves of a Frenet curve in E3 and E4", Turkish J. Math., vol. 38, pp. 1023-1037, 2014.
• S. Yılmaz, "Characterizations of some associated and special curves to type-2 Bishop frame in E3", Kırklareli University J. Eng. Sci., vol. 1, pp. 66-77, 2015.