On Bishop Frames of Any Regular Curve in Euclidean 3-Space

Yıl 2024, , 23 - 33, 27.02.2024

Sümeyye Gür Mazlum

https://doi.org/10.35414/akufemubid.1343172

Öz

Relationships between type-1 Bishop and Frenet, type-2 Bishop and Frenet, alternative and Frenet, N-Bishop and alternative frames of any regular curve in Euclidean 3-space are known. In this study, relationships between N-Bishop and Frenet frames and relationships between type-1 Bishop, type-2 Bishop and N-Bishop frames of any regular curve in Euclidean 3-space are given. In addition, pole vectors (unit vectors in the direction of Darboux vectors) belonging to these frames are computed. Last, pole and Darboux vectors belonging to these frames are compared with each other.

Anahtar Kelimeler

type-1 Bishop frame, Type-2 Bishop frame, alternatif frame, N-Bishop frame, Frenet frame, darboux vektörü

3-Boyutlu Öklid Uzayında Regüler Bir Eğrinin Bishop Çatıları Üzerine

Öz

3-boyutlu Öklid uzayında herhangi bir regüler eğrinin tip-1 Bishop ve Frenet, tip-2 Bishop ve Frenet, alternatif ve Frenet, N-Bishop ve alternatif çatıları arasındaki ilişkiler bilinmektedir. Bu çalışmada, 3-boyutlu Öklid uzayında herhangi bir regüler eğrinin N-Bishop ve Frenet çatıları arasındaki ilişkiler ve tip-1 Bishop, tip-2 Bishop ve N-Bishop çatıları arasındaki ilişkiler verilmiştir. Ayrıca bu çatılara ait pol vektörleri (Darboux vektörü yönündeki birim vektörler) hesaplanmıştır. Son olarak pol ve Darboux vektörleri birbirleriyle karşılaştırılmıştır.

Anahtar Kelimeler

tip-1 Bishop çatısı, tip-2 Bishop çatısı, alternatif çatı, N-Bishop çatısı, frenet çatısı, Darboux vektörü

Kaynakça

• Alıç, Ş. and Yılmaz, B., 2021. Smarandache Curves According to Alternative Frame in . Journal of Universal Mathematics, 4, 140–156. https://www.doi.org.tr/10.33773/jum.956862
• Bishop, R.L., 1975. There is more than one way to frame a curve. The American Mathematical Monthly, 82, 246–251.https://doi.org/10.1080/00029890.1975.11993807
• Bükcü, B., and Karacan, M.K., 2008. Special Bishop motion and Bishop Darboux rotation axis of the space curve. Journal of Dynamical Systems and Geometric Theories, 6, 27–34. https://doi.org/10.1080/1726037X.2008.10698542
• Bükcü, B. and Karacan, M.K., 2009. The slant helices according to Bishop frame. International Journal of Computational and Mathematical Sciences, 3, 67–70.
• Çakmak A. and Şahin, V., 2022. Characterizations of Adjoint Curves According to Alternative Moving Frame. Fundamental Journal of Mathematics and Applications, 5, 42–50. https://doi.org/10.33401/fujma.1001730
• Damar, E., Yüksel, N. and Vanlı, A.T., 2017. The ruled surfaces according to type-2 Bishop frame in International Mathematical Forum, 12, 133–143. https://doi.org/10.12988/imf.2017.610131
• Hacısalihoğlu, H.H., 1983. Diferansiyel Geometri. İnönü Üniversitesi Yayınları.
• Kelleci, A., Bektaş, M. and Ergüt, M., 2019. The Hasimoto surface according to bishop frame. Adıyaman Üniversitesi Fen Bilimleri Dergisi, 9, 13–22.
• Keskin, O. and Yaylı, Y., 2017. An application of N-Bishop frame to spherical images for direction curves. International Journal of Geometric Methods in Modern Physics, 14, 1750162. https://doi.org/10.1142/S0219887817501626
• Kılıçoğlu, Ş. and Hacısalihoğlu, H.H., (2013). On the ruled surfaces whose frame is the Bishop frame in the Euclidean 3-space. International Electronic Journal of Geometry, 6, 110–117.
• Kızıltuğ, S., Kaya, S. and Tarakcı, O., 2013. The slant helices according to type-2 Bishop frame in Euclidean 3-space. International Journal of Pure and Applied Mathematics, 2, 211–222. http://dx.doi.org/10.12732/ijpam.v85i2.3
• Masal, M. and Azak, A.Z., 2015. The Ruled Surfaces According to Type-2 Bishop Frame in the Euclidean 3-Space . Mathematical Sciences and Applications E-Notes, 3, 74–83. https://doi.org/10.36753/mathenot.421334
• Masal, M. and Azak, A., 2019. Ruled surfaces according to Bishop frame in the Euclidean 3-spaces. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 89, 415–424.
• Ouarab, S., Ouazzani, A. and Izıd, M., 2018. Ruled surfaces with alternative moving frame in Euclidean 3-space. International Journal of Mathematical Sciences and Engineering Applications, 12, 43–58.
• Samancı, H.K. and İncesu, M., 2020. Investigating a quadratic Bezier curve due to NCW and N-Bishop frames. Turkish Journal of Mathematics and Computer Science, 12, 120–127. https://doi.org/10.47000/tjmcs.704794
• Samancı, H.K. and Sevinç, M., 2022. N-Bishop Çatısına Göre Regle Yüzeylerin Bazı Karakterizasyonları. Karadeniz Fen Bilimleri Dergisi, 12, 113–134. https://doi.org/10.31466/kfbd.937683
• Scofield, P.D., 1995. Curves of constant precessions. The American mathematical monthly, 102, 531–537.
• Şenyurt, S., 2018. D-Smarandache Curves According to the Sabban Frame of the Spherical Indicatrix Curve. Turkish Journal of Mathematics and Computer Science, 9, 39–49.
• Şenyurt, S. and Kaya, G., 2018. NC and NW Smarandache Curves According to Alternative Frame. Turkish Journal of Mathematics and Computer Science, 10, 269–274.
• Şenyurt, S., Mazlum, S.G., Canlı, D. and Can, E., 2023. Some special Smarandache ruled surfaces according to alternative frame in . Maejo International Journal of Science and Technology, 17, 138–153.
• Uzunoğlu, B., Gök, İ. and Yaylı, Y., 2016. A New approach on curves of constant precession. Applied Mathematics and Computation, 275, 317–323. https://doi.org/10.1016/j.amc.2015.11.083
• Yılmaz, S. and Turgut, M., 2010. A new version of Bishop frame and an application to spherical images. Journal of Mathematical Analysis and Applications, 371, 764–776. https://doi.org/10.1016/j.jmaa.2010.06.0127
• Yılmaz, B. and Has, A., 2022. Obtaining fractional electromagnetic curves in optical fiber using fractional alternative moving frame. Optik, 260, 169067. https://doi.org/10.1016/j.ijleo.2022.169067
• Yılmaz, S. and Savcı, Ü.Z., 2017. A New Version Darboux Vector and Characterization Some Special Curves According to Type-2 Bishop Frame in Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 87, 355–362.https://doi.org/10.1007/s40010-017-0373-6