Helis Eğrisinin Küresel Göstergeleri Üzerine Bazı Uygulamalar

Yıl 2024, , 154 - 175, 30.06.2024

Süleyman Şenyurt , Sümeyye Gür Mazlum

https://doi.org/10.54370/ordubtd.1438188

Cited By: 1

Öz

Bu çalışmada, helis eğrisinin Frenet vektörlerinin lineer birleşimden elde edilen birim vektörlerin birim küre üzerinde çizdikleri eğrilerin Frenet elemanları hesaplanmıştır. Dahası bu eğrilere ait Sabban çatıları oluşturularak Smarandache eğrileri tanımlanmıştır. Son olarak bu Smarandache eğrilerinin geodezik eğrilikleri hesaplanmıştır.

Anahtar Kelimeler

Geodezik eğrilik, helis eğrisi, Sabban çatısı, Smarandache eğrisi

Some Applications on Spherical Indicatrices of the Helix Curve

Öz

In this study, the Frenet elements of the curves that are drawn on the unit sphere by the unit vectors obtained from linear combinations of Frenet vectors of the helix curve are calculated. Moreover, Sabban frames of these curves are created and Smarandache curves are defined. Finally, the geodesic curvatures of each Smarandache curve are calculated.

Anahtar Kelimeler

geodesic curvature, helix curve, sabban frame, smarandache curve

Kaynakça

• Abbena, E., Salamon, S., & Gray, A. (2017). Modern differential geometry of curves and surfaces with Mathematica. CRC Press (3rd Edition). https://doi.org/10.1201/9781315276038
• Alıç, Ş., & Yılmaz, B. (2021). Smarandache Curves according to alternative frame in . Journal of Universal Mathematics, 4(2), 140-156. https://doi.org/10.33773/jum.956862
• Ali, A. T. (2010). Special smarandache curves in the euclidean space. International Journal of Mathematical Combinatorics, 2, 30-36. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=710cb20458012e6eb683d3e29998cd0e57325b1e
• Bektaş Ö. and Yüce, S. (2013). Special smarandache curves according to darboux frame in . Romanian Journal of Mathematics and Computer Science, 3(1), 48-59. https://arxiv.org/pdf/1203.4830
• Çetin, M., Tunçer, Y., & Karacan, M. K. (2014). Smarandache curves according to Bishop frame in Euclidean 3-space. Gen. Math. Notes, 20(2), 50-66. https://fs.unm.edu/SN/MA-SmarandacheCurvesAccordingBishop.pdf
• Çetin, M. and Kocayiğit, H. (2013). On the quaternionic smarandache curves in euclidean 3-space. Int. J. Contemp. Math. Sciences, 8(3), 139-150. https://www.m-hikari.com/ijcms/ijcms-2013/1-4-2013/cetinIJCMS1-4-2013.pdf
• Do Carmo, M. P. (1976). Differential geometry of curves and surfaces. Prentice Hall, Englewood Cliffs, NJ.
• Gür Mazlum, S. (2023). On the evolute curves of any non unit speed curve in euclidean 3-space [Oral presentation]. 8th Asia Pacific International Modern Sciences Congress, Iksad Institute, Delhi, India.
• Koenderink, J. (1990). Solid Shape. MIT Press, Cambridge, MA. Şenyurt, S. (2018). D-smarandache curves according to the sabban frame of the spherical indicatrix curve. Turk. J. Math. Comput. Sci. 9, 39-49. https://dergipark.org.tr/en/download/article-file/610102
• Şenyurt, S., & Canlı, D. (2023). On the tangent indicatrix of special viviani’s curve and its corresponding smarandache curves according to sabban frame [Oral presentation]. 10th International Başkent Congress on Physical, Ankara, Türkiye.
• Şenyurt, S., & Çalışkan, A. (2015). An application according to spatial quaternionic Smarandache curve. Appl. Math. Sci., 9(5), 219-228. http://dx.doi.org/10.12988/ams.2015.411961
• Şenyurt, S., & Öztürk, B. (2018). Smarandache Curves of Salkowski Curve According Frenet Frame. Turkish Journal of Mathematics and Computer Science, 10, 190-201. https://dergipark.org.tr/en/download/article-file/615426
• Şenyurt, S., & Sivas, S. (2013). An application of smarandache curve. Ordu University Journal of Science and Tecnology, 3(1), 46-60. https://dergipark.org.tr/tr/pub/ordubtd/issue/11065/132165
• Şenyurt, S., Cevahir, C., & Altun, Y. (2019). Smarandache curves according to sabban frame belonging to mannheim curves pair. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math., 68(1), 500-523. https://doi.org/10.31801/cfsuasmas.431877
• Şenyurt, S., & Gür Mazlum, S. (2023). Some applications on the tangent indicatrix curve of the helix [Oral presentation]. 2nd International İzmir Congress on Life. İzmir, Türkiye.
• Şenyurt, S., Cevahir, C., & Altun, Y. (2020). On the Smarandache curves of spatial quaternionic involute curve. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 90, 827-837. https://doi.org/10.1007/s40010-019-00640-5
• Şenyurt, S., Canlı, D., Can, E., & Gür Mazlum, S. (2022). Some special smarandache ruled surfaces by frenet frame in -II. Honam Mathematical Journal, 44(4), 594-617. https://doi.org/10.5831/HMJ.2022.44.4.594
• Şenyurt, S., Ayvacı, K. H., & Canlı, D. (2023a). Smarandache curves according to flc-frame in euclidean 3-space. Fundamentals of Contemporary Mathematical Sciences, 4(1), 16-30. https://doi.org/10.54974/fcmathsci.1142404
• Şenyurt, S., Canlı, D., Can, E., & Gür Mazlum, S. (2023b). Another application of smarandache based ruled surfaces with the darboux vector according to frenet frame in . Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 72(4), 880-906. https://doi.org/10.31801/cfsuasmas.1151064
• Taşköprü, K., & Tosun, M. (2014). Smarandache curves according to Sabban frame on . Boletim da Sociedade Paranaense de Matematica, 32(1), 51-59. https://arxiv.org/pdf/1206.6229
• Turgut, M., & Yılmaz, S. (2008). Smarandache curves in minkowski space-time. International J. Math. Combin., 3, 51-55. https://www.mathcombin.com/upload/file/20150126/1422261423330082520.pdf#page=56