Another application of Smarandache based ruled surfaces with the Darboux vector according to Frenet frame in $E^{3}$

Year 2023, Volume: 72 Issue: 4, 880 - 906, 29.12.2023

Süleyman Şenyurt , Davut Canlı , Elif Çan , Sümeyye Gür Mazlum

https://doi.org/10.31801/cfsuasmas.1151064

Cited By: 1

Abstract

In this study, we first define the Smarandache curves derived from the Frenet vectors and the Darboux vector of any curve. Then, we construct new ruled surfaces along these Smarandache curves with the direction vectors obtained from the Frenet vectors and the Darboux vector, and give the equations of these surfaces. In addition, we calculate the Gaussian and mean curvatures of these surfaces separately and present the conditions to be minimal and developable for these surfaces. Finally, as an example, we obtain ruled surfaces whose base curves are Viviani’s curves and plot the graphics of these surfaces.

Keywords

Smarandache curve, ruled surfaces, mean curvature, Gaussian curvature, Viviani's curve

References

• Ali, A. T., Special Smarandache curves in the Euclidean space, Int. J. Math. Comb., 2 (2010), 30–36.
• Aslan, S., Bekar, M., Yaylı, Y., Ruled surfaces constructed by quaternions, J. Geom. Phys., 161 (2021), 1–9. https://doi.org/10.1016/j.geomphys.2020.104048
• Bektaş, M., On characterizations of general helices for ruled surfaces in the pseudo-Galilean space $G^{1}_{3}$-(Part-I), J. Math. of Kyoto Univ., 44(3) (2004), 523–528. https://doi.org/10.1215/kjm/1250283082
• Bektaş, Ö., Yüce, S., Special Smarandache curves according to Darboux frame in $E^{3}$, Romanian J. Math. Comp. Sci., 3 (2013), 48–59.
• Berk, A., A Structural Basis for Surface Discretization of Free Form Structures: Integration of Geometry, Materials and Fabrication, Ph.D Thesis, Michigan University, ABD, 2012.
• Çetin, M., Kocayiğit, H., On the quaternionic Smarandache curves in Euclidean 3-space, Int. J. Contemp. Math. Sci., 8(3) (2013), 139–150.
• Do-Carmo, P. M., Differential Geometry of Curves and Surfaces, IMPA, 1976.
• Fenchel, W., On the differential geometry of closed space curves, Bull. Am. Math. Soc., 57 (1951), 44–54.
• Gray, A., Abbena, E., Salamon, S., Modern Differential Geometry of Curves and Surfaces with Mathematica, Chapman and Hall/CRC, 2017.
• Gür Mazlum, S., Şenyurt S., Grilli, L., The dual expression of parallel equidistant ruled surfaces in Euclidean 3-space, Symmetry, 14 (2022), 1062. https://doi.org/10.3390/sym14051062
• Gür Mazlum, S., Bektaş, M., On the modified orthogonal frames of the non-unit speed curves in Euclidean 3-Space $E^{3}$, Turk. J. Sci., 7(2) 2022, 58–74.
• Hathout, F., Bekar, M., Yaylı, Y., Ruled surfaces and tangent bundle of unit 2-sphere, Int. J. Geom. Meth. Mod. Phys., 2 (2017). https://doi.org/10.1142/S0219887817501456
• Karaca, E., Singularities of the ruled surfaces according to RM frame and natural lift curves, Cumhur. Sci. J., 43(2) (2022), 308–315. https://doi.org/10.17776/csj.1057212
• Kılıçoğlu, S¸., Hacısalihoğlu, H., On the b-scrolls with time-like generating vector in 3-dimensional Minkowski space $E^{3}_{1}$ , Beykent Univ. J. Sci. and Tech., 3(2) (2008), 55–67.
• Li, Y., Liu, S., Wang, Z., Tangent developables and Darboux developables of framed curves, Topol. Appl. 301, (2021) 107526. https://doi.org/10.1016/j.topol.2020.107526
• Li, Y., Şenyurt, S., Özdura, A., Canlı, D., The characterizations of parallel q-equidistant ruled surfaces, Symmetry, 14 (2022), 1879. https://doi.org/10.3390/sym14091879
• Li, Y., Eren, K., Ayvacı, H., Ersoy, S., The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space, AIMS Math., 8(1) 2022, 2226–2239. https://doi.org/10.3934/math.2023115
• Ouarab, S., Smarandache ruled surfaces according to Frenet-Serret frame of a regular curve in $E^{3}$, Hindawi Abst. Appl. Analy., Article ID 5526536 (2021), 8 pages.
• Ouarab, S., Smarandache ruled surfaces according to Darboux Frame in $E^{3}$, Hindawi J. Math., Article ID 9912624 (2021), 10 pages. https://doi.org/10.1155/2021/9912624
• Ouarab, S., NC-Smarandache ruled surface and NW-Smarandache ruled surface according to alternative moving frame in $E^{3}$, Hindawi J. Math., Article ID 9951434 (2021), 6 pages. https://doi.org/10.1155/2021/9951434
• Öğrenmiş, A. O., Bektaş, M., Ergüt, M., On the helices in the Galilean space $G^{3}$, Iran J. Sci. Tech., 31(A2) (2007), 177–181.
• Pottmann, H., Eigensatz, M., Vaxman, A., Wallner, J., Architectural geometry, Comp. Graph. 47 (2015), 145–164. https://doi.org/10.1016/j.cag.2014.11.002
• Pressley, A., Elementary Differential Geometry, Springer Science & Business Media, 2010. https://doi.org/10.1007/978-1-84882-891-9
• Stillwell, J., Mathematics and Its History, (Vol. 3) New York, Springer, 2010.
• Struik, D. J., Lectures on Classical Differential Geometry, Addison-Wesley Publishing Company, 1961.
• Şenyurt, S., Sivas, S., An application of Smarandache curve, Ordu Univ. J. Sci. Tech., 3(1) (2013), 46–60.
• Şenyurt, S., Canlı, D., Some special Smarandache ruled surfaces by Frenet frame in $E^{3}$-I, Turk. J. Sci., 7(1) (2020), 31–42. https://doi.org/10.5831/HMJ.2022.44.4.594
• Şenyurt, S., Eren, K., Smarandache curves of spacelike anti-Salkowski curve with a spacelike principal normal according to Frenet frame, Gümüşhane Üniv. Fen Bil. Derg., 10(1) (2020), 251–260. https://doi.org/10.17714/gumusfenbil.621363
• Şenyurt, S., Eren, K., Smarandache curves of spacelike anti-Salkowski curve with a timelike principal normal according to Frenet frame, Erzincan Univ. J. Sci. Tech., 13(2) (2020), 404–416. https://doi.org/10.18185/erzifbed.621344
• Şenyurt, S., Eren, K., Smarandache curves of spacelike Salkowski curve with a spacelike principal normal according to Frenet frame, Erzincan Univ. J. Sci. Tech., 13(special issue -I) (2020), 7–17. https://doi.org/10.18185/erzifbed.590950
• Şenyurt, S., Eren, K., Some Smarandache curves constructed from a spacelike Salkowski curve with timelike principal normal, Punjab Univ. J. Math., 53(9) (2021), 679–690.
• Şenyurt, S., Canlı, D., Çan, E., Smarandache-based ruled surfaces with the Darboux vector according to Frenet frame in $E^{3}$, J. New Theory, 39 (2022), 8–18. https://doi.org/10.53570/jnt.1106331
• Şenyurt, S., Gür, S., Grilli, L., Gaussian curvatures of parallel ruled surfaces, Appl. Math. Sci., 14(4) (2020), 173–184, https://doi.org/10.12988/ams.2020.912175.
• Taşköprü, K., Tosun, M., Smarandache curves on $S^{2}$, Bol. Soc. Paran. Mat., 32(1) (2014), 51–59.
• Turgut, M., Yılmaz, S., Smarandache curves in Minkowski space-time, Int.l J. Math. Comb., 3 (2008), 51–55.
• Yayli, Y., Saracoglu, S., On developable ruled surfaces in Minkowski space, Adv. Appl. Clifford Algebr., 22 (2012), 499–510. https://doi.org/10.1007/s00006-011-0305-5