B-Lift curves and its ruled surfaces

In this paper, we have described the B-Lift curve in Euclidean space as a curve obtained by combining the endpoints of the binormal vector of a unit speed curve. Subsequently, we have explored the Frenet frames of the B-Lift curves. Moreover, we have introduced the tangent, normal and binormal surfaces of the B-Lift curve and examined the geometric invariants of these surfaces. Finally, we have investigated the singularities of these surface and visualized the surfaces with MATLAB program.

Keywords:

B-Lift, ruled surface, minimal surface developable surface, singular point,

PDF

___

Neill, B. O., Elementary Differential Geometry, Academic Press New York and London, 1966.
Ergün, E., Çalışkan, M., Ruled surface pair generated by a curve and its natural lift in $R^3$, Pure Mathematical Sciences, 1 (2012), 75-80.
Izumiya, S., Takeuchi, N., Special curves and ruled surfaces, Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry, 44 (1) (2003), 203-212.
Karaca, E., Çalışkan, M., Ruled surfaces and tangent bundle of pseudo-sphere of natural lift curves, Journal of Science and Arts, 52 (3) (2020), 573-586. https://doi.org/10.35378/gujs.588496
Kim, Y. H., Yoon, D. W., Classification of ruled surfaces in Minkowski 3-Spaces, Journal of Geometry and Physics, 49(1) (2004), 89-100. https://doi.org/10.1016/S0393-0440(03)00084-6
Yıldız, Ö. G., Akyiğit, M., Tosun, M., On the trajectory ruled surfaces of framed base curves in the Euclidean Space, Mathematical Methods in the Applied Sciences, 44 (9) (2021), 7463-7470. https://doi.org/10.1002/mma.6267
Hathout, F., Bekar, M., Yaylı, Y., Ruled surfaces and tangent bundle of unit 2-sphere, International Journal of Geometric Methods in Modern Physics, 14 (10) (2017), 1750145. https://doi.org/10.1142/S0219887817501456
Şentürk, G. Y., Yüce, S., Bertrand offsets of ruled surfaces with Darboux frame, Results in Mathematics, 72 (2017), 1151-1159. https://doi.org/10.1007/s00025-016-0571-6
Thorpe, J. A., Elementary Topics in Differential Geometry, Springer Verlag, New York, Heidelberg-Berlin, 1979.
Carmo, M. D., Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1976.
Izumiya, S., Takeuchi, N., Singularities of ruled surfaces in $R^3$, Math. Proc. Camb. Phil. Soc., 130(1) (2001), 1-11. https://doi.org/10.1017/S0305004100004643
Perez-Diaz, S., Blasco, A., On the computation of singularities of parametrized ruled surfaces, Advances in Applied Mathematics, 110 (2019), 270-298. https://doi.org/10.1016/j.aam.2019.07.005
Martins, R., Nuno-Ballesteros, J. J., Finitely determined singularities of ruled surfaces in R3, Math. Proc. Camb. Phil. Soc., 147 (2009), 701-733. https://doi.org/10.1017/S0305004109002618
Jia, X., Chen, F., Deng, J., Computing self-intersection curves of rational ruled surfaces, Computer Aided Geometric Design, 26 (3) (2009), 287-299. https://doi.org/10.1016/j.cagd.2008.09.005
Alghanemi, A., On geometry of ruled surfaces generated by the spherical indicatrices of a regular space curve II, International Journal of Algebra, 10 (4) (2016), 193-205. https://doi.org/10.12988/ija.2016.6321
Mond, D., Singularities of the tangent developable surface of a space curve, The Quarterly Journal of Mathematics, 40 (1) (1989), 79-91. https://doi.org/10.1093/qmath/40.1.79
Lancret, M. A., Memoire sur les courbes a double courbure, Memoires presentes a l’Institut 1, 1806.