On Vectorial Moment of the Darboux Vector

Year 2020, Volume: 8 Issue: 1, 144 - 151, 15.04.2020

Süleyman Şenyurt Hülya Şardağ Osman Çakır

Abstract

In this paper we define a new curve denoted by (c*). It is well known that any regular curve can be written by means of Frenet vectors and also via the vectorial moments. In a space we know a regular curve moves around an instantaneous rotation vector called as the Darboux vector. In this study we are interested in a curve plotted by the vectorial moment of the unit Darboux vector. The curve on which we worked generated by the vectorial moment of the unit Darboux vector satisfying the following condition that the curve is created by the vectorial moment of the unit Darboux vector whose components are of the Frenet vectors of a regular curve in Euclidean 3-space. We use c* to denote the vectorial moment vector of the unit Darboux vector and also c to denote the unit Darboux vector. We show that the new curve (c*) doesn't form a constant width curve pairs with the main curve. Then we calculate the Frenet apparatus of the regular curve (c*), drawn by the vectorial moment vector of c*. Also we point out that this new curve (c*) can be expressed as a linear combination of Frenet vectors. Further we assert that the principle normal and binormal of the curve (c*) doesn't form a constant width curve pairs with the main curve. Finally we draw a conclusion and compute the Frenet apparatus of the curve (c*) when the main curve is supposed to be an helix.

Keywords

Darboux vector; vectorial moment; constant width curve pair; regular curve

References

• [1] B. Altunkaya, L. Kula, General helices that lie on the sphere S2n in Euclidean space E2n+1, Univers. J. Math. Appl. 1(3) (2018), 166-170.
• [2] E. Azizpour, D. M. Ataei, Geometry of bracket-generating distributions of step 2 on graded manifolds, Univers. J. Math. Appl. 1(3) (2018), 196-201.
• [3] B. Altunkaya, L. Kula, Characterizations of slant and spherical helices due to pseudo-Sabban frame, Fundam. J. Math. Appl. 1(1) (2018), 49-56.
• [4] S. Senyurt, B. Oztu¨rk, Smarandache Curves According to Sabban Frame of the anti-Salkowski Indicatrix Curve, Fundam. J. Math. Appl., 2(2) (2019), 101 - 116.
• [5] S. Senyurt, Y. Altun, Smarandache Curves of the Evolute Curve According to Sabban Frame, Commun. Adv. Math. Sci. 3(1) (2020), 1 - 8.
• [6] T. Erisir, M. A. Gungor, Holditch-Type Theorem for Non-Linear Points in Generalized Complex Plane Cp, Univers. J. Math. Appl., 1(4) (2018), 239 - 243.
• [7] A. Zulfigar, Some Characterization of Curves of Constant Breadth in En Space, Turk J. Math., 25(2001), 433–444.
• [8] C. Bang-Yen, Constant ratio Hypersurface, Soochow J.Math., 27(2001), 353–362.
• [9] F. Werner , On The Differential Geometry of Closed Space Curves, Bulletin of American Mathematical Society, 57(1951), 44–54.
• [10] G. Herman, Higher Curvatures of Curves in Euclidean Space, The American Mathematical Monthly, 73(1966), 699–704.
• [11] d. C. Mantredo P., Differential geometry of curves and surfaces, Prentice-Hall Englewood Cliffs, NJ MATH Google Scholar, (1976).
• [12] T. Yılmaz, Vectorial moments of curves in Euclidean 3-space, International Journal of Geometric Methods in Modern Physics, 14(2017), 1750020.