On Tzitzeica Curves According to Q-Frame in Euclidean 3-Space

Abstract

Tzitzeica curve is a special spatial curve, which introduced Gheorghe Tzitzeica in 1911. Gheorghe Tzitzeica defined this curve as follows; for a spatial curve ϕ, "if the ratio of its torsion τ and the square of the distance d from the origin to the osculating plane at an arbitrary point of the curve is constant, then this spatial curve is a Tzitzeica curve in Euclidean space." Moreover Gheorghe Tzitzeica defined a special surface which is named Tzitzeica surface in 1907. In this surface, the asymptotic lines of Tzitzeica surfaces with negative Gaussian curvature are Tzitzeica curves. Also, the ratio of its Gaussian curvature K and the distance d from the origin to the tangent plane at any arbitrary point of the surface is constant. In this paper, we study the Tzitzeica curve via q-frame in E^3. Firstly, we redefine the Tzitzeica curve q-frame in Euclidean 3- space. Then, it is obtained some conditions for the Tzitzeica curve as a spherical curve. Finally, we investigate harmonic vector cases of the Tzitzeica curve according to q- frame in Euclidean space.

Keywords

• Tzitzeica curve
• q-frame
• Spherical curve
• Tzitzeica curve, q-frame, Spherical curve

References

1. Sariaydin, M.T & Yazla, A. (2022). On Tzitzeica curves according to Q-Frame in Euclidean 3-Space. The Eurasia Proceedings of Science, Technology, Engineering & Mathematics (EPSTEM), 17, 19-25.