Characterizations of Tzitzeica curves using Bishop frames

Abstract

In this study, the conditions of a regular curve to be Tzitzeica curve according to Type-1 and Type-2 Bishop frames in Euclidean 3-space are formulated. Also, the conditions for each Bishop spherical images of a regular curve to be Tzitzeica curves are obtained. Moreover, comprehensive investigations are given for a curve to be a Tzitzeica curve in the special cases of the curvatures of this curve. Also, the curves lying in different planes spanned by different elements of Type-1 and Type-2 Bishop frames are considered, and the conditions of these curves to be Tzitzeica curves are expressed in these cases.

On Framed Tzitzeica Curves in Euclidean Space

“…Corollary 2.1. i. [7] Let (γ, η 1 , η 2 ) : I → R 3 × ∆ η be a framed curve with a Bishop frame {ν, η 1 , η 2 }. Therefore, if…”

“…Moreover, these curves have the property of being centro-affine invariant [4]. The readers are referred to [1]- [7] for some interesting studies on the Tzitzeica curves and surfaces.…”

Characterizations of Tzitzeica Curves Using Conformable Frenet Frame

“…Eren K. and Ersoy E. investigate the conditions for a curve to be the Tzitzeica curve according to the Type-1 and Type-2 Bishop frame, and the conditions for being the Tzitzeica curve are expressed with respect to the curvatures of the curve. In addition, they obtained the necessary conditions for the spherical indicatrix curves defined according to both frames to be the Tzitzeica curves [13]. Fractional calculus represents a mathematical field that extends the traditional classical concepts of derivatives and integrals to non-integer orders.…”

“…Ozen K. E. and Tosun M. consider the Tzitzeica curves in 3-dimensional Euclidean space according to the PAF frame. The condition of being the Tzitzeica is characterized in terms of PAF apparatuses and some results are given [27]. Eren K. and Ersoy E. investigate the conditions for a curve to be the Tzitzeica curve according to the Type-1 and Type-2 Bishop frame, and the conditions for being the Tzitzeica curve are expressed with respect to the curvatures of the curve.…”