Characterizations of Tzitzeica curves using Bishop frames

摘要

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.