Comformable derivative curves in the plane

The theory of curves in differential geometry has become a widely attractive field of study. But recently, using the Uyumlu derivative in the theory of curves has given a different perspective to a large number of scientists. The aim of this doctoral thesis is to obtain the characterization of the basic properties of a plane curve in two-dimensional space using the Uyumlu derivative. In this doctoral thesis study, important definitions and theorems related to comformable derivative are given at the beginning. Afterward, the arc length of a plane curve is presented in detail using the Frenet-Serret equations and the fractional curvature Uyumlu derivative. Then, methods for finding multivariate functions corresponding to the bisector curvatures of two plane curves in two-dimensional space are given. In addition, the position vector and curvature of the congruent curve with α-Frenet frame are obtained using the Uyumlu derivative. Finally, congruence-compatible curves formed by congruent curves with the help of their own Frenet vectors are defined, their properties are studied and proved under some special cases. The suggestion about the method used during the thesis study, the contribution of the obtained results to the literature and the problems are presented in the last part.