Applications of fractional derivative to affine differential geometry of curves

Chain rule, the derivative formula of composite functions, is essential for the studies on differential geometry of parametric objects. This necessity poses an important obstacle for applying various fractional derivative operators existing in the literature to the studies on differential geometry of parametric curves. A simplification about the Caputo fractional derivative of composite functions was put forward by mathematicians to overcome this obstacle. Euclidean invariants of parametric plane curves, namely Frenet curvature and vectors, were considered through a technique using this simplification. Thanks to this technique, a new curvature of Frenet type could be defined, but it is not possible to introduce new vectors of Frenet type. Besides this is a remarkable contribution in itself, it is expected that the fractional derivative vectors of parametric curves are different from the classical ones in order to better observe geometric properties of fracitonal derivative. Inspired by this expectation the idea of studying equiaffine invariants of parametric plane curves via Caputo fractional derivative was put forward. This is because, by the mentioned technique, new equiaffine curvature and vectors of Frenet type could be obtained and hence a better geometric observation of fractional derivative could be presented. Following this strategy, in this thesis the equiaffine Frenet invariants of parametric space curves are re-analyzed via the technique. More clearly, new equiaffine Frenet curvatures and vectors are introduced. The relations between the new and classical invariants are provided. The examples are given by graphs and figures. Furthermore new perspectives and problems are expressed by the used technique in this thesis.