Conformable Differential Geometry of Some Special Curves in the Equi-Affine Space

This study investigates the conformable differential geometry of some special curves defined in the equi-affine space. The conformable derivative, a generalization of fractional calculus, is a flexible operator controlled by a parameter $\alpha$ that enables the modeling of nonlocal behavior in functions. This paper aims to offer a new perspective by combining the modern concept of derivative with equi-affine differential geometry. First, this paper introduces the conformable equi-affine arc length parameter and the corresponding conformable Frenet frame in the equi-affine space. The main focus is on characterizing special classes of curves, such as helices, slant helices, and rectifying curves, within the conformable equi-affine frame. These results expand the geometric applications of conformable calculus and provide a broader theoretical framework for curve analysis in equi-affine geometry. Finally, the accuracy of the results is observed with an example, and the curvatures are plotted as a function of $\alpha$ with MATLAB R2022b.