\(\mathcal{C}_\alpha\)-Rectifying Curves in a New Conformable Differential Geometry

In this study we reintroduce the theory of curves by incorporating local fractional calculus. We elucidate the condition for a naturally parametrized curve to be conformable, we define the orthonormal conformable frame of such a curve at any given point. Then we provide a comprehensive explanation of how these newly derived conformable geometric concepts are related to their classical counterparts. Furthermore, we introduce the concept of a conformable rectifying curve and provide its characterizations in terms of this differentiation with respect to arbitrary order. Some illustrative graphs are provided.