Fractal Frenet Equations for Fractal Curves: a Fractal Calculus Approach

Abstract

The formulation of Fractal Frenet equations, which are differential equations intended to characterize the geometric behavior of vector fields along fractal curves, is presented in this study. It offers a framework for calculating the length of such irregular curves by introducing a fractal analogue of arc length. The notion of a fractal unit tangent vector, which characterizes the local direction of the curve, and the fractal curvature vector, which depicts the bending behavior at each point, are two examples of fundamental geometric ideas that are extended to the fractal environment. Furthermore, the concept of fractal torsion is established to describe the three-dimensional spatial twisting of fractal curves. © 2025 Elsevier B.V., All rights reserved.