Fractal Calculus to Derive Fractal Frenet Equations for Fractal Curves

TL;DR

This paper develops a new mathematical framework called Fractal Frenet equations to describe the geometric properties of fractal curves, extending classical differential geometry concepts to fractal geometry.

Contribution

It introduces the concept of Fractal Frenet equations, providing a novel way to analyze curvature, torsion, and other geometric features of fractal curves.

Findings

Defined fractal analogues of tangent, curvature, and torsion vectors.

Applied the equations to fractal examples like the helix and snowflake.

Extended differential geometry tools to fractal curves.

Abstract

This paper introduces the concept of Fractal Frenet equations, a set of differential equations used to describe the behavior of vectors along fractal curves. The study explores the analogue of arc length for fractal curves, providing a measure to quantify their length. It also discusses fundamental mathematical constructs, such as the analogue of the unit tangent vector, which indicates the curve's direction at different points, and the analogue of curvature vector or fractal curvature vector, which characterizes its curvature at various locations. The concept of torsion, describing the twisting and turning of fractal curves in three-dimensional space, is also explored. Specific examples, like the fractal helix and the fractal snowflake, illustrate the application and significance of the Fractal Frenet equations.