The moving frame on the fractal curves

TL;DR

This paper introduces a method using moving frames and affine curvatures to uniquely identify and quantify fractal curves like Koch and Hilbert, revealing their iterative regularities and enabling direct generation from initial points.

Contribution

It develops a novel approach to describe fractal curves through affine curvatures, allowing for quantification, encoding, and direct iterative generation of fractals.

Findings

Affine curvatures characterize fractal regularities.

Fractal curves can be quantified and encoded as sequences.

Direct generation of fractals from initial points is possible.

Abstract

Using the moving frame and invariants, any discrete curve in $\R^3$ could be uniquely identified by its centroaffine curvatures and torsions. In this paper, depending on the affine curvatures of the fractal curves, such as Koch curve and Hilbert curve, we can clearly describe their iterative regularities. Interestingly, by the affine curvatures, the fractal curves can be quantified and encoded accordingly to a sequence. Hence, it is more convenient for future reference. Given three starting points, we can directly generate the affine Koch curve and affine Hilbert curve at the step $n, \forall n\in \mathbb{Z}^+$. Certainly, if the initial three points are standard, the curve is the traditional Koch curve or Hilbert curve. By this method, the characteristic of some fractal curves which look like irregular could be quantified, and the regularities would become more obvious.