Differentiability of fractal curves

TL;DR

This paper investigates the differentiability properties of plane self-affine curves, revealing conditions under which they are differentiable or continuously differentiable, and establishing the rarity of twice differentiable self-affine curves.

Contribution

It characterizes the differentiability of self-affine curves with two pieces, identifying parameter sets for differentiability and proving the absence of twice differentiable self-affine curves in the plane.

Findings

Open subset of parameters yields differentiability at all but countable points.

Parameter set of codimension one results in continuous differentiability.

No twice differentiable self-affine curves exist in the plane except parabolic arcs.

Abstract

While self-similar sets have no tangents at any single point, self-affine curves can be smooth. We consider plane self-affine curves without double points and with two pieces. There is an open subset of parameter space for which the curve is differentiable at all points except for a countable set. For a parameter set of codimension one, the curve is continuously differentiable. However, there are no twice differentiable self-affine curves in the plane, except for parabolic arcs.