How regular is the evolute of a plane curve?

TL;DR

This paper investigates how the smoothness of a plane curve relates to that of its evolute, revealing that generically the evolute's regularity is one order less than the original curve, even with limited differentiability.

Contribution

It provides new insights into the regularity relationship between plane curves and their evolutes, especially for curves with limited smoothness.

Findings

Evolutes are generally one order less smooth than their parent curves.

The regularity relationship holds even when the original curve is not highly differentiable.

The study extends understanding of evolute properties in less smooth contexts.

Abstract

We study the relationship between the smoothness of a plane curve and that of its evolute, especially in the cases where the parent curve is no more two or three times continuously differentiable, and exhibit the same kind of apparent improvement in regularity: in the generic local situation, the evolute has one order of regularity less than the parent curve.