Iterating evolutes and involutes

TL;DR

This paper investigates the iterative behavior of evolutes and involutes of plane curves, including their discretizations into polygons, analyzing their limiting dynamics and providing both theoretical insights and computer experiments.

Contribution

It introduces new analyses of evolutes and involutes for smooth and polygonal curves, including their discretizations and associated linear maps, with novel results on their dynamics and periodicity.

Findings

Discretized evolutes via circumcenters form vector bundle morphisms.

Discretized evolutes via incenters define averaging transformations.

Theoretical results supported by numerous computer experiments.

Abstract

We study iterations of two classical constructions, the evolutes and involutes of plane curves, and we describe the limiting behavior of both constructions on a class of smooth curves with singularities given by their support functions. Next we study two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices, or by the incenters of the triples of consecutive sides. The space of polygons is a vector bundle over the space of the side directions, and both kinds of evolutes define vector bundle morphisms. In both cases, we describe the linear maps of the fibers. In the first case, the induced map of the base is periodic, whereas, in the second case, it is an averaging transformation. We also study the dynamics of the related inverse constructions, the involutes of polygons. In addition to the theoretical study, we performed numerous computer experiments; some of the observations remain unexplained.