Iterating evolutes of spatial polygons and of spatial curves

TL;DR

This paper investigates the iterative properties of evolutes of spatial polygons and curves in Euclidean space, revealing linear transformations, homothety relations, and analogs of classical hypocycloids in three dimensions.

Contribution

It extends the study of evolute iterations from planar to spatial polygons and curves, providing new linear algebraic insights and continuous analogs.

Findings

Second evolute is a linear map with double eigenvalue multiplicities.

For n=m+2, the second evolute is homothetic to the original polygon.

Identifies spatial analogs of hypocycloids that are homothetic to their second evolutes.

Abstract

The evolute of a smooth curve in an m-dimensional Euclidean space is the locus of centers of its osculating spheres, and the evolute of a spatial polygon is the polygon whose consecutive vertices are the centers of the spheres through the consecutive (m+1)-tuples of vertices of the original polygon. We study the iterations of these evolute transformations. This work continues the recent study of similar problems in dimension two, see arXiv:1510.07742. Here is a sampler of our results. The set of n-gons with fixed directions of the sides, considered up to parallel translation, is an (n-m)-dimensional vector space, and the second evolute transformation is a linear map of this space. If n=m+2, then the second evolute is homothetic to the original polygon, and if n=m+3, then the first and the third evolutes are homothetic. In general, each eigenvalue of the second evolute map has double multiplicity. We also study curves, with cusps, in 3-dimensional Euclidean space and their evolutes. We provide continuous analogs of the results obtained for polygons, and present a class of curves which are homothetic to their second evolutes; these curves are spatial analogs of the classical hypocycloids.