# Moscow Mathematical Journal

Volume 17, Issue 4, October–December 2017 pp. 667–689.

Iterating Evolutes of Spacial Polygons and of Spacial Curves

**Authors**:
Dmitry Fuchs (1) and Serge Tabachnikov (2)

**Author institution:**(1) Department of Mathematics, University of California, Davis, CA 95616

(2) Department of Mathematics, Pennsylvania State University, University Park, PA 16802

**Summary: **

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 spacial 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. 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 non-zero eigenvalue of the second evolute map has even 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 spacial analogs of the classical
hypocycloids.

2010 Math. Subj. Class. 52C99, 53A04.

**Keywords:**Evolute, osculating sphere, hypocycloid, discrete differential geometry.

Contents Full-Text PDF