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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


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 (nm)-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.

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