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 Mosc. Math. J., 2017, Volume 17, Number 4, Pages 667–689 (Mi mmj652)

Iterating evolutes of spacial polygons and of spacial curves

Dmitry Fuchsa, Serge Tabachnikovb

a Department of Mathematics, University of California, Davis, CA 95616
b Department of Mathematics, Pennsylvania State University, University Park, PA 16802

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 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.

Key words and phrases: evolute, osculating sphere, hypocycloid, discrete differential geometry.

DOI: https://doi.org/10.17323/1609-4514-2017-17-4-667-689

Full text: http://www.mathjournals.org/.../2017-017-004-005.html
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Bibliographic databases:

MSC: 52C99, 53A04
Language:

Citation: Dmitry Fuchs, Serge Tabachnikov, “Iterating evolutes of spacial polygons and of spacial curves”, Mosc. Math. J., 17:4 (2017), 667–689

Citation in format AMSBIB
\Bibitem{FucTab17}
\by Dmitry~Fuchs, Serge~Tabachnikov
\paper Iterating evolutes of spacial polygons and of spacial curves
\jour Mosc. Math.~J.
\yr 2017
\vol 17
\issue 4
\pages 667--689
\mathnet{http://mi.mathnet.ru/mmj652}
\crossref{https://doi.org/10.17323/1609-4514-2017-17-4-667-689}