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
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.
MSC: 52C99, 53A04
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
\by Dmitry~Fuchs, Serge~Tabachnikov
\paper Iterating evolutes of spacial polygons and of spacial curves
\jour Mosc. Math.~J.
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