Moscow Mathematical Journal
Volume 17, Issue 4, October–December 2017 pp. 667–689.
Iterating Evolutes of Spacial Polygons and of Spacial Curves
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.
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:
Keywords: Evolute, osculating sphere, hypocycloid, discrete differential geometry.
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