TY - JOUR

T1 - Iterating evolutes of spacial polygons and of spacial curves

AU - Fuchs, Dmitry

AU - Tabachnikov, Serge

N1 - Funding Information:
Received December 28, 2016; in revised form May 16, 2017. The second named author was supported by NSF grant DMS-1510055 and by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.
Publisher Copyright:
© 2017 Independent University of Moscow.

PY - 2017/10/1

Y1 - 2017/10/1

N2 - The evolute of a smooth curve in an m-dimensional Eu-clidean 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 evo-lute 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.

AB - The evolute of a smooth curve in an m-dimensional Eu-clidean 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 evo-lute 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.

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U2 - 10.17323/1609-4514-2017-17-4-667-689

DO - 10.17323/1609-4514-2017-17-4-667-689

M3 - Article

AN - SCOPUS:85036583361

SN - 1609-3321

VL - 17

SP - 667

EP - 689

JO - Moscow Mathematical Journal

JF - Moscow Mathematical Journal

IS - 4

ER -