### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 667-689 |

Number of pages | 23 |

Journal | Moscow Mathematical Journal |

Volume | 17 |

Issue number | 4 |

State | Published - Oct 1 2017 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Moscow Mathematical Journal*,

*17*(4), 667-689.

}

*Moscow Mathematical Journal*, vol. 17, no. 4, pp. 667-689.

**Iterating evolutes of spacial polygons and of spacial curves.** / Fuchs, Dmitry; Tabachnikov, Sergei.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Iterating evolutes of spacial polygons and of spacial curves

AU - Fuchs, Dmitry

AU - Tabachnikov, Sergei

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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M3 - Article

AN - SCOPUS:85036583361

VL - 17

SP - 667

EP - 689

JO - Moscow Mathematical Journal

JF - Moscow Mathematical Journal

SN - 1609-3321

IS - 4

ER -