Iterating Evolutes and Involutes

Maxim Arnold, Dmitry Fuchs, Ivan Izmestiev, Serge Tabachnikov, Emmanuel Tsukerman

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

This paper concerns iterations of two classical geometric constructions, the evolutes and involutes of plane curves, and their discretizations: evolutes and involutes of plane polygons. In the continuous case, our main result is that the iterated involutes of closed locally convex curves with rotation number one (possibly, with cusps) converge to their curvature centers (Steiner points), and their limit shapes are hypocycloids, generically, astroids. As a consequence, among such curves only the hypocycloids are homothetic to their evolutes. The bulk of the paper concerns two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices (P-evolutes), or by the incenters of the triples of consecutive sides (A-evolutes). For equiangular polygons, the theory is parallel to the continuous case: we define discrete hypocycloids (equiangular polygons whose sides are tangent to hypocycloids) and a discrete Steiner point. The space of polygons is a vector bundle over the space of the side directions; our main result here is that both kinds of evolutes define vector bundle morphisms. In the case of P-evolutes, the induced map of the base is 4-periodic, and the dynamics reduces to the linear maps on the fibers. We prove that the spectra of these linear maps are symmetric with respect to the origin. The asymptotic dynamics of linear maps is determined by their eigenvalues with the maximum modulus, and we show that all types of behavior can occur: in particular, hyperbolic, when this eigenvalue is real, and elliptic, when it is complex. We also study P- and A-involutes and prove that the side directions of iterated A-involutes of polygons with odd number of sides behave ergodically; this generalizes well-known results concerning iterations of the construction of the pedal triangle. In addition to the theoretical study, we performed numerous computer experiments; some of the observations remain unexplained.

Original languageEnglish (US)
Pages (from-to)80-143
Number of pages64
JournalDiscrete and Computational Geometry
Volume58
Issue number1
DOIs
StatePublished - Jul 1 2017

Fingerprint

Hypocycloid
Polygon
Linear map
Equiangular
Steiner Point
Vector Bundle
Pedal triangle
Consecutive
Circumcentre
Discretization
Eigenvalue
Incentre
Convex Curve
Iteration
Curve
Rotation number
Computer Experiments
Odd number
Plane Curve
Cusp

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Cite this

Arnold, Maxim ; Fuchs, Dmitry ; Izmestiev, Ivan ; Tabachnikov, Serge ; Tsukerman, Emmanuel. / Iterating Evolutes and Involutes. In: Discrete and Computational Geometry. 2017 ; Vol. 58, No. 1. pp. 80-143.
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Arnold, M, Fuchs, D, Izmestiev, I, Tabachnikov, S & Tsukerman, E 2017, 'Iterating Evolutes and Involutes', Discrete and Computational Geometry, vol. 58, no. 1, pp. 80-143. https://doi.org/10.1007/s00454-017-9890-y

Iterating Evolutes and Involutes. / Arnold, Maxim; Fuchs, Dmitry; Izmestiev, Ivan; Tabachnikov, Serge; Tsukerman, Emmanuel.

In: Discrete and Computational Geometry, Vol. 58, No. 1, 01.07.2017, p. 80-143.

Research output: Contribution to journalArticle

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