TY - JOUR

T1 - Iterating Evolutes and Involutes

AU - Arnold, Maxim

AU - Fuchs, Dmitry

AU - Izmestiev, Ivan

AU - Tabachnikov, Serge

AU - Tsukerman, Emmanuel

N1 - Funding Information:
We are grateful to Branko Grünbaum for sending us his papers [, ], to Tim Hoffmann for sharing his lecture notes [] and for helpful remarks, and to Dmitry Khavinson for providing the reference [] and for a useful discussion. We are also grateful to numerous other colleagues for their interest and encouragement. This project was conceived and partially completed at ICERM, Brown University, during S. T.’s 2-year stay there and the other authors’ shorter visits. We are grateful to ICERM for the inspiring, creative, and friendly atmosphere. D. F. is grateful to Max-Planck-Institut in Bonn, where part of the work was done, for its invariable hospitality. Part of this work was done while I. I. held a Shapiro visiting position at the Pennsylvania State University; he is grateful to Penn State for its hospitality. I. I. was supported by ERC Advanced Grant 247029 and by an SNF Grant 200021_169391. S. T. was supported by NSF grants DMS-1105442 and DMS-1510055. E. T. was supported by NSF Graduate Research Fellowship under Grant No. DGE 1106400.
Publisher Copyright:
© 2017, Springer Science+Business Media New York.

PY - 2017/7/1

Y1 - 2017/7/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00454-017-9890-y

DO - 10.1007/s00454-017-9890-y

M3 - Article

AN - SCOPUS:85017623507

VL - 58

SP - 80

EP - 143

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 1

ER -