Monatshefte für Mathematik

, Volume 185, Issue 1, pp 43–60 | Cite as

Closed cycloids in a normed plane

Article

Abstract

Given a normed plane \(\mathcal {P}\), we call \(\mathcal {P}\)-cycloids the planar curves which are homothetic to their double \(\mathcal {P}\)-evolutes. It turns out that the radius of curvature and the support function of a \(\mathcal {P}\)-cycloid satisfy a differential equation of Sturm–Liouville type. By studying this equation we can describe all closed hypocycloids and epicycloids with a given number of cusps. We can also find an orthonormal basis of \({\mathcal C}^0(S^1)\) with a natural decomposition into symmetric and anti-symmetric functions, which are support functions of symmetric and constant width curves, respectively. As applications, we prove that the iterations of involutes of a closed curve converge to a constant and a generalization of the Sturm–Hurwitz Theorem. We also prove versions of the four vertices theorem for closed curves and six vertices theorem for closed constant width curves.

Keywords

Minkowski geometry Sturm–Liouville equations Evolutes Hypocycloids Curves of constant width Sturm–Hurwitz theorem Four vertices theorem Six vertices theorem 

Mathematics Subject Classification

52A10 52A21 53A15 53A40 

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Copyright information

© Springer-Verlag Wien 2017

Authors and Affiliations

  • Marcos Craizer
    • 1
  • Ralph Teixeira
    • 2
  • Vitor Balestro
    • 3
    • 4
  1. 1.Departamento de MatemáticaPUC-RioRio de JaneiroBrazil
  2. 2.Departamento de Matemática AplicadaUFFNiteróiBrazil
  3. 3.Instituto de Matemática e EstatísticaUFFNiteróiBrazil
  4. 4.CEFET/RJNova FriburgoBrazil

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