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.
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Communicated by A. Constantin.
Marcos Craizer wants to thank CNPq for financial support during the preparation of this manuscript.
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Craizer, M., Teixeira, R. & Balestro, V. Closed cycloids in a normed plane. Monatsh Math 185, 43–60 (2018). https://doi.org/10.1007/s00605-017-1030-5
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DOI: https://doi.org/10.1007/s00605-017-1030-5
Keywords
- Minkowski geometry
- Sturm–Liouville equations
- Evolutes
- Hypocycloids
- Curves of constant width
- Sturm–Hurwitz theorem
- Four vertices theorem
- Six vertices theorem