Closed Cycloids in a Normed Plane

TL;DR

This paper studies $ extit{P}$-cycloids in normed planes, describing their properties through differential equations, classifying closed hypocycloids and epicycloids, and establishing geometric theorems related to vertices and involutes.

Contribution

It introduces the concept of $ extit{P}$-cycloids, analyzes their differential equations, classifies closed hypocycloids and epicycloids, and proves new geometric theorems for closed curves.

Findings

Classification of all closed hypocycloids and epicycloids with a given number of cusps.

Decomposition of support functions into symmetric and anti-symmetric parts.

Convergence of involutes of closed curves and generalized Sturm-Hurwitz and vertices theorems.

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.