Closed cycloids in a normed plane

Abstract

Given a normed plane P, we call P-cycloids the planar curves which are homothetic to their double P-evolutes. It turns out that the radius of curvature and the support function of a 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 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.Mathematics Subject Classification (2010). 52A10, 52A21, 53A15, 53A40.

Concepts of curvatures in normed planes

“…As we will see, this can be regarded as the inverse of the radius of the osculating circle (which is a 2nd-order contact Minkowski circle attached to the curve at the considered point). This curvature type was also used by Craizer [10], Craizer et al [12] and Ghandehari [16,17]. Guggenheimer [20] works with an osculating anti-circle, obtaining, as we shall see, the isoperimetric curvature.…”

Isoperimetric Inequalities in Normed Planes

“…for some scalar function r(t). We remark that when u is smooth, the class C includes all the convex smooth curves ( [4]), and when u is polygonal, the class C consists of all polygons with sides parallel to those of u ( [5]).…”

A new geometric view on Sturm–Liouville eigenvalue problems

“…Notice that this is unique up to translation. Now we investigate the conditions for the solutions of (3.2) under which the corresponding cycloid is a closed curve from a viewpoint slightly different to that used in [8]. This approach is somehow the inspiration for Theorem 6.1.…”

“…Remark 6.1. In [8] it was proved that if λ = 1 is a double eigenvalue of the Sturm-Liouville problem, then the indicatrix (constructed from the solutions) is locally strictly convex.…”

“…This can also be extended to normed planes, but the bi-evolute is then calculated with respect to the anti-norm. This approach was taken in [8], where the classical Sturm-Liouville theory was used to investigate the existence and the properties of closed cycloids (see also [9] for the discrete version).…”