Discrete Cycloids from Convex Symmetric Polygons

Abstract

Cycloids, hipocycloids and epicycloids have an often forgotten common property: they are homothetic to their evolutes. But what if use convex symmetric polygons as unit balls, can we define evolutes and cycloids which are genuinely discrete? Indeed, we can! We define discrete cycloids as eigenvectors of a discrete double evolute transform which can be seen as a linear operator on a vector space we call curvature radius space. We are also able to classify such cycloids according to the eigenvalues of that transform, and show that the number of cusps of each cycloid is well determined by the ordering of those eigenvalues. As an elegant application, we easily establish a version of the four-vertex theorem for closed convex polygons. The whole theory is developed using only linear algebra, and concrete examples are given.

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]).…”

“…Then an inner product can be defined in C such that C sym 0 and C cw 0 become orthogonal and there exist orthonormal basis of cycloids for both subspaces, where a cycloid here means an eigenvector of the double evolute operator. In a normed plane with polygonal unit ball, by considering the class C of polygons with sides parallel to the unit ball, one can describe a similar discrete construction ( [5]).…”

A new geometric view on Sturm–Liouville eigenvalue problems

“…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).…”