# Discrete Cycloids from Convex Symmetric Polygons

## TL;DR

This paper introduces a discrete theory of cycloids based on convex symmetric polygons, using linear algebra to classify and analyze their properties, including cusps and a polygonal four-vertex theorem.

## Contribution

It defines discrete cycloids as eigenvectors of a linear double evolute transform, providing a novel algebraic framework for polygonal cycloids and their classification.

## Key findings

- Discrete cycloids are eigenvectors of a linear operator.
- Number of cusps is determined by eigenvalue ordering.
- A polygonal four-vertex theorem is established.

## 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.