2017/02/02 by Marcos Craizer, Craizer, Marcos, Ralph Teixeira +3 · 1 voice
Computer Science · Engineering · Mathematics · #39A06 #39A14 #39A23 #52C05 #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications #Topological and Geometric Data Analysis #math.DG #msc:39A06 #msc:39A14 #msc:39A23 #msc:52C05
paper · pdf · doi:10.48550/arxiv.1702.00522
23 pages, 7 figures
openalex publication_date 2017/02/02 · arxiv published 2017/02/02 · arxiv created 2017/02/07 · arxiv updated 2017/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.