Discrete cycloids from convex symmetric polygons

M. Craizer, R. Teixeira and V. Balestro, Discrete cycloids from convex symmetric polygons, Discrete and Computational Geometry 60(4), pp. 859-884, 2018. (arXiv link) (journal link)

Comments: A convex centrally symmetric polygon P in a plane naturally induces a discrete curvature concept for polygons whose sides are respectively parallel to P. Then, discrete evolutes are defined to be the polygon joining the centers of the “osculating” polygons, and discrete cycloids can be characterized as the eigenvectors of a certain linear transformation (which “plays the role” of a Sturm-Liouville operator). Interestingly, the discrete theory can be developed in a very analogous way to the continuous counterpart. This suggests that the discrete curvature considered is a very natural one.