Closed cycloids in a normed plane

M. Craizer, R. Teixeira and V. Balestro, Closed cycloids in a normed plane, Monatshefte für Mathematik 185(1), pp. 43-60, 2018. (arXiv link) (journal link)

Comments: The centers of curvature of a curve in a smooth Minkowski plane can be defined in a completely analogous manner as in the usual Euclidean case. The curve obtained joining these points is, therefore, the Minkowski evolute of the original curve. Doing this process again, but now with respect to the dual norm, yields the bi-evolute of a curve. Cycloids are then defined to be the curves which are homothet to their bi-evolutes. Their curvature radius functions are characterized as the eigenvectors of a certain Sturm-Liouville operator, and geometric data can be obtained from analysis of this operator.