Regular Polygon Rolling on a Catenary

On a linked set of catenaries, all regular polygons can roll without slipping — except for the triangle, which intersects the curve near its corners. A square can roll at a constant angular velocity (blue outlines). However, if it also moves at a constant horizontal velocity (red outlines), it must cut into the curve. The same applies to a pentagon. However, hexagons can simultaneously roll at a constant horizontal and a constant angular velocity, as can any other regular polygon with more sides.