Developable roller

A developable roller is a kind of convex curved 3D solid that can roll indefinitely on a flat plane without stopping such that eventually every point on its surface will come in contact with the plane. The surface of a developable roller consists of a single curved "face" that always touches the plane at a line segment, which may vary in length.

Classes
There are several known classes of developable rollers.

Prime polysphericons
The ($p$,$k$)-polysphericon is constructed by taking the solid of rotation of a regular $p$-gon, bisecting it along an $p$-gon, rotating one of the halves by 2$k&pi;/n$, and recombining the two halves along their $p$-gonal faces. A (2$p$,$k$)-polysphericon is a developable roller iff $$\gcd(p,k)=1$$. A ($p$,$l$)-polysphericon where $p$ is odd, is never a developable roller. The result can still have a single developable face, however it will not roll indefinitely because the solid of rotation has a circular face where the axis of rotation intersects an edge which results in culs-de-sac preventing the polysphericon from rolling.

The polysphericons which are developable rollers are called prime polysphericons. The prime polysphericons correspond 1-to-1 with the regular polygons (including the degenerate digon) with the $n/k$-gon corresponding to the (2$n$,$k$)-polysphericon.

Polycons
The polycons are an infinite class of developable rollers that generalize the construction of the sphericon. The $n$-polycon can be constructed as follows:
 * Start with a cone with apex angle equal to the internal angle of a regular 2$n$-gon.
 * Arrange $n$ copies of the cone such that their apices lie on the vertices of a regular $n$-gon and their axes pass through the origin.
 * Take the intersection of the cones.
 * Divide the result in half along the plane which intersects the apices of the cones.
 * Rotate one of the halves by /$n$.
 * Recombine the two halves to form the polycon.

This construction always results in a developable roller of $n$ > 1. The smallest polycon is the sphericon with $n$=2. The edges of polycons are conic sections, with the sphericon's edges being circular, the hexacon's edges being parabolic and all other polycons having hyperbolic edges.

Platonicons
The platonicons are a class of developable rollers based on the platonic solids.