Which convex bodies roll straight? Let $K$ be a convex body in $\mathbb{R}^3$.
Suppose $K$ is held at some position and orientation on an inclined plane,
and released.
Let there be sufficient friction so that it rolls without slippage.
My question is:

Q. If $K$ rolls along a straight line, i.e., if the point of contact along
the inclined plane is a single straight line, what can we conclude about the shape of $K$?
In other words, which $K$, when properly oriented, roll straight?


         
   
(Figure from Which convex bodies roll along closed geodesics?)

It seems that if $K$ is a smooth surface of revolution about an axis $X$,
and $K$ has reflective symmetry about a plane orthogonal to $X$ (as in the
above illustration), then $K$ rolls straight.
But perhaps a wider class of bodies also roll straight.
Perhaps reflective symmetry is not necessary; perhaps equal moments of inertia
about $X$ in the two halves suffice? Or would any asymmetry cause a wobble
in the footprint?
I would be interested in learning of any class of shapes that roll straight, especially
non-symmetric shapes.
 A: Have you considered the curves of constant width? These are curves where the minimum distance between two parallel and tangential lines are the same no matter what the orientation of the line is.
These when turned into cylinders will roll straight when a flat plate is set on them. Examples of such curves are the circle, which is standard, and the reuleaux triangle, which is not. It is a theorem that such curves are always convex and that by Barbier's theorem their circumference is pi x diameter. Moreover, a tangent is always perpendicular. Also, by the Blashke-Lesbegue theorem, the Releaux triangles have the least area of any curve of constant width.
Wikipedia has a good page on them as well as surfaces of constant width. These include the Meissner solids and a flat plate on several of these will roll straight in any direction. There are also visualisations on youtube that you might find useful.
A: Assuming that we are considering idealized physical objects, then a criterion for convex bodies to roll on straight lines would be the existence of closed planar geodesics of which the containing plane is orthogonal to an inertial axis and that also contains the center of gravity.
Instable equilibria do not pose a problem; in the real world a sufficiently high rotational velocity will provide the necessary stability.
Another physical effect is that the curve on which the body rolls may depend on speed whenever the center of gravity is not contained in the plane in which the closed "contact geodesic" is contained: take a cone that is rotating around its axis of symmetry; its rolling motion will become more linear with increasing rotational velocity.
What also must be guaranteed for linear rolling motion is that the convex body must be oriented in a way that renders the inertial axis parallel to the (tangent of the) plane's height-lines at the initial point of contact.
