# 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.

• What if we take the picture above and attach a small "T" to the object above on the left and the same "T" on the right rotated by 90° ? Dec 17, 2021 at 13:26
• Note that some shapes can roll straight for certain speeds only. Dec 17, 2021 at 15:11
• @AntonPetrunin: Interesting! Can you sketch an example? Dec 17, 2021 at 15:25
• Better think in terms of total energy. You can take a ball and push it inside in a neighborhood of closed path (say small circle). So if the energy is near minimum, you can only run along this circle. But if energy is higher, then you can run straight since assuming you keep a mirror symmetry of the ball. Dec 17, 2021 at 15:40
• Do you wish to include cylindrical shapes, for example an extruded ellipse $E$, or should there be a single point of contact at all times? Also for $E$: Is the center of gravity allowed to change in height, because this requires a minimal kinetic energy? Dec 20, 2021 at 8:49

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.

• Nice point re extruded constant-width curves to $\mathbb{R}^3$. Dec 18, 2021 at 1:23
• @Joseph O'Rourke: I only learnt about them recently. I'm surprised that they're not better known! Dec 18, 2021 at 1:34
• @Mozibur Ullah You say " It is a theorem that such curves are always convex". Which theorem are you referring to? We can define "constant width" for certain nonconvex curves". For instance a hypocycloid with three cusps is of constant width 0: see page 3 of this paper hal.archives-ouvertes.fr/hal-03265440/document Dec 21, 2021 at 19:51

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.

• if one is also interested in rolling motions with periodic sideways deviation from the straight line, then the Oloid and the Sphericon are worthy examples Dec 25, 2021 at 16:57