The cycloids generalize nicely to curves on the sphere. They can be physically generated by letting one movable cone roll on a fixed cone, keeping their tips together, and tracing the motion of a point in the plane of the base of the rolling cone. Like so:
Varying the shapes of the cones will gives differently shaped cycloids, most of which will not close. When they do, they have a tremendously appeal (to me) as 3-dimensional designs, like this tent frame
or this candle holder:
In a future post I will use the following curves for another construction. Each is without self-intersection
and together they form a nice cage that from the side has an organic appearance.
I’d be interested to learn how such objects could be manufactured, say as pieces of jewelry. How does one bend metal tubes accurately?
The images are created using explicit formulas for the cycloids, but rendering approximations of spherical sweeps about cubic splines in PoVRay.