cycloid
Curve traced by a point on a rolling circle; solves both tautochrone and brachistochrone.
Definition
A cycloid is the path traced by a point on the rim of a circle as it rolls along a straight line without slipping. In parametric form, x = r(t − sin t) and y = r(1 − cos t), where r is the radius of the rolling circle and t is the angle through which it has turned.
The cycloid has two remarkable physical properties. First, it is the tautochrone: a frictionless bead sliding down a cycloidal track reaches the bottom in the same time regardless of where it starts. Second, it is the brachistochrone: of all smooth curves connecting two points at different heights, the cycloid is the one along which a bead under gravity descends in the least time. Johann Bernoulli posed the brachistochrone challenge in 1696; Newton, Leibniz, l'Hôpital, and Jakob Bernoulli all solved it.
These properties made the cycloid central to seventeenth- and eighteenth-century mathematics and directly influenced the development of the calculus of variations.