A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage. It is an example of a roulette, a curve generated by a curve rolling on another curve.

The inverted cycloid (a cycloid rotated through 180°) is the solution to the brachistochrone problem (i.e., it is the curve of fastest descent under gravity) and the related tautochrone problem (i.e., the period of an object in descent without friction inside this curve does not depend on the object's starting position).