A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line. It is an example of a roulette, a curve generated by a curve rolling on another curve.
The cycloid 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).
Read more about Cycloid: History, Equations, Area, Arc Length, Cycloidal Pendulum, Related Curves, Use in Architecture, Use in Violin Plate Arching