The shape of the limiting curve that eliminates the amplitude dependence of a pendulum's period is a cycloid. By deriving the equations of motion for a particle moving under gravity along a curved path, the document shows that the trajectory is a cycloid. Specifically, (1) the equation of motion along the curved path is the same as that of a simple pendulum if the path is a cycloid, (2) integrating the equations of motion yields parametric equations defining a cycloid, and (3) therefore, the limiting curve that gives a period independent of amplitude is the evolute of a cycloid.