That curve upon which a particle moving under the action of gravity (or any given force) will reach the lowest (or some fixed) point in the same time, from whatever point it starts: A tautochrone or "isochrone" curve is the curve for which the time taken by a frictionless particle sliding down it under uniform gravity to its lowest point is independent of its starting point. The curve is a cycloid, and the time is equal to π times the square root of the radius over the acceleration of gravity.