As is well-known, the cycloid is the curve described by a point P rigidly attached to a circle C that rolls, without sliding, on a fixed line AB (fig. 1). The full arc ABD has a length equal to 8r (r = the radius of the generating circle), and the surface included between one complete arc and the fixed line is 3pr2 . The cycloid curve is "brachistochrone", i.e. a curve of least time: given two points A, B in a vertical plane, a heavy point will take the least time to travel from A to B if it is displaced along an arc of a cycloid. It is also an "isochrone" curve, i.e. a curve of equal time. A heavy point which travels along an arc of cycloid placed in a vertical position with the concavity pointing upwards will always take the same amount of time to reach the lowest point, independent of the point from which it was released.

Illustration taken from the Enciclopedia Treccani.