Institute and Museum of History of Science, Florence, ITALY
|The cycloid curve
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.
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