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Isochrone curveis the curve for which the time taken by an object sliding without friction in 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.- A ball set on an Isocrone (or Tautochrone) curve will reach the bottom at the same length of time no matter where you place the ball, so long as there is no impeding friction.

[Gif] - Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time. The blue arrows show the points’ acceleration along the curve. On the top is the time-position diagram.

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