The Tautochrone Curve

The tautochrone is the curve in which a ball, positioned anywhere on the curve, will take the same time to arrive at its lowest point.

tautochrone_curve
Source: Wikipedia

In a physical model where uniform gravity is the only force acting on the ball, the tautochrone curve is a cycloid (the curve described by a fixed point in a rolling circle), as correctly proven by Huygens in 1673 [BOYDE] using pure geometric arguments. We are going to present two different ways to arrive at this result.

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