The cycloid is the solution to the brachistochrone problem (i.e. it is the curve of fastest descent under gravity) and the related tautochrone problem (i.e. the period of a ball rolling back and forth inside this curve does not depend on the ballâ€™s starting position).

The cycloid through the origin, generated by a circle of radius r, consists of the points (x, y), with

If its length is equal to that of half the cycloid, the bob of a pendulum suspended from the cusp of an inverted cycloid, such that the "string" is constrained between the adjacent arcs of the cycloid, also traces a cycloid path. Such a cycloidal pendulum is isochronous, regardless of amplitude. This is because the path of the pendulum bob traces out a cycloidal path (presuming the bob is suspended from a supple rope or chain); a cycloid is its own involute curve, and the cusp of an inverted cycloid forces the pendulum bob to move in a cycloidal path.

Combined the above two equations:

So

and

The time required to travel from the top of the cycloid to the bottom is

Please check out http://mathworld.wolfram.com/TautochroneProblem.html or http://en.wikipedia.org/wiki/Cycloid for more information.

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