Pendulum Contained by Cycloid moves along Cycloidal Path

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Let a pendulum with a flexible rod be suspended from a point $P$.

Let the rod be contained by a pair of bodies shaped as the arcs of a cycloid such that $P$ is the cusp between those two arcs.


Then the bob is constrained to move such that its path traces the arc of a cycloid.


From Evolute of Cycloid is Cycloid, the evolute of a cycloid is another cycloid.

From Curve is Involute of Evolute, the involute of a cycloid is another cycloid as well.

But by the definition of involute, the path defined by the pendulum as described is the involute of the cycloid.

Hence the result.


Historical Note

The discovery that a Pendulum Contained by Cycloid moves along Cycloidal Path was made by Christiaan Huygens during his work on developing a reliable and accurate pendulum clock.

Unfortunately, the technique proved impractical, as the energy losses caused by the mechanics of the system compromise the pendulum's ability to swing reliably for a practical length of time.