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Let a point $A$ be joined by a wire to a lower point $B$.

Let the wire be allowed to be bent into whatever shape is required.

Let a bead be released at $A$ to slide down without friction to $B$.

The shape of the wire so that the bead takes least time to descend from $A$ to $B$ is called the brachistochrone.

Also see

Historical Note

The brachistochrone was investigated by Jacob Bernoulli, using the techniques of calculus.

Linguistic Note

The word brachistochrone comes from the Greek βράχιστος χρόνος (brakhistos khrónos), which means shortest time.