Definition:Brachistochrone

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

Suppose we are allowed to bend the wire into whatever shape we want.

Suppose that a bead is allowed to slide down without friction from $A$ to $B$.

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

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

Also see

 * Brachistochrone is Cycloid