Brachistochrone curve

A Brachistochrone curve is the fastest path for a ball to roll between two points that are at different heights. A ball can roll along the curve faster than a straight line between the points.
The curve will always be the quickest route regardless of how strong gravity is or how heavy the object is. However, it might not be the quickest if there is friction.
The curve can be found using calculus of variations and optimal control.