Immagine you have two points, A and B.

What path minimizes travel time from A to B? A straight Line? A polygonal line? A piece of a circle? A piece of parabole? A cycloid?

Naively, and heuristically, most of people would answer that the shortest path would be the fastest. Thus a straight line connecting A and B should the fair and simple solution.

In physics and mathematics this path, connecting A and B, is called brachistochrone curve (from two Ancient Greek words  “βράχιστος” and “χρόνος” meaning  “shortest” and “time”), or curve of fastest descent. Of course we assume that a bead can slide frictionlessly under the influence of a uniform gravitational field.

The problem was posed by Johann Bernoulli in 1696.

What he found, interestingly, is that the fastest path is not the shortest one! Here you see beads sliding from A to B on the different paths (straight path, parabolic path, circular path, cycloid path).

The fastest path was the cycloid one. For a detailed explanation I highly suggest to follow the following video of Prof. Steven Strogats.