The cycloid

A curve with very peculiar properties is the cycloid, that is the curve described by a point on a rotating circumference. The cycloid can be seen by attaching a light bulb to the wheel of a bicycle, possibly in the dark, or just rolling a circle on which we have marked a point.

Although simple to describe, the cycloid is a relatively modern curve. Among the first, if not the first, to examine it was Galileo, who observed how elegantly it described the arch of a bridge. During the whole seventeenth century, the cycloid was an object of study by the best mathematicians, who determined its length, its enclosed area (which Galileo had first guessed as being three times the generating circle, as many geometers later demonstrated independently, among them Torricelli), its balance point and other connected quantities.

But surprises lay in wait until the end of the century, when the cycloid became the solution of two important mathematical problems.

One deals with the fall of weights, and it marks the birth of a totally new branch of mathematics: the calculus of variations.

Let us suppose that we want to send a ball from a point A to a lower point B, but not along the vertical, We can imagine building a profile which unites the points A and B, and of making the ball roll along it. Obviously, there are infinite such profiles. We then ask ourselves: is there one that minimises the trajectory time?

At first sight, one might think that the solution is the straight line joining A and B. However, if we stop and think, we will see that the straight line is indeed the shortest line between two points, but not necessarily the fastest: it could be best to make the ball start more vertically, so that it would instantly gain enough speed to compensate for the longer path.

This problem of the brachistochronous curve, or curve of the shortest time (from the Greek brachistos, shortest, and chronos, time), was posited by Johann Bernoulli as a challenge to the mathematicians of his time, and was solved among others by Newton and Leibniz - the fastest curve is a cycloid. Bernoulli was so proud of his discovery, that he had on the title page of his Collected Works the drawing of a dog trying - in vain - to catch up with a cycloid, and the motto: Supra invidiam, above envy.

We have compared the cycloid with the straight line: letting two balls fall at the same time along those two curves, we can see how the time for the cycloid is considerably shorter than the one for the straight line.