Cycloid

The cycloid is the locus of a point on the rim of a circle of radius rolling along a straight line. It was studied and named by Galileo in 1599. Galileo attempted to find the area by weighing pieces of metal cut into the shape of the cycloid. Torricelli, Fermat, and Descartes all found the area. The cycloid was also studied by Roberval in 1634, Wren in 1658, Huygens in 1673, and Johann Bernoulli in 1696. Roberval and Wren found the arc length (MacTutor Archive). Gear teeth were also made out of cycloids, as first proposed by Desargues in the 1630s (Cundy and Rollett 1989).

In 1696, Johann Bernoulli challenged other mathematicians to find the curve which solves the brachistochrone problem, knowing the solution to be a cycloid. Leibniz, Newton, Jakob Bernoulli and L'Hospital all solved Bernoulli's challenge. The cycloid also solves the tautochrone problem, as alluded to in the following passage from Moby Dick: "[The try-pot] is also a place for profound mathematical meditation. It was in the left-hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along a cycloid, my soapstone, for example, will descend from any point in precisely the same time" (Melville 1851). Because of the frequency with which it provoked quarrels among mathematicians in the 17th century, the cycloid became known as the "Helen of Geometers" (Boyer 1968, p. 389).

The cycloid catacaustic when the rays are parallel to the y-axis is a cycloid with twice as many arches. The radial curve of a cycloid is a circle. The evolute and involute of a cycloid are identical cycloids.

If the cycloid has a cusp at the origin and its humps are oriented upward, its parametric equation is

			(1)
			(2)

Humps are completed at values corresponding to successive multiples of , and have height and length . Eliminating in the above equations gives the Cartesian equation

(3)

which is valid for and gives the first half of the first hump of the cycloid. An implicit Cartesian equation is given by

(4)

The arc length, curvature, and tangential angle for the first hump of the cycloid are

			(5)
			(6)
			(7)

For the first hump,

(8)

For a single hump of the cycloid, the arc length and area under the curve are therefore

			(9)
			(10)