History

The cycloid was first studied by Nicholas of Cusa and later by Mersenne. It was named by Galileo in 1599. In 1634 G.P. de Roberval showed that the area under a cycloid is three times the area of its generating circle. In 1658 Christopher Wren showed that the length of a cycloid is four times the diameter of its generating circle. The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th century mathematicians.

Equations

The cycloid through the origin, created by a circle of radius r, consists of the points (x, y) with

x = r(t - sin t),

y = r(1 - cos t),

where t is a real parameter; rt is the x-coordinate of the center of the rolling circle.

Using degrees, a more modifiable form of the equation can be found: x = a(cos(270-t)) + 2πrt/360

y = a(sin(270-t)) + r

This curve is differentiable everywhere except at the cusps where it hits the x-axis, with the derivative tending toward $infty$ or $-infty$ as one approaches a cusp. It satisfies the differential equation

left(frac{dy}{dx}right)^2 = frac{2r-y}{y}.

Area

One arch of a cycloid generated by a circle of radius r can be parametrized by

x = r(t - sin t),,

y = r(1 - cos t),,

with

0 le t le 2 pi.,

Since

frac{dx}{dt} = r(1- cos t),

we find the area under the arch to be

A=int_{t=0}^{t=2 pi} y , dx = int_{t=0}^{t=2 pi} r^2(1-cos t)^2 , dt

left. r^2 left(frac{3}{2}t-2sin t + frac{1}{2} cos t sin tright) right|_{t=0}^{t

2pi} =3 pi r^2.

Arc length

The arc length S of one arch is given by

int_{0}^{2 pi} left(left(frac{dy}{dt}right)^2+left(frac{dx}{dt}right)^2right)^{1/2}, dt=int_{0}^{2 pi} 2r sin(t/2) , dt = 8r.

Cycloidal pendulum

If its length is equal to that of half the cycloid, the bob of a pendulum suspended from the cusp of an inverted cycloid, such that the "string" is constrained between the adjacent arcs of the cycloid, also traces a cycloid path. Such a cycloidal pendulum is isochronous, regardless of amplitude.

An up-side-down cycloid is called a tautochrone which is a path of a cycloidal pendulum.

Related curves

Several curves are related to the Cycloid. When we relax the requirement that the fixed point be on the edge of the circle, we get the curtate cycloid and the prolate cycloid. In the former case, the point tracing out the curve is inside the circle, and, in the latter case, it is outside. A trochoid refers to any of the cycloid, the curtate cycloid and the prolate cycloid. If we further allow the line on which the circle rolls to be an arbitrary circle then we get the epicycloid (circle rolling on outside of another circle, point on the rim of the rolling circle), the hypocycloid (circle on the inside, point on the rim), the epitrochoid (circle on the outside, point anywhere on circle), and the hypotrochoid (circle on the inside, point anywhere on circle).

All these curves are roulettes with a circle rolled along a uniform curvature. The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute. If q is the product of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the curve:evolute similitude ratio is 1+2q.

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

References

An application from physics: Ghatak, A. & Mahadevan, L. Crack street: the cycloidal wake of a cylinder tearing through a sheet. Physical Review Letters, 91, (2003). http://link.aps.org/abstract/PRL/v91/e215507