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A cycloid generated by a rolling circle
A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line.It is an example of a roulette, a curve generated by a curve rolling on another curve.

The cycloid is the solution to the brachistochrone problem (i.e. it is the curve of fastest descent under gravity) and the related tautochrone problem (i.e. the period of a ball rolling back and forth inside this curve does not depend on the ball’s starting position).

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

A cycloid generated by a circle of radius r = 2
The cycloid through the origin, generated 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, corresponding to the angle through which the rolling circle has rotated, measured in radians. For given t, the circle's centre lies at x = rt, y = r.

Solving for t and replacing, the Cartesian equation would be

x = r \cos^{-1} \left(1-\frac{y}{r}\right)-\sqrt{y(2r-y)}


The first arch of the cycloid consists of points such that

0 \le t \le 2 \pi.\,


The cycloid 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

\begin{align}
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-2\sin t + \frac{1}{2} \cos t \sin t\right) \right|_{t=0}^{t=2\pi} \\&= 3 \pi r^2.\end{align}

Arc length

The arc length S of one arch is given by
\begin{align}
S &= \int_{t=0}^{t=2 \pi} \left(\left(\frac{dy}{dt}\right)^2+\left(\frac{dx}{dt}\right)^2\right)^{1/2} \, dt \\&= \int_{t=0}^{t=2 \pi} 2r \sin\left(\frac{t}{2}\right) \, dt \\&= 8r.\end{align}

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. This is because the path of the pendulum bob traces out a cycloidal path (presuming the bob is suspended from a supple rope or chain); a cycloid is its own involute curve, and the cusp of an inverted cycloid forces the pendulum bob to move in a cycloidal path.

The 17th Century Dutch mathematician Christiaan Huygens discovered this property of the cycloid and applied it to the design of more accurate clocks for use in navigation.

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.

Use in architecture

Cycloidal arches at the Kimbell Art Museum
The cycloidal arch was used by architect Louis Kahn in his design for the Kimbell Art Museummarker in Fort Worth, Texas.It was also used in the design of the Hopkins Center in Hanover, New Hampshire.

See also



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




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