Cycloid

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Cycloid_animated.gif
Cycloid (red) generated by a rolling circle

A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line.

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 upside down 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 it does not depend on the ball's starting position). The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th century mathematicians.

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Cycloid_r=2.png
Graph of cycloid generated by a circle of radius r=2

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, equal to the center of the rolling circle.

If seen as a function y(x), it is arbitrary often differentiable everywhere except at the cusps where it hits the x-axis; the slope at the cusps is infinite. It satisfies the differential equation

[itex]\left(\frac{dy}{dx}\right)^2 = \frac{2r-y}{y}[itex]

Related curves

Several curves are related to the cycloid. When we relax the requirement that the fixed point be on the rim 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 (a straight line is a circle of infinite radius) 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.

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