Cycloid

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A cycloid is the parametric 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 an object in descent without friction inside this curve does not depend on the ball's starting position).

Contents

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[1].

Equations

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

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

The first arch of the cycloid consists of points such that

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. The map from t to (xy) is a differentiable curve or parametric curve of class C and the singularity where the derivative is 0 is an ordinary cusp.

The cycloid satisfies the differential equation:

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