Cycloid

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.

Related Topics:
Locus - Circle - Straight line

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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.

Related Topics:
Nicholas of Cusa - Mersenne - Galileo - 1599 - 1634 - G.P. de Roberval - 1658 - Christopher Wren

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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.

Related Topics:
Brachistochrone problem - Tautochrone problem - Mathematician

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

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:x = r(t - sin t)

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:y = r(1 - cos t)

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where t is a real parameter, equal to the center of the rolling circle.

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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

Related Topics:
Function - Differentiable - Cusp - Slope - Differential equation

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:left(rac{dy}{dx} ight)^2 = rac{2r-y}{y}

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