# Cycloid

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**cycloid** is the shape 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 cycloid is the solution to the brachistochrone problem and the related tautochrone problem. The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th century mathematicians.

Equations (parametric):

- x = a(t - sin t)
- y = a(1 - cos t)

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).

## References