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A cycloid is a curve that a point on the perimeter of a wheel traces when rolling along the  axis without slipping. If the radius of the rolling wheel is  , then the cycloid may be presented in the parametric form
where  expresses the angle rotated by the wheel around its center.
In what follows, a blue curve indicates a cycloid (or a portion thereof) and red line segments indicate radii of the wheel.
Below is a picture of the wheel on the  axis with  .
As the wheel rolls,  increases. To obtain the cycloid, we keep track of the path along which the fixed point of the wheel has travelled.
After the wheel has completed a full turn, the cycloid takes a sharp turn due to the fact that the point hits the  axis, then begins travelling upwards again.
Thus, below is the graph of a cycloid for  .
The graph of a cycloid for any  can be obtained by replacing  with  and  with  on the  axis of the graph above.
The length of one arc of the cycloid formed by one revolution of the circle (e.g.
 ) is
Therefore, the length of one arc of the cycloid is  , i.e. four times the diameter of the circle.
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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
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