Let be a point on a circle of radius
Consider the curve, which is traced out by the point as the circle rolls along the -axis. We will allow that our circle begins to trace the curve with the point at the origin.
Such a curve is called a cycloid.
Now, we can find the parametric equation fir the cycloid as follows:
Let the parameter be the angle of rotation of for our given circle. Note that when the point is at the origin.
Next consider the distance the circle has rolled from the origin after it has rotated through radians, which is given by
And so we can see that the center of the circle is given by .
Now, letting the coordinates of P be we have that
Hence, we have
which gives us the parametric equations of the cycloid.