Let C be a circle of radius with center
at *(0,a)*. As we move the circle along a horizontal line L the center
of the circle will stay *a* units above the x-axis, but its x-coordinate
will be translated by the distance that the circle is rolled. Suppose that
the radius from *(0,a)* to *(0,0)* is rotated clockwise as we
roll the circle from left to right. Let the angle of rotation be represented
by *-t*, *t *> 0. It follows that the circle has been rolled
a distance of *t***a*.
The following diagram shows this situation.

From the triangle we have that

Hence the parametric equations of the cycloid are

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