The Equations of a Cycloid

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 p = a  sin(-t) and q = a cos(-t). Thus the point (0,0) has moved to the point at the end of the blue radius which has coordinates (a*t - a sin(t), a(1-cos(t)). (Verify.)

Hence the parametric equations of the cycloid are

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