The curve you are talking about is known as the cycloid. To parametrize this curve, consider the curve at the instant 0 and after t radian. I would start the cycloid at the point (0,0) and then see how the point moved after the circle described a rotation of t radians.
(i.e. investigate how the coordinate's position varies when t does )
One of the things to notice is that, after t radians of rotation, the center of your circle will have moved rt units. This is also the mesure of the arc of circle between your point (x,y) and the point of the circle that touches the horizontal straight line. All of this is due to the fact that the circle rolls without "sliding" on the line.
The fun part is eliminating the parameter...
Edit : I considered the angle t as being the angle between the point (x,y), the center of the circle and the point that touches the straight line. You can also consider a different angle t and the result will also be the same. This choice avoids tricky sign "problems".