These illustrations use cubic Bezier curves in an attempt to approximate a cycloid.
The upper illustration has blue control points 20px outside the baseline start & end points.
The middle illustration places the control points 20px inside the baseline end points.
The bottom illustration locates the 2 control points directly above the start and end points.
All control points are the same distance above the baseline - equivalent to the circle diameter (80px).
The moving circle produces a true cylcoid with the red circle - note that in all cases there are slight variations in the paths.