Cycloid

The parametric curve that is without the limits is known as a cycloid.  In its general form the cycloid is,

X = r (θ - sin θ)

Y = r (1- cos θ)

 The cycloid presents the following situation. Refer a wheel of radius r.  Let the point in which the wheel touches the ground basically be called P.  Then start rolling the wheel to the right.  Like the wheel rolls to the right trace out the path of the point that is P.  The path which the point P traces out is called a cycloid and is specified by the equations above. In these equations we can think of θ as the angle by which the point P has rotated.

 Now here is a cycloid sketched out with the wheel shown at several places. The blue dot is the point P on the wheel that we were using to draw out the curve.

From this diagram we can see that one arch of the cycloid is traced out in the range  0 < θ < 2π.   This makes sense while you consider that the point P will be back on the ground later it has rotated by an angle of 2π.