Let us say that the particle is at the left extreme point of the diameter (horizontal). call this point origin O.
The displacement vector is the line joining this point O and the current position P (x,y) or (r, theta) of the particle on the circle.
X = a + a cos Ф = 2a cos² Ф/2
S = √(2 a X)
S = 2 a cos (wt / 2)
Position vector r or S is given by : r = 2a cos Ф/2
where Ф is the angle made by line joining P and center of circle C, with the
Ф = w t