CYCLOID animation
Because the circle has rolled in contact with the x-axis,
we see that the distance it has rolled from the origin is
|OB| = arc PB = r*t
Therefore the center of the circle is C(r*t, r). Let the coordinates of P be (x, y).
Then from the figure, we see that
x = |OB| - |PQ| = r*t - r*sin t = r(t - sin t)
y = |BC| - |QC| = r - r*cos t = r(1- cos t)
Therefore parametric equations of the cycloid are
x(t) = r(t - sin t) = r*t - r*sin t
y(t) = r(1 - cos t) = r - r*cos t for t in R