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