Parametrization of the curve

Question

A cycloid is a flat curve that is traced by point of the rim of a circle while the circle rolls without slippage on the line. Show that if the line is the axis $x$ and the circle has radius $a>0$, then the cycloid can be parametrized by $$\gamma (t)=a(t-\sin t, 1-\cos t)$$ Could you give me some hints how we could show that? I don't really have an idea what I could do...

Answer 1

I give you here (with weak English) an explanation of the parametric you ask. In the figure below, your point $P(x,y)$ has started from the position $P_0$ at the coordinate origin.

By definition of cycloid the arc $\widehat{PQ}$ subtended by the angle $t$ (which is choose as parameter!) and the segment $\overline{P_0Q}$ have the same length equal to $at$. Now all is easy: $$x=\overline {P_0Q}-\overline{SQ}=at-a\space cos(t-\frac{\pi}{2})=a(t-sin\space t)$$ $$y=\overline{SP}=\overline{SR}+\overline{RP}=a+a\space sin(t-\frac{\pi}{2})=a(1-cos\space t)$$