A Cycloid is given by $$\left\{\begin{matrix} & x(t) = 3 \cdot (t-\sin t)\\ & y(t) = 3\cdot(1-\cos t) \end{matrix}\right.$$
I need to find the parametrized curve for the Normal line $x(s,t)$, $y(s,t)$ that passes through the point $ (x(t), y(t))$, where $s$ is the parameter for the Normal line.
$$ X(s,t) = x(t) -\dot y(t)\, s $$ $$ Y(s,t) = y(t) +\dot x(t)\, s $$
where $x,y$ are those of the cycloid, and $X,Y$ are those of the normal.
Update:
The derivatives are: $$ \dot x(t) = 3-3\cos t $$ $$ \dot y(t) = +3\sin t $$
Therefore: $$ X(s,t) = 3(t-\sin t) -3s\sin t = 3t - 3\sin t (1+s) $$ $$ Y(s,t) = 3(1-\cos t) +3s(1-\cos t) = 3(1-\cos t)(1+s). $$