[ Edited : I mistakenly asked, initially, for a parametric equation of point $R$; no parameter is involded in the definition of $R$ ; but my problem remains ]
Link to my construction using Desmos : https://www.desmos.com/calculator/kb9xxtx7mc
Let $\mathscr C$ be a circle centered at $O=(0,0)$ and defined parametrically as $(f(t), g(t))$ , with $f(t)=3cos(t)$ and $g(t)=3sin(t)$ ( with $t$ from $0$ to $2\pi$).
Let $P$ be a point moving on circle $\mathscr C$ , such that $P= (f(a), g(a))$, with angle $a$ ranging from $0$ to $2 \pi$.
Let $Q$ be a point located on line $OP$ , at $2$ units outside circle $\mathscr C$.
So, $Q= (n, m) = (f(a)+2cos(a), g(a)+2sin(a))$.
The equation of a little circle $\mathscr C'$, turning around $\mathscr C$, having a $2$ units radius and centered at $Q=(n,m)$ is : $(x-n)^2 +(y-m)^2=4$.
My question is what should be the definiion of a point $R$ that turns on circle $\mathscr C'$, in such a way that $R$ makes one turn on $\mathscr C'$ when $\mathscr C'$ makes one turn around $\mathscr C$, and also that $R$ goes down when $\mathscr C'$ goes up, so that $R$ seems to cause the rotation of $\mathscr C'$.
The problem I encounter is that, in this Desmos construction : https://www.desmos.com/calculator/kb9xxtx7mc , the point $R$ does not move in the right sense .
I defined $R$ as follows : $R= \Large ($$ n+2cos(-a+\pi) , m+2sin(-a+\pi)$$\Large )$.
I tried, in the parametric equation of pointt $R$, to use the opposite of the angle I'm currectly using, namely I tried the angle $(a-\pi)$ but that did not work.
The problem is that if you move along $\mathscr C'$ with the same angular speed, but a phase offset of $\pi$, then $P$ and $R$ will be the same. Your smaller circle is rotating around the big circle. If you want to see the point on the circumference rotate another circle, then you need to increase the speed, to use $2a$ instead of $a$.