I was wondering about the parameterization of a roulette on Wikipedia. A roulette is a curve formed by a point associated to one curve as it rolls upon another fixed curve. Wikipedia says, if $f$ is the fixed curve, $r$ is the rolling curve and $p$ is a point associated to $r$ with
- $r(0) = f(0)$
- $r'(0) = f'(0)$
- $|r'(t)| = |f'(t)|=1 \quad \forall \ t$
then the mapping of the roulette $z(t)$ is $$z : \mathbb{R} \rightarrow \mathbb{C}, \quad t \mapsto f(t) + (p-r(t)) \frac{f'(t)}{r'(t)}.$$
I don't understand the second summand, especially $\frac{f'(t)}{r'(t)}$.
I think $f(t)$ is the point of contact between both curves. And the distance from $f(t)$ and $z(t)$ is exactly $|p-r(t)|$. But what is $\frac{f'(t)}{r'(t)}$?
You are correct that $f(t)$ is the point of contact between the curves at time $t$, as $f$ is the fixed curve. Also, $|p-r(t)|$ is the distance of the generator from $f(t)$. More specifically, $p-r(t)$ is the location of the generating point on the rolling curve at time $t$. (As the curve "rolls" forward according to $r$, the "fixed" generating point needs to move backwards, along $-r$.) So, we have accounted for the (relative) movement of $p$ along the rolling curve $r$, and the movement of the rolling curve along the fixed curve $f$. The only thing left to include is "rolling" itself, that is, the rotation of $r$ as it moves.
As the curve $r$ rolls along $f$, it will rotate to be "facing" the same direction as $f$, which can be represented as the angle part of the polar coordinates of $f'(t)$. So we will need to add the angle of $f'(t)$ to the angle of $p-r(t)$. But $p$ moves backwards along $r$ as it rolls, so we will also need to correct for this by subtracting the angle of $r'(t)$.
When using the polar coordinate form of complex numbers, multiplying has the effect of multiplying the radius and adding the angle, and dividing has the effect of dividing the radius and subtracting the angle. Since $|r'(t)|=|f'(t)|=1$, multiplying or dividing by either will only change the angle, not the radius. So, to add the angle of $f'(t)$ to $p-r(t)$ and subtract the angle of $r'(t)$, we can multiply by $f'(t)$ and divide by $r'(t)$ to get $(p-r(t))\dfrac{f'(t)}{r'(t)}$.
In short, $\dfrac{f'(t)}{r'(t)}$ tells you in which direction to put $z(t)$, relative to $f(t)$.