Say I have a very simple polar equation $r=\sin(2\theta)$ and I want to somehow transform/rewrite it into a pair of parametric equations where they both dependent on a third factor $x$ such that the distance traversed by point $(r,\theta)$ is linearly related to the variable $x$. How would I do so and how can I apply it to more complex polar equations (generalize)?
Background
I am building a Sisyphus table where there is a motor turning an arm (this is the $\theta$ motor). On the arm is a linear actuator that traverses from 0 to 1 ($r$ motor). The linear actuator has a magnet that moves a ball bearing across a layer of sand and leaves an indent in the sand as it moves through. The hardware and firmware is figured out, all I need is a script that can feed the $\theta$ and $r$ to the firmware. I would like the ball bearing to be moving at a constant speed instead of at a variable speed that is somewhat related to the current radius. Thus far, I have figured out that I have to somehow compensate for the radius along the curve but overall I am pretty lost.