Fix a reference frame in $\mathbb{R}^2$. Suppose you have an ellipse $\mathcal{E}$, a point $P \in \mathcal{E}$ and a real number $d$, where it is implicitly understood that $d>0$ means movement along a counter-clockwise direction ($d<0$ clockwise).
The problem is: find a function which, given in input ($P$, $d$, $\mathcal{E}$), gives, as output, a new point, $Q$, such that: $Q \in \mathcal{E}$ and the (signed) arc-length between $P$ and $Q$ is $d$ (the direction is implicitly specified by the sign of $d$).
In short: find the point $Q$ obtained by moving a point $P$ of "distance" $d$ along an ellipse.
It may be that this problem has no closed-form solution (I suspect it due to the fact that the arc-length of an ellipse has no closed-form), in which case useful approximations can be discussed.
