Finding the arc length between a point on a parabola and its vertex is a well documented process:
\begin{align} h&=\frac{p}{2}\\ q&=\sqrt{f^2+h^2}\\ s&=\frac{hq}{f}+f\ln\left(\frac{h+q}{f}\right) \end{align}Where $f$ is the focal length and $p$ is the perpendicular distance from the point to the axis of symmetry of the parabola.
How do i derive the inverse of this formula? One that would allow finding the destination point given the arc length starting from the vertex. I tried and quickly failed to isolate $p$ as the deadly combination of square roots and logarithms quickly grows out of control at least for my abilities.