The problem of comparing the lengths of two paths of line segments connecting points in $\mathbb{Q}^2$ is not known to be in $\text{P}$, nor even in $\text{NP}$.
Does requiring that the paths begin and end on the same point change anything about our state of knowledge?
What if additionally the paths are not permitted to self-intersect, that is, they are polygons?
Finally, what if the polygons are required to be convex?
My understanding of the obstruction is that the length of a path with $n$ segments is the root of a $2^n$-degree polynomial and we can't rule out that those might end up being so close together that we have to compute exponentially many digits. Superficially, we're in the same situation with bumpy wheels as with wild scribbles.
