There is an arc a to b. The length is say 1 unit. There is an another arc c to d, drawn just below a to b, such that it sticks with it (ie. No vertical y space).
Let us say i have my world coordinate system, where measurements with precision of more than 2 decimal "CANNOT" be found out. For example i can have a measurement 1.55 (2 decimal places) but not 1.551 (3 decimal places). That is 2 decimal places is the limit of complexity i can dive into.
Now, in such a world, can i say by intuition that the arc that goes from c to d (which by intuition must be smaller than a to b) must be of length 0.99 unit? (0.99 is something immediate before "1", when i say i have restricted my self to 2 decimal places)
If a rectifyable arc (e.g. a polyline in a lattice) from $a$ to $b$ passes through $c\ne a$ and $d\ne b$ on its way (in this order) and the points can be distinguished in your world, then the arc from $c$ to $d$ is shorter by at least $0.02$ units ($0.01$ per end).
However, if the underlying world is not a lattice with grid size $0.01$, but rather there is some "true" distance between $a$ and $c$ that you only measure as $0.01$, say, because it is truely between $0.005$ and $0.015$, then the path $c$ to $d$ is shorter only by at least $0.01$ units ($0.05$ per end). Another big caveat: The result of rectifying an arbitrary curve with imperfect measuring tools may depend on the starting point in a nontrivial way, i.e. we cannot fully predict, how close your imperfect rectification is to the true arc length; it may even be the case that the measuring obtained for the truely shorter $cd$ exceeds the measuring obtained for the truely longer $ab$!