Yeah, one could draw a tight spiral in any of the open spaces of unbounded length. But "discretization" is probably a good angle to interpret it, since such a counterexample would be discretized to a point.

It also has the corner cases of curves that exactly align with the overlay and hence never cross any lines.

For increased accuracy, one can displace the overlay slightly and measure again, and take the average of multiple measurements.

> uncomputable

[citation needed]

There exist plenty of mathematical curves whose lengths are undefined. The term to search is "non-rectifiable curve".

This tool is for measuring curves drawn on a piece of paper, not abstract curves.

Oh, that doesn't make it any easier: https://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Brita...

Well, it doesn't in theory, but it does in practice. Natural phenomena are only fractal over a certain range of scales. Outside that range they follow normal rules.

That doesn't fit on a piece of paper, either.

A cartographer might disagree.