How can curves approximate curves, but lines cannot approximate lines?

Question

This is my first question on StackExchange, and i have very little knowledge of mathematics to tell if it is a physics issue or a math issue, so any info will be useful.

Main question

Let's say i want to find the length of a line. Take the following image for reference (1): it is a right triangle, and a naive/wrong way of finding the length of the hypotenuse. Is simple to show how this way of thinking can go wrong with a counterexample, if both smaller sides of the triangle have length $a$, the hypotenuse has length $a\sqrt{2}$. While the "zigzagging" gives the answer $2a$.

Again, i have a shamefully poor background on Real Analysis, so i could not explain the following main question:

How can one of these two "curvy" lines approximate the other?

My physics professor used this "zigzagging" argument (but with curves) to show how to calculate the variation of entropy in a thermodynamic process. And i just could not stop thinking about this problem in the first figure.

Some context

I had this problem in my mind when i saw this on my thermodynamics book (the book is in Portuguese), it showed how to approximate a small line element with fragments that make up a small Carnot Cycle. I don't think this is important to the discussion, so i will just put info regarding the context, just for curiosity: page 22 of this book (there should be an equivalent in English literature).

This question has some explanation about the staircase approximation. — Tesla Daybreak, Jun 20 '21 at 16:20
Your professor was NOT computing the length of a curve, but was rather evaluating a more complicated integral along a curve. Hence it is possible (and can be proved) that both curves give the same result. — Intelligenti pauca, Jun 21 '21 at 15:05

score 1 · Answer 1 · answered Jun 20 '21 at 16:23

Note that the fact that the zigzag lines converge to the diagonal in some sense does not imply that the lengths of the zigzag lines converge to the length of the diagonal. For the length of a curve, we have to first come up with a suitable definition of length, and that definition would be:

elementary for line segments
sum of its line segments for polylines
for general curves: supremum (if it exists) over all polylines with all its nodes on the curve in correct order.

And while the zigzags are polylines, they have only about half their nodes on the curve we want to measure.

Is a "node" on a curve just a point on the curve? – Joe Jun 20 '21 at 19:54 — Joe, Jun 20 '21 at 19:54

How can curves approximate curves, but lines cannot approximate lines?

Main question

Some context

1 Answers1