Can you define arc length using a piece of string?

In calculus, how we calculate the arc length of a curve is by approximating the curve with a series of line segments, and then we take the limit as the number of line segments goes to infinity. This is a perfectly valid approach to calculating arc length, and obviously it will allow you calculate correctly the length of any (rectifiable) curve. But it's obviously not the way people intuitively think about the length of a curve.

Here is how they introduced arclength to us in elementary school. If you want to measure the length of a straight line segment, use a ruler. If you want to measure the length of a curve, overlay the curve with a piece of string, then straighten the string and measure it with a ruler.

So I was wondering if it's possible to make a definition of arc length that preserves the spirit of that definition. Without using the calculus-based definition of length, is there any way to define what it means for one curve to be a "length-preserving deformation" of another curve? If that's possible, we could construct equivalence classes of curves that are length-preserving deformations of one another, and we can define the length associated with an equivalence class to be the length of the straight line that's in the class.

Is there anything in topology that would allow us to make such a definition? We'd need to account for the Euclidean metric somehow, since, e.g. in Taxicab geometry the circumference of a circle is $8r$ rather than $2\pi r$ (which is why your friends keep sending you that dumb $\pi = 4$ picture).

Any help would be greatly appreciated.

Thank You in Advance.

Solution 1:

It's somewhat simpler, I think, to characterize maps that don't increase length rather than those that preserve it.

A map $f: X \to Y$ (where $X$ and $Y$ are metric spaces, with metric denoted $d$ in both cases) is said to be contractive if $d(f(x),f(y)) \le d(x,y)$ for all $x, y \in X$.

EDIT (following Jim Belk's remark) The length of a curve $C$ is the infimum of $L$ such that there exists a contractive map from $[0,L]$ onto $C$.