Can you define arc length using a piece of string?

Question

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.

Answer 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 $C$.

Answer 2

One neat way to make this precise is using the language of nonstandard analysis. Very generally, given two compact metric spaces $X$ and $Y$, say a map $f:X\to Y$ is length-preserving if whenever $a,b\in {}^*X$ are infinitely close, $\frac{d(a,b)-d(f(a),f(b))}{d(a,b)}$ is infinitesimal. That is, $f$ preserves infinitesimal distances up to an infinitesimally smaller error. For differentiable parametrizations of curves in $\mathbb{R}^n$, this recovers the usual characterization of length-preserving parametrizations as those such that the derivative has norm $1$ everywhere.

Answer 3

In isometric mappings lengths, and some curvatures of all metrics are definable through the first fundamental form of surface theory. In definition and practice we say these are inextensible strings. Length preserving nets are referred to as funicular.

Length is seen as the fundamentally unchanging/invariant physical entity and dimension during bending ( "overlaying" in the question ) and twisting. This may appear to be a negative definition but is has a positive association with visual/sensory experience.

In topology strings areas and volumes are extensible and there are some preserved topological invariants like the Euler characteristic.

EDIT1:

Is there any way to define what it means for one curve to be a "length-preserving deformation" of another curve?

If I may be allowed to digress from the physical world, the invariant soul is defined as an example of what or how it does not change, its invariant character is defined relative to environmentally forced stimuli. That it cannot be destroyed by fire, cut by sword, wetted by water etc. is not at all a negative definition.

A simple physics/geometric example of the of an unwinding in-extensible unstretchable string length $ 4 \pi a$ of 2 turns on cylinder when forced deformed/punished to unwind along helices ( continuous bending of geodesics of zero geodesic curvature and constant normal curvature $\kappa_n$ under constant twisting rate geodesic torsion $\tau_g$ ) on a cylinder radius $a$ with no change in length can be demonstrated. Here the definition came from what (the intrinsic length) that could not changed under some action and modalities forcing changes.