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.

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    $\begingroup$ It's putting the cart before the horse. How can you talk about preserving length before you define what length is? $\endgroup$ – Robert Israel Jul 13 '15 at 1:10
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    $\begingroup$ @RobertIsrael Well, "length-preserving" is just a name. I'm trying to see if possible to define a certain equivalence relation between curves without relying on the concept of length, and then after we've defined the notion of length it will turn out that that equivalence relation corresponds to the notion of having the same length." I'm not doing anything unusual here. Historically, this is how cardinality was defined. We call two sets "equinumerous" if there exists a one-to-one correspondence between them. And then we define the number of elements of a set using the concept of equinumerous. $\endgroup$ – Keshav Srinivasan Jul 13 '15 at 1:17
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    $\begingroup$ "But defining "stretching" geometrically is what we need to think about here." Yeah, that's what I'm after. Just as it's possible to define the concept of bijection without relying on the notion of cardinality, I'd like to define "deformation without stretching or shrinking" without relying on arc length. $\endgroup$ – Keshav Srinivasan Jul 13 '15 at 1:20
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    $\begingroup$ So in more "dignified" terms, what you're looking for is a natural way to characterize the length-preserving mappings between curves, without presupposing a quantitative concept of arc length. That's an interesting question, +1. $\endgroup$ – hmakholm left over Monica Jul 13 '15 at 1:24
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    $\begingroup$ Reading through the proposed answers and the various objections to them, I am beginning to wonder whether this question is answerable. All of the infinite processes of calculus, standard or nonstandard, are being ruled out. It seems that the question is being reducied to: how does one define arc length of curves, with neither calculus nor ruler measurements? If this boils down to asking for arc length of any smooth curve on any differentiable manifold, that is an impossible task. $\endgroup$ – Lee Mosher Jul 15 '15 at 17:47

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 $C$.

  • $\begingroup$ What's the metric on the curve $C$? If the curve is planar and you're using the restriction of the Euclidean metric (which seems to be the natural choice if you haven't defined length yet), then this won't work. What you want to do instead is take the infimum of all $L$ such that there exists a contractive map from $[0,L]$ onto $C$. $\endgroup$ – Jim Belk Jul 13 '15 at 4:42
  • $\begingroup$ Oops, of course you're right. $\endgroup$ – Robert Israel Jul 13 '15 at 6:26
  • $\begingroup$ Thanks, your definition does a good job of specifying what "deformation of a line segment without stretching" means, and that suffices to define length. But I have two lingering questions. First of all, is it possible to define what "D is a deformation of a curve C without stretching" means without invoking a definition of length? And second of all, is it possible to define "deformation without shrinking" without invoking a definition if length? $\endgroup$ – Keshav Srinivasan Jul 13 '15 at 15:29
  • $\begingroup$ Are there examples where we must say "infimum" rather than "minimum"? $\endgroup$ – Akiva Weinberger Sep 8 '15 at 3:51
  • $\begingroup$ I think not, using compactness to get a convergent subsequence... $\endgroup$ – Robert Israel Sep 9 '15 at 4:28

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.

  • $\begingroup$ It seems to me that this is just the nonstandard analysis equivalent of the standard calculus definition of arc length - instead of approximating the curve by $N$ finite line segments and taking the limit as $N$ goes to infinity, we're just breaking the curve up into infinitely many infinitesimal line segments. It doesn't really capture the intuition of deforming a continuous string. $\endgroup$ – Keshav Srinivasan Jul 14 '15 at 3:07
  • $\begingroup$ Well, this is not a definition of arc length itself but rather what it means for a map to preserve arc length (indeed, this should more generally define maps between compact manifolds with boundary embedded in $\mathbb{R}^n$ which are isometries with respect to the intrinsic metric). If $X$ is a curve representing the current state of a string and $f:X\to Y$ is a bijection to some other state of the string, this is a definition of when such a deformation is length-preserving, without actually measuring length. $\endgroup$ – Eric Wofsey Jul 14 '15 at 3:36
  • $\begingroup$ Yeah, but it seems to me that it's implicitly using the (nonstandard analogue of the) calculus-based notion of arc length: we're basically saying that a deformation is length-preserving If the lengths of the infinitesimal line segments don't change to much, in order to satisfy the condition that the sum of the lengths of the infinitesimal line segments doesn't change too much. But the average person doesn't think about motions of a string in terms of infinitesimal line segments (or in terms of finite line segments which approximate the curve). $\endgroup$ – Keshav Srinivasan Jul 14 '15 at 3:53

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