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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.

RobPratt
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Keshav Srinivasan
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    It's putting the cart before the horse. How can you talk about preserving length before you define what length is? – Robert Israel Jul 13 '15 at 01:10
  • Instead of taking a limit, I've seen it defined as taking a supremum, whose definition is simpler than that of a limit. – Michael Hardy Jul 13 '15 at 01:11
  • For doing physical measurements, the piece of string works provided you know that it doesn't stretch as you bend it, and provided you can make the string accurately follow the curve you want to measure. But defining "stretching" geometrically is what we need to think about here. – Michael Hardy Jul 13 '15 at 01:13
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    @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. – Keshav Srinivasan Jul 13 '15 at 01:17
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    "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. – Keshav Srinivasan Jul 13 '15 at 01:20
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    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. – hmakholm left over Monica Jul 13 '15 at 01:24
  • I think the "approximate by line segments" definition is actually closer to the "string" intuition than you realize. Physically, when you wrap a string around something, what is roughly going on is that the string is a chain of a whole bunch of little rigid pieces, and by bending the junctures in the chain you can bend the string to approximately fit the curve. (At least, I assume that is what is going on; someone who actually knows physics should correct me if this is totally wrong.) – Eric Wofsey Jul 13 '15 at 06:41
  • @EricWofsey Well, it might be closer to the physics of strings when you take atoms and molecules into account, but I daresay that when the average person is thinking about this stuff, their intuition is based on a continuous, infinitely divisible piece of string. – Keshav Srinivasan Jul 13 '15 at 13:19
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    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. – Lee Mosher Jul 15 '15 at 17:47
  • @LeeMosher Well, Robert Israel's answer is a good start; it defines what "deformation without stretching" means without trying to approximate the curve by either finite or infinitesimal line segments. So it seems possible that "deformation without stretching or shrinking" can also be defined. – Keshav Srinivasan Jul 15 '15 at 17:52
  • Light travels along the shortest path geodesic curves, gravity bends light, something about isometries of spacetime continuum... – DVD Jul 16 '15 at 11:31
  • @KeshavSrinivasan: This is an isometric mapping preserving lengths, curvature of all metrics definable through the first fundamental form of surface theory. In definition and practice we say these are inextensible strings. – Narasimham Sep 05 '22 at 22:08
  • @KeshavSrinivasan Most people would have the intuition of constant-speed parametrization of a curve. If you walk at the same speed, the distance you cover in an hour is the same regardless of the shape of the path. Then two curves have the same length if their unit-speed parametrizations are defined on intervals of the same length. Don't know that you can fully formalize this intuition without some calculus, though. – dxiv Mar 14 '23 at 00:18
  • What if we say that two curves are equivalent if there's a homotopy $f(s,t)$ (with the two curves being $f(s,0)$ and $f(s,1)$) such that $\partial_t\lVert\partial_s f(s,t)\rVert=0$ ? – mr_e_man Mar 15 '23 at 14:35
  • I like the question, but I'm not sure I've fully understood your criteria. Two comments above (by dxiv and by Lee Mosher) seem to imply that you ruled out calculus, but I don't see where you did. Would a calculus-based answer based on bijections between parametrized continuous curves that move at the same speed everywhere be admissible as long as it doesn't use lengths of finite line segments? (And if not, what is it in the text of the question that rules that out?) – joriki Mar 19 '23 at 12:53
  • Hmm – I see now that in your comment under Eric Wofsey's answer you do exclude even relying on the preservation of infinitesimal lengths. In that case, it's rather unclear to me what you'd allow. Also, this doesn't seem in the spirit of the original question to my mind. The teacher for good reason told you to use a string, not a rubber band. The difference between a string and a rubber band is that the string can't locally change its length, and I think the people undefiled by calculus whose intuition you want to preserve have intuition about this difference between a string and a rubber band. – joriki Mar 19 '23 at 13:00

3 Answers3

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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 $C$.

Robert Israel
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  • 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 $C$. – Jim Belk Jul 13 '15 at 04:42
  • Oops, of course you're right. – Robert Israel Jul 13 '15 at 06:26
  • 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? – Keshav Srinivasan Jul 13 '15 at 15:29
  • Are there examples where we must say "infimum" rather than "minimum"? – Akiva Weinberger Sep 08 '15 at 03:51
  • I think not, using compactness to get a convergent subsequence... – Robert Israel Sep 09 '15 at 04:28
  • The important distance, which we want not to stretch, is the distance along the curve, right? So the metric $d$ cannot be any ordinary metric of the space which the curve is imbedded in. And then we seem to be back where we started: $d(x,y)$ is the length of the curve from $x$ to $y$, but how to define this? – Lorents Mar 23 '23 at 08:05
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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.

Eric Wofsey
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  • 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. – Keshav Srinivasan Jul 14 '15 at 03:07
  • 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. – Eric Wofsey Jul 14 '15 at 03:36
  • 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). – Keshav Srinivasan Jul 14 '15 at 03:53
  • @KeshavSrinivasan The idea in the answer above might capture some intuition in that "no stretching" means "all points keep their distance along the string". – Lorents Mar 21 '23 at 14:38
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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.

Narasimham
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