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Let $\gamma(t)$ be a smooth curve in normal Euclidean space. Then the length of $\gamma$ is defined as

$$\int ds =\int_0^1 \sqrt{x'(t)^2 + y'(t)^2} dt$$

The formula for $ds$ is usually justified by cutting the curve up into many pieces, and approximating each piece by the chord joining the endpoints. The length of $\gamma$ is then "approximately" the sum of the lengths of the chords.

I am trying to provide better justifications for the form of $ds$ without resorting to the intuition that each chord "approximates" the respective curve segment, once the curve segments get short enough.

I am hoping to say something like "$ds$ has this form because our space is Euclidean". In fact, an arbitrarily defined $ds$ induces a metric $d(x, y) = \inf_\gamma \int_{\gamma} ds$, where $\gamma$ ranges over all curves from $x$ to $y$. It is easy to verify that the standard $ds$ gives rise to the Euclidean metric. More promisingly, if the underlying space is instead Manhattan, then $ds$ would have been $|x'(t)| + |y'(t)|$.

So let's define an alternative $ds$ as $ds = \sqrt{x'(t)^2 + y'(t)^2 + \kappa(t)^2}$ where $\kappa(t)$ is the curvature of the curve. It seems like this $ds$ works just as fine for any smooth curve, and it induces the same Euclidean metric $d(x, y)$, because straight line segments have zero curvature and thus is not affected by the added curvature term. Intuitively, this $ds$ distorts curves with nonzero curvature.

I also tried to work out properties of this weird $ds$, and I failed to find any obvious contradiction or problem.

  • Are there any problems with this weird $ds$, as in, does it exhibit any contradictory or counterintuitive behavior if used to define lengths of smooth curves?
  • Are there any justification for $ds$ not containing any higher order terms (such as $\kappa$)?
  • If there are no problems, it seems like materially different $ds$ can induce the same metric $d$. Does this imply that $d : X \times X \to \mathbb{R}$ is insufficient at describing a metric space $X$?
user1641237
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    Key properties that we want for arclength: 1) The length should not change if you move the curves around, 2) The union of two disjoint curves is equal to the sum of the lengths, 3) If you stretch the curve in all directions by a common factor, the length should also stretch by the same factor. In other words, the curve $\hat{\gamma} = 2\gamma$ should have twice the length of $\gamma$. Your proposed formula satisfies 1) and 2) but not 3). Your overall idea is a good one, and I believe it does work. – Deane Jun 19 '25 at 23:59
  • @Deane Yeah, I had in my mind (1) and (2). Your criteria (3) definitely rules out any trickery involving curvature (or higher order analogues thereof), but I think x''^2 and y''^2 seems to work, though such terms no longer have obvious intuitive interpretations. I am also thinking about general manifolds/metric spaces instead of normed vector spaces, where it is not as obvious how something like (3) can be applied... – user1641237 Jun 20 '25 at 00:15
  • There's also 4) The length should be independent of the parameterization of the curve. $(x'')^2+(y'')^2$ does not satisfy this. – Deane Jun 20 '25 at 03:05
  • As for the length of a curve in a metric space, there is a straightforward definition without using chords but is obviously equivalent to that definition in Euclidean space. The catch is that the length of the shortest curve between two points is not necessarily equal to the distance between them. If it always is, then the space is called a length space. See the Wikipedia article https://en.wikipedia.org/wiki/Intrinsic_metric – Deane Jun 20 '25 at 03:09
  • The idea is interesting. You are just defining another function. Where I see a problem is integration.Try it for $x(t)=t$, $y(t)=t^2$ – Claude Leibovici Jun 20 '25 at 07:15

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