Intrinsic definition of arc length [closed]

Question

Is there an intrinsic way of defining the arc length of a curve in $\mathbb{R}^{3}$, that is without resorting to a parametrization of the curve?

Answer 1

When a definition applies to a much larger class of spaces $\ X\subseteq \mathbb R^3,\ $ then such definition rather is not directly related to any parametrization for the smaller class of the parametrized curves (which can be rectified). Several of variants of Hausdorff dimension for dimension 1 would be a possible answer. Another could be

$$\lim_{r\rightarrow +0}\ \frac {V(X+B(r))}{\pi\cdot r^2}$$

where $\ X+B(r)\ =\ \{y\in\mathbb R^3:\ \exists_{x\in X}\ d(x\ y)\le r\},\ \ d\ $ is the euclidean distance, and $\ V\ $ stands for volume (see also, optionally, the detailed terminology below). This will certainly work properly for the piecewise $\ C^1$-curves, but most likely for all rectifiable curve (see the justification below); and of course--by definition--for many other spaces in the Euclidean space. Again, variations of this definition are possible, e.g. using cubes with edges $\ \left[\frac k{2^n};\frac{k+1}{2^n}\right],\ $ etc.

TERMINOLOGY

• $\ B(r)\ :=\ \{x\in\mathbb R^3: |x|\le r\}\ $ is the closed ball of radius $\ r,\ $ centered at the origin of $\ \mathbb R^3$;
• $\ X+Y\ :=\, \ \{x+y:\ x\in X\ \ \&\ \ y\in Y\}\quad $ for $\ X\ Y\subseteq\mathbb R^3$.

JUSTIFICATION of the volume formula for the length

In the case of a finitely piece-wise linear curve, the above volume is a sum of the respective cylinders around the intervals plus/minus a negligible error when the radius approaches $\ 0.\ $ The general case of rectifiable curves is obtained by $\ \epsilon/\delta\ $ (:-) which I am ready to provide if asked to.

Answer 2

Assuming the curve $S$ is ``reasonable'' say, it is semialgebraic or, more generally, definable in an $o$-minimal category, then you can define the length using Crofton's formula. $\DeclareMathOperator{\Graff}{Graff}$ $\newcommand{\bR}{\mathbb{R}}$

Define $\Graff_2(\bR^3)$ to be the set of affine planes in $\bR^3$. Up to a multiplicative constant, there is a unique measure on $\Graff_2(\bR^3)$ invariant under the group of isometries of $\bR^3$. Pick one such measure $\mu$. Then

$$ C(\mu) {\rm length}\;(C)= \int_{\Graff_2(\bR^3)} \#(L\cap C) \mu(dL), $$

where $C(\mu)$ is a universal, explicit, positive constant that depends linealrly on the invariant measure $\mu$.

Remark. Federer's book on geometric measure theory contains varies concepts of measure and dimension one can associate to a subset of $\bR^n$: Hausdorff measure, integralgeometric measure (using Crofton's formula). For ``reasonable'' sets these concepts coincide. In particular, for reasonable $1$-dimensional sets they yield various definitions of length that do not use parametrizations of the curve.

Answer 3

You can use the Darboux sums (named after the mathematician Gaston Darboux). If you have a totally ordered metric space $M$ with a minimum and a maximum ($\alpha$ and $\omega$) (this is, in particular, the case of a curve with an injective parametrization, but you do not have to refer to it). Call subdivision a sequence of points $s=(x_i)_{0\leq i\leq n}$ linearly ordered $(\forall i<n)(x_i<x_{i+1})$ and joining the endpoints $x_0=\alpha,\ x_n=\omega$. For every subdivision $s$, form the Darboux sum \begin{equation} l(s)=\sum_{i=1}^n d(x_i,x_{i-1})\ . \end{equation} Take the usual order of refinement between subdivisions, if these quantities converge for the net of refinement order, the limit the length of the linearly ordered metric space.

$$ length(M)=lim_{s\nearrow} l(s)\ . $$

Call rectifiable a (linearly ordered) metric space such that $length(M)<+\infty$. This notion is elementary, encompasses all others I know and has very nice properties :

• if a (linearly ordered) metric space is rectifiable then all its intervals $[u,v]$ are so
• length is additive : if $u<v<w$, then $$ length([u,w])=length([u,v])+length([v,w]) $$

Answer 4

The most general definition of arc length will involve summing the distances between nearby points (taken as a limit, obviously) along the section of the curve. The arc length will be independent of parameterization chosen, so any parameterization can be chosen without loss of generality. Explicit parameterization can only be avoided if the curve is actually a function on one axis, over the section in question. (Or piecewise.) For example you could use $\intop\sqrt{1+\left(\frac{dy}{dx}\right)^2+\left(\frac{dz}{dx}\right)^2}dx$ but only if y = f(x), z = g(x) are (single-valued) functions of x. My advice is to not be afraid of parameterization; just use it. It's more general. So I think the short answer to your question is "no".