Rectifiable curve

2020 Mathematics Subject Classification: Primary: 53A04 Secondary: 53A35 [MSN][ZBL]

General definition

A rectifiable curve is a curve having finite length (cf. Line (curve)). More precisely, consider a metric space $(X, d)$ and a continuous function $\gamma: [0,1]\to X$. $\gamma$ is a parametrization of a rectifiable curve if there is an homeomorphism $\varphi: [0,1]\to [0,1]$ such that the map $\gamma\circ \varphi$ is Lipschtz. We can think of a curve as an equivalence class of continuous maps $\gamma:[0,1]\to X$, where two maps $\gamma$ and $\gamma'$ are equivalent if and only if there is an homeomorphism $\varphi$ of $[0,1]$ onto itself such that $\gamma'=\gamma\circ \varphi$. Each element of the equivalence class is a parametrization of the curve and thus a rectifiable curve is a curve which has a Lipschitz continuous parametrization.

Consider now the family $\Pi$ of finite ordered subsets of $[0,1]$, i.e. of points $0 \leq t_0< t_1<t_2<\ldots < t_N \leq 1$. Given a continuous $\gamma:[0,1]\to X$ and an element $\pi = \{t_0, \ldots, t_N\} \in \Pi$ consider the number \[ s (\pi, \gamma) = \sum_{i=1}^N d (\gamma (t_{i-1}), \gamma (t_i))\, . \] A continuous function $\gamma:[0,1]\to X$ parametrize a rectifiable curve if and only if the following number is finite \[ L (\gamma) = \sup_{\pi\in\Pi}\, s (\pi, \gamma)\, . \] The number $L (\gamma)$ is the length of the curve and it is independent of the parametrization.

Euclidean setting

A primary example are rectifiable curves in the euclidean space, where $X$ is given by $\mathbb R^n$ and the distance $d$ is the usual euclidean one: $d(x,y)=|x-y|$. In this case, if $\gamma$ is a Lipschitz parametrization, the length of $\gamma$ can be expressed through the usual integral formula \begin{equation}\label{e:lunghezza} L (\gamma) = \int_0^1 |\dot{\gamma} (t)|\, dt\, , \end{equation} where $\dot{\gamma} (t)$ is the derivative of $\gamma$ at $t$ (recall that, by Rademacher theorem the Lipschitz function $\gamma$ is differentiable at almost every $t$). The formula \eqref{e:lunghezza} can be suitably generalized to metric spaces introducing an appropriate notion of metric derivative (see [AGS]).

Relation to Hausdorff measure and rectifiable sets

The images of rectifiable curves are primary examples of rectifiable sets. The following classical theorem characterizes the images of rectifiable curves.

Theorem 1 A set $E\subset \mathbb R^n$ is the image of a rectifiable curve if and only if it is compact, connected and it has finite Hausdorff $1$-dimensional measure $\mathcal{H}^1$.

When the curve is not self-intersecting (i.e. its parametrizations are injective), the length of the curve is then the $\mathcal{H}^1$ measure of its image. More generally, in the presence of self-intersections $\mathcal{H}^1 (\gamma ([0,1]))$ and $L (\gamma)$ can be related to each other through the Area formula.

Comments

Since any compact interval is homeomorphic to $[0,1]$ in the definition above we could have taken a generic compact interval $[a,b]$. Other obvious variants of these definitions can be obtained using open intervals or $\mathbb S^1$ as domains for the parametrizations $\gamma$. In the latter case the resulting objects are called closed curves and closed rectifiable curves. Some authors use the names arc and rectifiable arc (resp. closed and open) when the domain of definition of the parametrization is an interval (resp. closed and open).

References

[AGS]	L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zurich. Birkhäuser, 2005.
[Fa]	K. J. Falconer, "The geometry of fractal sets". Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986.
[Jo]	C. Jordan, "Cours d'analyse" , Gauthier-Villars (1883)
[Sc]	L. Scheeffer, "Allgemeine Untersuchungen über Rectification der Curven" Acta Math. , 5 (1885) pp. 49–82