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A curve having a finite length (cf. [[Line (curve)|Line (curve)]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r0801301.png" /> be a continuous parametric curve in three-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r0801302.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r0801303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r0801304.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r0801305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r0801306.png" />, are continuous functions on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r0801307.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r0801308.png" /> be a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r0801309.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013010.png" /> be the sequence of points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013011.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013012.png" />. Also, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013013.png" /> be the polygonal arc inscribed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013014.png" /> having vertices at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013015.png" />. The length of this arc is
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{{MSC|53A04|53A35}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013016.png" /></td> </tr></table>
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[[Category:Analysis]]
  
where
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013017.png" /></td> </tr></table>
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===General definition===
 +
A rectifiable curve is a curve having finite length (cf. [[Line (curve)|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 [[Lipschitz Function|Lipschitz]]. 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.
  
Then
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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
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\[
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s (\pi, \gamma) = \sum_{i=1}^N d (\gamma (t_{i-1}), \gamma (t_i))\, .
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\]
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A continuous function $\gamma:[0,1]\to X$ parametrize a rectifiable curve if and only if the following number is finite
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\[
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L (\gamma) = \sup_{\pi\in\Pi}\,  s (\pi, \gamma)\, .
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\]
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The number $L (\gamma)$ is the [[Length|length]] of the curve and it is independent of the parametrization (cp. with Section 3.2 of {{Cite|Fa}} for the Euclidean case).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013018.png" /></td> </tr></table>
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===Euclidean setting===
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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
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\begin{equation}\label{e:lunghezza}
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L (\gamma) = \int_0^1 |\dot{\gamma} (t)|\, dt\, ,
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\end{equation}
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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 {{Cite|AGS}}).  
  
is called the length of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013019.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013020.png" /> does not depend on the parametrization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013023.png" /> is called a rectifiable curve. A rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013024.png" /> has a tangent at almost every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013025.png" />, i.e. for almost all parameter values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013026.png" />. The study of rectifiable curves was initiated by L. Scheeffer [[#References|[1]]] and continued by C. Jordan [[#References|[2]]], who proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013027.png" /> is rectifiable if and only if the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013029.png" />, are of bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013030.png" /> (cf. [[Function of bounded variation|Function of bounded variation]]).
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====Relation to Hausdorff measure and rectifiable sets====
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The images of rectifiable curves are primary examples of [[Rectifiable set|rectifiable sets]]. The following classical theorem characterizes the images of rectifiable curves (cp. with Exercise 3.5 of {{Cite|Fa}}).
  
====References====
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'''Theorem 1'''
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Scheeffer,  "Allgemeine Untersuchungen über Rectification der Curven"  ''Acta Math.'' , '''5'''  (1885)  pp. 49–82</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Jordan,  "Cours d'analyse" , Gauthier-Villars  (1883)</TD></TR></table>
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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 measure|Hausdorff $1$-dimensional measure]] $\mathcal{H}^1$.  
  
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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 (cp. with Lemma 3.2 of {{Cite|Fa}}). 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]].
  
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===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).
  
====Comments====
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===References===
All the above works completely similarly for curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080130/r08013032.png" />.
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{|
 +
|-
 +
|valign="top"|{{Ref|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. {{MR|2401600}} {{ZBL|090.35002}}
 +
|-
 +
|valign="top"|{{Ref|Fa}}|| K. J.  Falconer, "The geometry of fractal sets". Cambridge Tracts in  Mathematics, 85. Cambridge University Press, Cambridge, 1986. {{MR|0867284}} {{ZBL|0587.28004}}
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|-
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|valign="top"|{{Ref|Jo}}|| C. Jordan,   "Cours d'analyse" , Gauthier-Villars  (1883) {{ZBL|19.0252.01}} {{ZBL|24.0247.03}} {{ZBL|25.0452.02}}
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|-
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|valign="top"|{{Ref|Sc}}||  L. Scheeffer,  "Allgemeine Untersuchungen über Rectification der Curven" ''Acta Math.'' , '''5'''  (1885)  pp. 49–82
 +
|}

Latest revision as of 13:56, 14 December 2012

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 Lipschitz. 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 (cp. with Section 3.2 of [Fa] for the Euclidean case).

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 (cp. with Exercise 3.5 of [Fa]).

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 (cp. with Lemma 3.2 of [Fa]). 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. MR2401600 Zbl 090.35002
[Fa] K. J. Falconer, "The geometry of fractal sets". Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986. MR0867284 Zbl 0587.28004
[Jo] C. Jordan, "Cours d'analyse" , Gauthier-Villars (1883) Zbl 19.0252.01 Zbl 24.0247.03 Zbl 25.0452.02
[Sc] L. Scheeffer, "Allgemeine Untersuchungen über Rectification der Curven" Acta Math. , 5 (1885) pp. 49–82
How to Cite This Entry:
Rectifiable curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rectifiable_curve&oldid=17729
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article