View source for Rectifiable curve

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

<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>

where

<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>

Then

<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>

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]]).

====References====
<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>



====Comments====
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" />.