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A numerical characteristic of the extension of lines in a metric space. The length of a segment of a straight line is the distance between its ends, measured by means of some segment accepted as the unit length. The length of a broken line is the sum of the lengths of its parts. The length of a simple arc is the least upper bound of the lengths of the broken lines inscribed in this arc. Any continuous curve has a length, finite or infinite. If its length is finite, the curve is known as rectifiable. The length of a planar curve defined in rectangular coordinates by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l0581401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l0581402.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l0581403.png" /> having a continuous derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l0581404.png" />) is given by the integral
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<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/l/l058/l058140/l0581405.png" /></td> </tr></table>
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A numerical characteristic of the extension of lines in a metric space. The length of a segment of a straight line is the distance between its ends, measured by means of some segment accepted as the unit length. The length of a broken line is the sum of the lengths of its parts. The length of a simple arc is the least upper bound of the lengths of the broken lines inscribed in this arc. Any continuous curve has a length, finite or infinite. If its length is finite, the curve is known as rectifiable. The length of a planar curve defined in rectangular coordinates by an equation  $  y = f( x) $,
 +
$  a \leq  x \leq  b $(
 +
$  f $
 +
having a continuous derivative  $  f ^ { \prime } $)
 +
is given by the integral
 +
 
 +
$$
 +
= \int\limits _ { a } ^ { b }  \sqrt {1 + [ f ^ { \prime } ( x) ]  ^ {2} }  d x .
 +
$$
  
 
If the curve is given in parametric form
 
If the curve is given in parametric form
  
<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/l/l058/l058140/l0581406.png" /></td> </tr></table>
+
$$
 +
= x ( t) ,\  y  = y ( t) ,\  t _ {1} \leq  t \leq  t _ {2} ,
 +
$$
  
 
its length is given by
 
its length is given by
  
<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/l/l058/l058140/l0581407.png" /></td> </tr></table>
+
$$
 +
= \int\limits _ { t _ {1} } ^ { {t _ 2 } } \sqrt {[ x  ^  \prime  ( t) ]  ^ {2} + [ y  ^  \prime  ( t) ]  ^ {2} }  d t .
 +
$$
  
The length of a rectifiable curve does not depend on the parametrization. The length of a spatial curve given in parametric form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l0581408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l0581409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814011.png" />, is given by the formula
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The length of a rectifiable curve does not depend on the parametrization. The length of a spatial curve given in parametric form $  x = x ( t) $,  
 +
$  y = y ( t) $,  
 +
$  z = z ( t) $,  
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$  t _ {1} \leq  t \leq  t _ {2} $,  
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is given by the formula
  
<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/l/l058/l058140/l05814012.png" /></td> </tr></table>
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$$
 +
= \int\limits _ { t _ {1} } ^ { {t _ 2 } } \sqrt {[ x  ^  \prime  ( t) ]  ^ {2} + [ y  ^  \prime  ( t) ]  ^ {2} + [ z  ^  \prime  ( t) ]  ^ {2} }  d t .
 +
$$
  
In the case of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814013.png" />-dimensional space,
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In the case of an $  n $-
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dimensional space,
  
<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/l/l058/l058140/l05814014.png" /></td> </tr></table>
+
$$
 +
= \int\limits _ { t _ {1} } ^ { {t _ 2 } } \sqrt {\sum _ {i = 1 } ^ { n }  [ x _ {i}  ^  \prime  ( t) ]  ^ {2} }  d t .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814015.png" /> be a continuously-differentiable curve, given by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814017.png" />, on a continuously-differentiable surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814018.png" />. Then the length of an arc of the curve counted from the point corresponding to the parameter value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814019.png" /> is equal to
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Let $  \gamma $
 +
be a continuously-differentiable curve, given by functions $  u = u ( t) $,  
 +
$  v= v ( t) $,  
 +
on a continuously-differentiable surface $  \mathbf r = \mathbf r ( u , v ) $.  
 +
Then the length of an arc of the curve counted from the point corresponding to the parameter value $  t = t _ {0} $
 +
is equal to
  
<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/l/l058/l058140/l05814020.png" /></td> </tr></table>
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$$
 
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s ( t , t _ {0} )  = \int\limits _ {t _ {0} } ^ { t }  | \mathbf r  ^  \prime  ( t)
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814021.png" /> is the [[First fundamental form|first fundamental form]] of the surface. The length of a continuously-differentiable curve given by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814023.png" />, in a Riemannian space with metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058140/l05814024.png" /> is
+
|  dt  = \int\limits _ {\gamma ( P _ {0} , P ) } | d \mathbf r ( u , v
 
+
) |  = \int\limits _ {t _ {0} } ^ { t }  \sqrt I ,
<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/l/l058/l058140/l05814025.png" /></td> </tr></table>
+
$$
  
 +
where  $  I $
 +
is the [[First fundamental form|first fundamental form]] of the surface. The length of a continuously-differentiable curve given by functions  $  x  ^ {i} = x  ^ {i} ( t) $,
 +
$  t _ {1} \leq  t \leq  t _ {0} $,
 +
in a Riemannian space with metric tensor  $  g _ {ik} $
 +
is
  
 +
$$
 +
s  =  \int\limits _ { t _ {1} } ^ { {t _ 2 } }  d s  =  \int\limits _ { t _ {1}  } ^ { {t _ 2 } } \sqrt {g _ {ik}
 +
\frac{d x  ^ {i} }{dt}
 +
 +
\frac{d x  ^ {k} }{dt}
 +
}  d t .
 +
$$
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)  {{MR|0903026}} {{MR|0895392}} {{MR|0882916}} {{MR|0882541}} {{ZBL|0619.53001}} {{ZBL|0606.51001}} {{ZBL|0606.00020}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Blumenthal,  K. Menger,  "Studies in geometry" , Freeman  (1970)  {{MR|0273492}} {{ZBL|0204.53401}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Busemann,  "The geometry of geodesics" , Acad. Press  (1955)  {{MR|0075623}} {{ZBL|0112.37002}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)  {{MR|0903026}} {{MR|0895392}} {{MR|0882916}} {{MR|0882541}} {{ZBL|0619.53001}} {{ZBL|0606.51001}} {{ZBL|0606.00020}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Blumenthal,  K. Menger,  "Studies in geometry" , Freeman  (1970)  {{MR|0273492}} {{ZBL|0204.53401}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Busemann,  "The geometry of geodesics" , Acad. Press  (1955)  {{MR|0075623}} {{ZBL|0112.37002}} </TD></TR></table>

Latest revision as of 22:16, 5 June 2020


A numerical characteristic of the extension of lines in a metric space. The length of a segment of a straight line is the distance between its ends, measured by means of some segment accepted as the unit length. The length of a broken line is the sum of the lengths of its parts. The length of a simple arc is the least upper bound of the lengths of the broken lines inscribed in this arc. Any continuous curve has a length, finite or infinite. If its length is finite, the curve is known as rectifiable. The length of a planar curve defined in rectangular coordinates by an equation $ y = f( x) $, $ a \leq x \leq b $( $ f $ having a continuous derivative $ f ^ { \prime } $) is given by the integral

$$ s = \int\limits _ { a } ^ { b } \sqrt {1 + [ f ^ { \prime } ( x) ] ^ {2} } d x . $$

If the curve is given in parametric form

$$ x = x ( t) ,\ y = y ( t) ,\ t _ {1} \leq t \leq t _ {2} , $$

its length is given by

$$ s = \int\limits _ { t _ {1} } ^ { {t _ 2 } } \sqrt {[ x ^ \prime ( t) ] ^ {2} + [ y ^ \prime ( t) ] ^ {2} } d t . $$

The length of a rectifiable curve does not depend on the parametrization. The length of a spatial curve given in parametric form $ x = x ( t) $, $ y = y ( t) $, $ z = z ( t) $, $ t _ {1} \leq t \leq t _ {2} $, is given by the formula

$$ s = \int\limits _ { t _ {1} } ^ { {t _ 2 } } \sqrt {[ x ^ \prime ( t) ] ^ {2} + [ y ^ \prime ( t) ] ^ {2} + [ z ^ \prime ( t) ] ^ {2} } d t . $$

In the case of an $ n $- dimensional space,

$$ s = \int\limits _ { t _ {1} } ^ { {t _ 2 } } \sqrt {\sum _ {i = 1 } ^ { n } [ x _ {i} ^ \prime ( t) ] ^ {2} } d t . $$

Let $ \gamma $ be a continuously-differentiable curve, given by functions $ u = u ( t) $, $ v= v ( t) $, on a continuously-differentiable surface $ \mathbf r = \mathbf r ( u , v ) $. Then the length of an arc of the curve counted from the point corresponding to the parameter value $ t = t _ {0} $ is equal to

$$ s ( t , t _ {0} ) = \int\limits _ {t _ {0} } ^ { t } | \mathbf r ^ \prime ( t) | dt = \int\limits _ {\gamma ( P _ {0} , P ) } | d \mathbf r ( u , v ) | = \int\limits _ {t _ {0} } ^ { t } \sqrt I , $$

where $ I $ is the first fundamental form of the surface. The length of a continuously-differentiable curve given by functions $ x ^ {i} = x ^ {i} ( t) $, $ t _ {1} \leq t \leq t _ {0} $, in a Riemannian space with metric tensor $ g _ {ik} $ is

$$ s = \int\limits _ { t _ {1} } ^ { {t _ 2 } } d s = \int\limits _ { t _ {1} } ^ { {t _ 2 } } \sqrt {g _ {ik} \frac{d x ^ {i} }{dt} \frac{d x ^ {k} }{dt} } d t . $$

Comments

References

[a1] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French) MR0903026 MR0895392 MR0882916 MR0882541 Zbl 0619.53001 Zbl 0606.51001 Zbl 0606.00020
[a2] L. Blumenthal, K. Menger, "Studies in geometry" , Freeman (1970) MR0273492 Zbl 0204.53401
[a3] H. Busemann, "The geometry of geodesics" , Acad. Press (1955) MR0075623 Zbl 0112.37002
How to Cite This Entry:
Length. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Length&oldid=47612
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article