Length

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

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

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