Rectifiable curve
A curve having a finite length (cf. Line (curve)). Let be a continuous parametric curve in three-dimensional Euclidean space
, that is,
,
, where the
,
, are continuous functions on the interval
. Let
be a partition of
and let
be the sequence of points on
corresponding to
. Also, let
be the polygonal arc inscribed in
having vertices at
. The length of this arc is
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where
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Then
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is called the length of the curve .
does not depend on the parametrization of
. If
, then
is called a rectifiable curve. A rectifiable curve
has a tangent at almost every point
, i.e. for almost all parameter values
. The study of rectifiable curves was initiated by L. Scheeffer [1] and continued by C. Jordan [2], who proved that
is rectifiable if and only if the functions
,
, are of bounded variation on
(cf. Function of bounded variation).
References
[1] | L. Scheeffer, "Allgemeine Untersuchungen über Rectification der Curven" Acta Math. , 5 (1885) pp. 49–82 |
[2] | C. Jordan, "Cours d'analyse" , Gauthier-Villars (1883) |
Comments
All the above works completely similarly for curves in ,
.
Rectifiable curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rectifiable_curve&oldid=27378