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
where
Then
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).