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Rectifiable curve

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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).

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 , .

How to Cite This Entry:
Rectifiable curve. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Rectifiable_curve&oldid=17729
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article