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The plane transcendental curve describing the form of a homogeneous flexible string of fixed length and with fixed ends attained under the action of gravity (see Fig.).

Figure: c020790a

In Cartesian coordinates its equation is

The length of an arc beginning at the point is

The radius of curvature is

The area bounded by an arc of the catenary, two of its ordinates and the -axis is

If an arc of a catenary is rotated around the -axis, it forms a catenoid.


[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)



[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
How to Cite This Entry:
Catenary. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article