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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.).
 
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.).
  
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In Cartesian coordinates its equation is
 
In Cartesian coordinates its equation is
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$$
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y= \frac{a}{2} \left( e^{x/a} + e^{-x/a} \right) = a \cosh \frac{x}{a}
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020790/c0207901.png" /></td> </tr></table>
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The length of an arc beginning at the point $x=0$ is
 
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$$
The length of an arc beginning at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020790/c0207902.png" /> is
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l= \frac{1}{2} \left( e^{x/a} - e^{-x/a} \right) = a \sinh \frac{x}{a}
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020790/c0207903.png" /></td> </tr></table>
 
  
 
The radius of curvature is
 
The radius of curvature is
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$$
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r= a \cosh^2 \frac{x}{a}
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020790/c0207904.png" /></td> </tr></table>
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The area bounded by an arc of the catenary, two of its ordinates and the $y$-axis is
 
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$$
The area bounded by an arc of the catenary, two of its ordinates and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020790/c0207905.png" />-axis is
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S = a \sqrt{ y_2^2 - a^2 } - a \sqrt{ y_1^2 - a^2 } = a^2 \left( \sinh \frac{x_2}{a} - \sinh \frac{x_1}{a}  \right)
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020790/c0207906.png" /></td> </tr></table>
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020790/c0207907.png" /></td> </tr></table>
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If an arc of a catenary is rotated around the $x$-axis, it forms a [[Catenoid|catenoid]].
 
 
If an arc of a catenary is rotated around the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020790/c0207908.png" />-axis, it forms a [[Catenoid|catenoid]].
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>
 
 
 
====Comments====
 
 
 
====References====
 
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR></table>

Latest revision as of 07:00, 3 June 2016

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 $$ y= \frac{a}{2} \left( e^{x/a} + e^{-x/a} \right) = a \cosh \frac{x}{a} $$

The length of an arc beginning at the point $x=0$ is $$ l= \frac{1}{2} \left( e^{x/a} - e^{-x/a} \right) = a \sinh \frac{x}{a} $$

The radius of curvature is $$ r= a \cosh^2 \frac{x}{a} $$

The area bounded by an arc of the catenary, two of its ordinates and the $y$-axis is $$ S = a \sqrt{ y_2^2 - a^2 } - a \sqrt{ y_1^2 - a^2 } = a^2 \left( \sinh \frac{x_2}{a} - \sinh \frac{x_1}{a} \right) $$

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

References

[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: http://encyclopediaofmath.org/index.php?title=Catenary&oldid=16508
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article