Catenary

A catenary is a type of curve. An ideal chain hanging between two supports and acted on by a uniform gravitational force makes the shape of a catenary.^[1] (An ideal chain is one that can bend perfectly, cannot be stretched and has the same density throughout.^[2]) The supports can be at different heights and the shape will still be a catenary.^[3] A catenary looks a bit like a parabola, but they are different.^[4]

Plots of ${\displaystyle y=a\cosh \left({\frac {x}{a}}\right)}$ with ${\displaystyle a=0.5,1,2}$. The variable ${\displaystyle x}$ is on the horizontal axis and ${\displaystyle y}$ is on the vertical axis.

A chain hanging like this forms the shape of a catenary approximately

The equation for a catenary in Cartesian coordinates is^[2][5]

${\displaystyle y=a\cosh \left({\frac {x}{a}}\right)}$

where ${\displaystyle a}$ is a parameter that determines the shape of the catenary^[5] and ${\displaystyle \cosh }$ is the hyperbolic cosine function, which is defined as^[6]

${\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}}$.

Hence, we can also write the catenary equation as

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

The word "catenary" comes from the Latin word catena, which means "chain".^[6] A catenary is also called called an alysoid and a chainette.^[1]

References

1. "Catenary". Wolfram Research. Retrieved 2016-10-30.
2. "The Catenary - The "Chain" Curve". California State University. Archived from the original on 2017-06-30. Retrieved 2019-01-01.{{cite web}}: CS1 maint: bot: original URL status unknown (link)
3. Rosbjerg, Bo. "Catenary" (PDF). Aalborg University. Retrieved 2016-10-30.
4. "Catenary and Parabola Comparison". Drexel University. Retrieved 2016-11-05.
5. "Equation of Catenary". Math24.net. Archived from the original on 2016-10-21. Retrieved 2016-10-30.
6. Stroud, K. A.; Booth, Dexter J. (2013). Engineering Mathematics (7th ed.). Palgrave Macmillan. p. 438. ISBN 978-1-137-03120-4.