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]

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]

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