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. (An ideal chain is one that can bend perfectly, cannot be stretched and has the same density throughout.) The supports can be at different heights and the shape will still be a catenary. A catenary looks a bit like a parabola, but they are different.

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

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

where $a$ is a parameter that determines the shape of the catenary and $\cosh$ is the hyperbolic cosine function, which is defined as

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

Hence, we can also write the catenary equation as

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

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