Catenary

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

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.

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

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

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