# Catenary

Curve that an idealized hanging chain or cable assumes

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

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