Catenary

File:Catenary-pm.svg

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

File:Kette Kettenkurve Catenary 2008 PD.JPG

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

[math]\displaystyle{ y = a \cosh \left(\frac{x}{a}\right) }[/math]

where [math]\displaystyle{ a }[/math] is a parameter that determines the shape of the catenary and Template:Math is the hyperbolic cosine function, which is defined as

[math]\displaystyle{ \cosh x = \frac {e^x + e^{-x}} {2} }[/math].

Hence, we can also write the catenary equation as

[math]\displaystyle{ y = \frac{a\left(e^\frac{x}{a} + e^{-\frac{x}{a}}\right)}{2} }[/math].

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