Catenary

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.

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 [math]\displaystyle{ \cosh }[/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.

Catenary Media

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  A chain hanging from points forms a catenary.

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  The silk on a spider's web forming multiple elastic catenaries.

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  Antoni Gaudí's catenary model at Casa Milà

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  Analogy between an arch and a hanging chain and comparison to the dome of Saint Peter's Basilica in Rome (Giovanni Poleni, 1748)

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  Simple suspension bridges are essentially thickened cables, and follow a catenary curve.

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  Stressed ribbon bridges, like the Leonel Viera Bridge in Maldonado, Uruguay, also follow a catenary curve, with cables embedded in a rigid deck.

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  A heavy anchor chain forms a catenary, with a low angle of pull on the anchor.

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  Diagram of forces acting on a segment of a catenary from c to r. The forces are the tension T0 at c, the tension T at r, and the weight of the chain (0, −ws). Since the chain is at rest the sum of these forces must be zero.