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Catenary

Object type: Plane curve

Definition

The catenary is the graph of the hyperbolic cosine, or, more generally, the function x \mapsto a \cosh {\left(\frac{x}{a}\right)} for some a > 0. In the image below, a = 5.

Catenary with a = 5

Curvature

The curvature function for the catenary (parameterised using x as parameter) is the bump function \kappa(x) = \frac{1}{a} \text{sech}^2 {\left(\frac{x}{a}\right)}.

Examples

An ideal chain (cable, rope, wire) placed in a (constant) gravitational field suspended at its ends will obtain the shape of the catenary. Below this is demonstrated by the author himself.

A chain

Let us superimpose one parabola (blue) and one catenary (red):

A chain with graphs