Equation of Catenary

Theorem

Cartesian Form

Let a catenary be embedded in a cartesian plane so that the $y$-axis passes through the lowest point of the catenary.

Formulation 1

The catenary is described by the equation:

$y = \dfrac {e^{a x} + e^{-a x} } {2 a} = \dfrac {\cosh a x} a$

where $a$ is a constant.

The lowest point of the catenary is at $\tuple {0, \dfrac 1 a}$.

Formulation 2

The catenary is described by the equation:

$y = \dfrac a 2 \paren {e^{x / a} + e^{-x / a} } = a \cosh \dfrac x a$

where $a$ is a constant.

The lowest point of the chain is at $\tuple {0, a}$.

Cesàro Form

Formulation 1

The catenary can be described by the Cesàro equation:

$a = \kappa \paren {a^2 + s^2}$

where:

$s$ is the arc length

$\kappa$ is the curvature

$a$ is a constant.

Formulation 2

The catenary can be described by the Cesàro equation:

$a \rho = a^2 + s^2$

where:

$s$ is the arc length

$\rho$ is the radius of curvature

$a$ is a constant.

Whewell Form

The catenary can be described by the Whewell equation:

$s = a \tan \psi$

where:

$s$ is the arc length

$\psi$ is the turning angle

$a$ is a constant.