Catenary Involute

The parametric equation for a Catenary is

$${\bf r}(t) = a\left[{\matrix{t\cr \cosh t\cr}}\right],$$ (1)

so

$${d{\bf r}\over dt} = a\left[\begin{array}{c}1\\ \sinh t\end{array}\right]$$ (2)

$$\left\vert{d{\bf r}\over dt}\right\vert = a\sqrt{1+\sinh^2 t} = a\cosh t$$ (3)

and

$$\hat {\bf T} = {{d{\bf r}\over dt}\over \left\vert{d{\bf r}\... ...t[{\matrix{\mathop{\rm sech}\nolimits t\cr \tanh t\cr}}\right]$$ (4)

$$ds^2=\vert d{\bf r}^2\vert = a^2(1+\sinh^2 t)\,dt^2 = a^2\cosh^2\,dt^2$$ (5)

$${ds\over dt} = a\cosh t.$$ (6)

Therefore,

$$s=a\int \cosh t\,dt = a\sinh t$$ (7)

and the equation of the Involute is

$$x = a(t-\tanh t)$$ (8)

$$y = a\mathop{\rm sech}\nolimits t.$$ (9)

This curve is called a Tractrix.