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Catenary Involute

\begin{figure}\begin{center}\BoxedEPSF{CatenaryInvolute.epsf scaled 600}\end{center}\end{figure}

The parametric equation for a Catenary is

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

$\displaystyle {d{\bf r}\over dt}$ $\textstyle =$ $\displaystyle a\left[\begin{array}{c}1\\  \sinh t\end{array}\right]$ (2)
$\displaystyle \left\vert{d{\bf r}\over dt}\right\vert$ $\textstyle =$ $\displaystyle a\sqrt{1+\sinh^2 t} = a\cosh t$ (3)

\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]
\end{displaymath} (4)

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

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

s=a\int \cosh t\,dt = a\sinh t
\end{displaymath} (7)

and the equation of the Involute is
$\displaystyle x$ $\textstyle =$ $\displaystyle a(t-\tanh t)$ (8)
$\displaystyle y$ $\textstyle =$ $\displaystyle a\mathop{\rm sech}\nolimits t.$ (9)

This curve is called a Tractrix.

© 1996-9 Eric W. Weisstein