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Parabola Involute

\begin{figure}\begin{center}\BoxedEPSF{parabola_involute.epsf}\end{center}\end{figure}


\begin{displaymath}
{d{\bf r}\over dt} = \left[{\matrix{1\cr 2t\cr}}\right]
\end{displaymath} (1)


\begin{displaymath}
\hat{\bf T} = {1\over\sqrt{1+4t^2}} \left[{\matrix{1\cr 2t\cr}}\right]
\end{displaymath} (2)


\begin{displaymath}
ds^2=\vert d{\bf r}\vert^2 = (1+4t^2)\,dt^2
\end{displaymath} (3)


\begin{displaymath}
ds=\sqrt{1+4t^2}\,dt
\end{displaymath} (4)


\begin{displaymath}
s=\int \sqrt{1+4t^2}\,dt = {\textstyle{1\over 2}}t\sqrt{1+4t^2}+{\textstyle{1\over 4}}\sinh^{-1}(2t).
\end{displaymath} (5)

So the equation of the Involute is
$\displaystyle {\bf r}_i$ $\textstyle =$ $\displaystyle {\bf r}-s\hat{\bf T} = \left[\begin{array}{c}t\\  t^2\end{array}\...
...t)\over \sqrt{1+4t^2}}
\left[\begin{array}{c}1\\  2t\end{array}\right]\nonumber$  
  $\textstyle =$ $\displaystyle {1\over 2\sqrt{1+4t^2}} \left[\begin{array}{c}t-{\textstyle{1\over 2}}\sinh^{-1}(2t)\\  -\sinh^{-1}(2t)\end{array}\right].$ (6)




© 1996-9 Eric W. Weisstein
1999-05-26