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Logarithmic Spiral Pedal Curve

\begin{figure}\begin{center}\BoxedEPSF{LogarithmicSpiralPedal.epsf scaled 700}\end{center}\end{figure}

The Pedal Curve of a Logarithmic Spiral with parametric equation

$\displaystyle f$ $\textstyle =$ $\displaystyle e^{at}\cos t$ (1)
$\displaystyle g$ $\textstyle =$ $\displaystyle e^{at}\sin t$ (2)

for a Pedal Point at the pole is an identical Logarithmic Spiral
$\displaystyle x$ $\textstyle =$ $\displaystyle {(a\sin t+\cos t)e^{at}\over 1+a^2}$ (3)
$\displaystyle y$ $\textstyle =$ $\displaystyle {(\sin t-a\cos t)e^{at}\over 1+a^2},$ (4)

so
\begin{displaymath}
r=\sqrt{x^2+y^2}={e^{at}\over\sqrt{1+a^2}}.
\end{displaymath} (5)




© 1996-9 Eric W. Weisstein
1999-05-25