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)

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

(5)