Hyperbola Inverse Curve

$\begin{figure}\begin{center}\BoxedEPSF{HyperbolaInverseCenter.epsf scaled 560}\end{center}\end{figure}$

$\displaystyle x$	$\textstyle =$	$\displaystyle {2k\cos t\over a[3-\cos(2t)]}$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle {k\sin(2t)\over a[3-\cos(2t)]}$	(2)

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

$\displaystyle x$	$\textstyle =$	$\displaystyle a+{4k\cos t\sin^2({\textstyle{1\over 2}}t)\over a[5-4\cos t+\cos(2t)-2\sin(2t)]}$	(3)
$\displaystyle y$	$\textstyle =$	$\displaystyle a+{k(\tan t-1)\over a[(\sec t-1)^2+(\tan t-1)^2]}$	(4)

$\begin{figure}\begin{center}\BoxedEPSF{HyperbolaInverseFocus.epsf scaled 750}\end{center}\end{figure}$

$\displaystyle x$	$\textstyle =$	$\displaystyle ae={k\cos t(1-e\cos t)\over a(\cos t-e)^2}$	(5)
$\displaystyle y$	$\textstyle =$	$\displaystyle {\sqrt{e^2-1}\,k\sin(2t)\over 2a(\cos t-e)^2}$	(6)

is a Limaçon, where

$\begin{figure}\begin{center}\BoxedEPSF{HyperbolaInverseSq3Vertex.epsf scaled 560}\end{center}\end{figure}$

For a Hyperbola with $a=\sqrt{3}\,b$ and Inversion Center at the Vertex, the Inverse Curve

$\displaystyle x$	$\textstyle =$	$\displaystyle b+{2k\cos t(\sqrt{3}-\cos t)\over b[9-4\sqrt{3}\cos t+\cos(2t)-2\sin(2t)]}$	(7)
$\displaystyle y$	$\textstyle =$	$\displaystyle b+{k(\tan t-1)\over b[(\sqrt{3}\sec t-1)^2+(\tan t-1)^2]}$	(8)

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 203, 1972.