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Hyperbola Inverse Curve

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

For a Hyperbola with $a=b$ with Inversion Center at the center, the Inverse Curve

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

is a Lemniscate.


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

For an Inversion Center at the Vertex, the Inverse Curve

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

is a Right Strophoid.


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

For an Inversion Center at the Focus, the Inverse Curve

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

is a Limaçon, where $e$ is the Eccentricity.


\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)

is a Maclaurin Trisectrix.


References

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




© 1996-9 Eric W. Weisstein
1999-05-25