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Parabola Inverse Curve

The Inverse Curve for a Parabola given by

$\displaystyle x$ $\textstyle =$ $\displaystyle a t^2$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle 2at$ (2)

with Inversion Center $(x_0,y_0)$ and Inversion Radius $k$ is
$\displaystyle x$ $\textstyle =$ $\displaystyle x_0+{k(at^2-x_0)\over (at^2+x_0)^2+(2at-y_0)^2}$ (3)
$\displaystyle y$ $\textstyle =$ $\displaystyle y_0+{k(2at-y_0)\over (at^2+x_0)^2+(2at-y_0)^2}.$ (4)

\begin{figure}\begin{center}\BoxedEPSF{ParabolaInverseFocus.epsf scaled 550}\end{center}\end{figure}

For $(x_0,y_0)=(a,0)$ at the Focus, the Inverse Curve is the Cardioid

$\displaystyle x$ $\textstyle =$ $\displaystyle a+{k(t^2-1)\over a(1+t^2)^2}$ (5)
$\displaystyle y$ $\textstyle =$ $\displaystyle {2kt\over a(1+t^2)^2}.$ (6)

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

For $(x_0,y_0)=(0,0)$ at the Vertex, the Inverse Curve is the Cissoid of Diocles

$\displaystyle x$ $\textstyle =$ $\displaystyle {k\over a(4+t^2)}$ (7)
$\displaystyle y$ $\textstyle =$ $\displaystyle {2k\over at(4+t^2)}.$ (8)

© 1996-9 Eric W. Weisstein