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Parabola Inverse Curve

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The inverse curve for a parabola given by

x=at^2
(1)
y=2at
(2)

with inversion center (x_0,y_0) and inversion radius k is

x=x_0+(k(at^2-x_0))/((at^2+x_0)^2+(2at-y_0)^2)
(3)
y=y_0+(k(2at-y_0))/((at^2+x_0)^2+(2at-y_0)^2).
(4)
ParabolaInverseVertex

For (x_0,y_0)=(0,0) at the parabola vertex, the inverse curve is the cissoid of Diocles

x=k/(a(4+t^2))
(5)
y=(2k)/(at(4+t^2)).
(6)
ParabolaInverseFocus

For (x_0,y_0)=(a,0) at the focus, the inverse curve is the cardioid

x=a+(k(t^2-1))/(a(1+t^2)^2)
(7)
y=(2kt)/(a(1+t^2)^2).
(8)

See also

Inverse Curve, Inversion, Parabola

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Cite this as:

Weisstein, Eric W. "Parabola Inverse Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ParabolaInverseCurve.html

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