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 = [画像:x_0+(k(at^2-x_0))/((at^2+x_0)^2+(2at-y_0)^2)]

(3)

y = [画像: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 = [画像:k/(a(4+t^2))]

(5)

y = [画像:(2k)/(at(4+t^2)).]

(6)

ParabolaInverseFocus

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

x = [画像:a+(k(t^2-1))/(a(1+t^2)^2)]

(7)

y = [画像:(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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