Parabola Inverse Curve

The Inverse Curve for a Parabola given by

$$x = a t^2$$ (1)

$$y = 2at$$ (2)

with Inversion Center $(x_0,y_0)$ and Inversion Radius $k$ is

$$x = x_0+{k(at^2-x_0)\over (at^2+x_0)^2+(2at-y_0)^2}$$ (3)

$$y = y_0+{k(2at-y_0)\over (at^2+x_0)^2+(2at-y_0)^2}.$$ (4)

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

$$x = a+{k(t^2-1)\over a(1+t^2)^2}$$ (5)

$$y = {2kt\over a(1+t^2)^2}.$$ (6)

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

$$x = {k\over a(4+t^2)}$$ (7)

$$y = {2k\over at(4+t^2)}.$$ (8)