Hi,

Please see the attached images.

I am trying to invert an ellipse about one of its foci. I am not sure if I'm doing it right. My resulting inversion (image on the right) should look like the textbook image on the left, but seems to differ slightly.

In both images the limacon intercepts the ellipse at the point where the circle of inversion intercepts the ellipse. But in my inversion this point also happens to be in line with the center of the ellipse (as indicated by the dotted line). This is different from the textbook case.

Am I doing something wrong, or is this difference just owing to the different parameters of the two ellipses?

The ellipse I want to invert is: $\displaystyle \frac{(x+b)^2}{a^2}+\frac{y^2}{b^2}=1$

$\displaystyle b$, the semi-minor axis = $\displaystyle \frac{\sqrt{2}}{2}$

$\displaystyle a$, the semi-major axis = $\displaystyle 1$

The focus about which I want to invert is located at the origin (0,0).

To invert all of the points on the ellipse about the focus I am

using this equation:

$\displaystyle \frac{(x+b)^2}{a^2}+\frac{y^2}{b^2}=(x^2+y^2)^2$

(I'm not 100% sure if this is the correct equation for inversion)