Can someone help me express this parametric curve as an implicit curve $f(x,y) = C$?

Question

I have the following parametric equation in polar coordinates: $$ r(\theta) = r_0 + \Delta r\cos(2\theta),$$ where $\Delta r$ is some perturbation to the radius $r_0$. Can anyone help me express the above as an implicit curve $f(x,y)=C$?

Answer 1

$$ \cos \theta =\dfrac{x}{r}$$ $$ \sqrt{x^2+y^2}-r_o =\Delta r\, ( \dfrac{2x^2}{x^2+y^2}-1)$$ $$ \sqrt{x^2+y^2}-r_o =\Delta r \,( \dfrac{x^2-y^2}{x^2+y^2})$$