I wanted to draw a rose curve: $r = \cos 2\theta $ then if I wanted to write the equation in Cartesian coordinates: $$ \sqrt{x^2 + y^2} = x^2 - y^2 \text{ or } x^2 + y^2 = (x^2 - y^2)^2 $$ Except this is not the case in fact the Wikipedia page gives a cube on the left side. And I wonder why that is: $$ (x^2 + y^2)^3 = (x^2 - y^2)^2 $$ Is there an algebra mistake on my part?
We want $\sqrt{x^2+y^2}=r=\cos2\theta=\cos^2\theta-\sin^2\theta=\frac{x^2-y^2}{r^2}$. It all trickles down from your forgetting an $r^{-2}$ factor there.