Identifying a surface $\rho^2\cos(2\phi)-1=0$

Question

I need convert this spherical expression, to a rectangular form (specific surface). $$\rho^2\cos(2\phi)-1=0$$ Thanks for a while.

Answer 1

If $\phi$ is the polar and $\theta$ the azimuthal coordinate, using double angle trigonometric identity your surface is described by $$\rho^2\cos(2\phi)=\rho^2\cos^2(\phi)-\rho^2\sin^2(\phi)=1,$$i.e. $$\rho^2\cos^2(\phi)-\rho^2\sin^2(\phi)(\sin^2\theta+\cos^2\theta)=1$$ which transforming to Cartesian coordinates yields the hyperboloid $$z^2-(y^2+x^2)=1$$.

Answer 2

$(x^2+y^2)\cos(2\arctan(y/x))-1 = (x^2+y^2).\frac{x^2-y^2}{x^2+y^2} -1 = x^2-y^2-1$