Spheric equation to Cartesian equation

Question

I have this equation $$\rho^2=2 \sin(2\theta)\cos(\phi)$$ and I want to convert it into a Cartesian equation. I’ve already applied an trigonometric identity and I have this$$\rho^2=4\sin(\theta)\cos(\theta)\cos(\phi)$$ and I don’t know what else to do

Answer 1

I suppose, that you have following spherical coordinates: $x=r \sin \phi \cos \theta$, $y=r \sin \phi \sin \theta$, $z = r \cos \phi$

From these we have $$y^2+z^2=r^2\sin^2 \phi \sin^2 \theta + r^2 \cos^2 \phi=\\=r^2\left( \sin^2 \phi \sin^2 \theta +1- \sin^2 \phi \right)=r^2\left( 1+ \sin^2 \phi \left(\sin^2 \theta -1 \right) \right)= r^2 \big(1- \sin^2 \theta \cos^2 \phi \big) \ \ \ (1)$$ and $$x^2 + y^2=r^2\sin^2 \phi \cos^2 \theta + r^2\sin^2 \phi \sin^2 \theta=r^2 \sin^2 \phi \ \ \ (2)$$

Your equation is possible to write as $$r^4=16\sin^2(\theta)\cos^2(\theta)\cos^2(\phi) = 16(1- \cos^2 \theta)\cos^2 \theta (1-\sin^2 \phi)$$

Then we will have equation in $x,y,z$. Happy simplification.