I have the following equations in $x$, $y$, $z$ coordinates:
$x = R_0[f(\theta,\phi)\sin(\theta)\cos(\phi) + \partial_{\theta}f(\theta,\phi)\cos(\theta)\cos(\phi) - \partial_{\phi}f(\theta,\phi)\sin(\phi)/\sin(\theta)]$
$y = R_0[f(\theta,\phi)\sin(\theta)\sin(\phi) + \partial_{\theta}f(\theta,\phi)\cos(\theta)\sin(\phi) + \partial_{\phi}f(\theta,\phi)\cos(\phi)/\sin(\theta)]$
$z = R_0[f(\theta,\phi)\cos(\theta) - \partial_{\theta}f(\theta,\phi)\sin(\theta)]$
with $f(\theta,\phi) = 1 + \frac{4\epsilon_e}{1-4\epsilon_e}[\cos^4(\theta) + \sin^4(\theta)(1 -2\sin^2(\phi)\cos^2(\phi))]$
where $\epsilon_e = 0.047, R_0 = 15 $. Here, $\theta,\phi$ are spherical angles
Now, I would like to plot $(x,y)$ for $\epsilon_e = 0.047$ and $0$
This is what I have tried:
f[\[Theta]_, \[Phi]_] :=
1 + ((4 Subscript[\[Epsilon], e])/(
1 - 3 Subscript[\[Epsilon], e]))[(
Cos^4)[\[Theta]] + (
Sin^4)[\[Theta]] (1 - 2 (Sin^2)[\[Phi]] (Cos^2)[\[Phi]] )]
x = Subscript[R, 0][
f[\[Theta]_, \[Phi]_] Sin[\[Theta]] Cos[\[Phi]] +
Diff[f[\[Theta]_, \[Phi]_], \[Theta]] Cos[\[Theta]] Cos[\[Phi]] -
Diff[f[\[Theta]_, \[Phi]_], \[Phi]] Sin[\[Phi]]/Sin[\[Theta]]]
y = Subscript[R, 0][
f[\[Theta]_, \[Phi]_] Sin[\[Theta]] Sin[\[Phi]] +
Diff[f[\[Theta]_, \[Phi]_], \[Theta]] Cos[\[Theta]] Sin[\[Phi]] +
Diff[f[\[Theta]_, \[Phi]_], \[Phi]] Cos[\[Phi]]/Sin[\[Theta]]]
z = Subscript[R, 0][
f[\[Theta]_, \[Phi]_] Cos[\[Theta]] -
Diff[f[\[Theta]_, \[Phi]_], \[Theta]] Sin[\[Theta]]]
Can anyone suggest how to proceed forward to plot in XY-2D plane ? Do I have to use SphericalPlot or ListPlot?