I'm trying to realized 3d models of stones. My idea was to create a 2D random angular distribution with opportune correlation, namely
$R(\theta,\varphi)=rand(\theta,\varphi,c_l)$
where $\theta\in[-\pi,\pi], \varphi\in[-\pi/2,\pi/2]$.
Once the surface is generated, I convert it to a 3D cartesian surface computing for every point of my domain $\mathbf{x}=(x,y,z)$ the following standard conversion
$\begin{cases}x=rcos(\theta)cos(\varphi)\\y=rsin(\theta)cos(\varphi)\\z=rsin(\varphi)\\\end{cases}$
$\theta^*=atan(y/x)$
$\varphi^*=atan(z/\sqrt{x^2+y^2})$
$r=\sqrt{(x^2+y^2+z^2)}$
The condition to determine if the point lies inside the rock is trivially
$r<R(\theta^*,\varphi^*)\implies\mathbf{x}\in\text{ sphere}$
This is the result I get:
As you can see in proximity of the z axis (red) there is some kind of distortion coming from the spherical to cartesian transformation. Is there any way to reduce it?
Your approach is inherently dependent on the coordinate system. I guess I'd try to find some approach which is invariant under changes of the coordinate system. For that I'd first have a look at the related topic of sphere point picking.
For example, you could use that to place a number of bups on the sphere, each with a random weight. So for $1\le i\le n$ you'd choose $-1<z_i<1, -\pi<\theta_i<\pi, -1<w_i<1$ uniformly and independently, and then turn that into $x_i=\sqrt{1-z_i^2}\cos\theta_i, y_i=\sqrt{1-z_i^2}\sin\theta_i$. Then for every point on $(x,y,z)$ the sphere, you could do something like this:
$$R(x,y,z)=C+\frac{1}{n\sqrt{x^2+y^2+z^2}}\sum_{i=1}^n\max(0, xx_i+yy_i+zz_i)w_i$$
The idea is that the closer one of these $n$ bumps is to a given direction, the stronger it influences the radius in that direction. Only half the sphere will be considered for this (i.e. you ignore negative dot products), which avoids too strong a correlation between antipodal points. There are a number of parameters to tune. You might want to adjust the offset $C$, and you also might want to adjust the distribution for these $w_i$. You might tune the number $n$, and so on. I haven't tried and of this, and I'm not sure that it will actually look stone-like. I am however certain that its appearance will be independent of the choice of coordinate system, and I simply hope that the rest works out as well.
If you try this approach, feel free to edit my answer and provide some images.