[Tex/LaTex] Gaussian ellipsoids using tikz

Question

I'm trying to draw illustrations of Gaussian ellipsoids using shaded ellipses.
In simple terms what I'm after is a way to have a light shade throughout the outer edge of the ellipse and a darker shade towards the centre of the ellipse (i.e the contours of a 2D Gaussian distribution should have the same shade proportional to the density). I've tried radial shading but it's not the effect I'm looking for.

Here is a simple example (which does not produce the shading result I want).

  \begin{tikzpicture}
    \def\particles{(20,-3),(22,-5),(22,-7),(19,-8) }
     \pgfsetfillopacity{0.6};

    \foreach \point in \particles{
      \shade[rotate around={30:\point},inner color=green] \point ellipse (1 and 2);         
      \draw[fill=black] \point circle (2mm);
    } 
  \end{tikzpicture}

Can anyone please suggest a method to draw the Gaussian ellipsoids?

Answer 1

What you could do is layer lots of ellipses on top of each other, each one a little bit smaller and darker than the previous one. This gives the illusion of a smooth gradient, providing there are enough ellipses. In the code below I'm using ten ellipses, but you can adjust that to your liking.

Code

\documentclass{article}
\usepackage{tikz}
\begin{document}
  \begin{tikzpicture}
    \def\particles{(20,-3),(22,-5),(22,-7),(19,-8) }

    \foreach \point in \particles{
      \foreach\i in {0,0.1,...,1} {
        \fill[opacity=\i,green,rotate around={30:\point}] \point ellipse ({1-\i} and {2-2*\i});         
      }
      \fill[black] \point circle (2mm);
    } 
  \end{tikzpicture}
\end{document}

Result

---

In the code below, I used 100 ellipses. I also adjusted the opacity by a factor of 0.02. There's no method to this, I just fiddled with the numbers until it looked "right".

\documentclass{article}
\usepackage{tikz}
\begin{document}
  \begin{tikzpicture}
    \def\particles{(20,-3),(22,-5),(22,-7),(19,-8) }
    \foreach \point in \particles{
      \foreach\i in {0,0.01,...,1} {
        \fill[opacity=\i*0.02,green,rotate around={30:\point}] \point ellipse ({1-\i} and {2-2*\i});         
      }
      \fill[black] \point circle (2mm);
    } 
  \end{tikzpicture}
\end{document}