# Gaussian ellipsoids using tikz

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?

• Welcome to TeX.SE. While code snippets are useful in explanations, it is always best to compose a fully compilable MWE that illustrates the problem including the \documentclass and the appropriate packages so that those trying to help don't have to recreate it. This is especially important for tikz as there are numerous libraries. – Peter Grill May 21 '13 at 6:36
• Probably the answer to this question will help. – Tom Bombadil May 21 '13 at 7:44

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}


You can also use PGFPlots and adapt the approach from Draw a bivariate normal distribution in TikZ to draw gaussian ellipses. Note that this is much slower than John's approach.

\documentclass{standalone}

\usepackage{pgfplots}
\usepgfplotslibrary{patchplots}

\begin{document}

\pgfplotsset{
colormap={whitered}{color(0cm)=(white); color(1cm)=(orange!75!red)}
}

\begin{tikzpicture}[
rotate=30,
declare function={mu1=1;},
declare function={mu2=2;},
declare function={sigma1=0.5;},
declare function={sigma2=1;},
declare function={normal(\m,\s)=1/(2*\s*sqrt(pi))*exp(-(x-\m)^2/(2*\s^2));},
declare function={bivar(\ma,\sa,\mb,\sb)=
1/(2*pi*\sa*\sb) * exp(-((x-\ma)^2/\sa^2 + (y-\mb)^2/\sb^2))/2;}]
\begin{axis}[
colormap name=whitered,
width=15cm,
view={0}{90},
enlargelimits=false,
domain=0:2,
y domain=0:4,
samples=17,
axis equal image,
hide axis,
]