Fill an ellipse with random dots

Question

For a paper, I must draw an ellipse, which is filled by random dots in a coordinate system.

Of course it isn't a problem to draw an ellipse and the coordinate system. Also I know the function "rand" to create random dots in combination with "only marks". How is it possible to create these (random) dots in an ellipse?

Answer 1

Here's a very, very easy one. Some (or, in worst case, all) points might be invisible as they are hidden. You can alter \pgfmathsetseed to get another distribution or comment it out to get a new distribution on every run.

Code

\documentclass[tikz, border=2mm]{standalone}

\begin{document}

\begin{tikzpicture}

\draw (0,0) ellipse (4 and 2);
\clip (0,0) ellipse (4 and 2);
\pgfmathsetseed{24122015}
\foreach \p in {1,...,50}
{ \fill (4*rand,2*rand) circle (0.05);
}

\end{tikzpicture}

\end{document}

Output

---

Edit 1: I played around with it a little, as I wanted to keep all points inside of the ellipse. The second version generates a random x value and then computes a random y value in such a way that it is inside. As there is less "y space" for extremal values of x, the points cluster at the end points of the major axis. The third is an ellipse in polar form: first, an angle is randomly chosen and the radius is computed to be on the inside of the ellipse. Here, the points cluster at the minor axis and the center, a so called Gorthaur¹-Distribution.

Code

\documentclass[tikz, border=2mm]{standalone}

\begin{document}

\begin{tikzpicture}
    \draw (0,0) ellipse (4 and 2);
    \clip (0,0) ellipse (4 and 2);
    \pgfmathsetseed{24122015}
    \foreach \p in {1,...,1000}
    { \fill[black]  (4*rand,2*rand) circle (0.05);
    }
\end{tikzpicture}

\begin{tikzpicture}
    \draw (0,0) ellipse (4 and 2);
    \clip (0,0) ellipse (4 and 2);
    \pgfmathsetseed{24122015}
    \foreach \p in {1,...,1000}
    { \pgfmathsetmacro{\x}{4*rand}
        \pgfmathsetmacro{\y}{rand*0.5*sqrt(16-pow(\x,2))}
        \fill[black]    (\x,\y) circle (0.05);
    }
\end{tikzpicture}

\begin{tikzpicture}
    \fill[inner color=black, outer color=yellow!20!black] (0,0) ellipse (4 and 2);
    \clip (0,0) ellipse (4 and 2);
    \pgfmathsetseed{24122015}
    \foreach \p in {1,...,1000}
    { \pgfmathsetmacro{\t}{360*rnd}
        \pgfmathsetmacro{\r}{rnd*4*2/(sqrt(pow(2*cos(\t),2)+pow(4*sin(\t),2)))}
        \pgfmathsetmacro{\c}{abs(\r)/4*100}
        \fill[yellow!\c!red]    (\t:\r) circle (0.05);
        \typeout{\t, \r, \c}
    }
\end{tikzpicture}

\end{document}

Output

¹: I totally made this up. Gorthaur is a name Sauron, the villain in the lord of the rings, used in earlier ages ;-) As I usually watch the trilogy with my family over christmas I felt this little joke was in order.

Answer 2

Whenever there's a question about filling something with random dots, there has to be an answer using JLDiaz's Poisson Disc Sampling code:

\documentclass{article}
\usepackage{tikz}
\usepackage{poisson}
\begin{document}
\edef\mylist{\poissonpointslist{8}{4}{0.1}{20}}
\begin{tikzpicture}
\begin{scope}
    \clip (4,2) ellipse (4 and 2);
    \foreach \x/\y in \mylist {
        \fill (\x,\y) circle(1pt);
    }
\end{scope}
\draw (4,2) ellipse (4 and 2);
\end{tikzpicture}
\end{document}

---

And here's the approach from Filling specified area by random dots in TikZ: 150 equally distributed dots that all lie in the ellipse:

The equation for the top half of the ellipse is

sqrt( 2^2 * (1 - x^2/(4^2) )

The antiderivative of the inverse is

2 * asin(x/4)

and the scaling factor is 4/pi.

\documentclass[tikz, border=2mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\pgfmathsetseed{3}

\begin{document}

\begin{tikzpicture}
\begin{axis}[
    hide axis,
    axis equal,
    declare function={a(\x) = sqrt( 2^2 * (1 - \x^2/(4^2) );},
    declare function={b(\x) = -sqrt( 2^2 * (1 - \x^2/(4^2) );},
    declare function={f(\x) = 2 * rad(asin(x/4)) * 4 / pi;}
]
\addplot [only marks, samples=150, domain=-4:4] ({f(x) },{rand * ( a(f(x)) - b(f(x)) ) / 2} );
\draw (0,0) ellipse [x radius=4, y radius=2];
\end{axis}

\end{tikzpicture}

\end{document}

---

Here's the code for an ellipse with x radius=2 and y radius=1:

\documentclass[tikz, border=2mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\pgfmathsetseed{3}

\begin{document}

\begin{tikzpicture}
\begin{axis}[
    %hide axis,
xmin=-4, xmax=4,
    axis equal,
    declare function={a(\x) = sqrt( 1^2 * (1 - \x^2/(2^2) );},
    declare function={b(\x) = -sqrt( 1^2 * (1 - \x^2/(2^2) );},
    declare function={f(\x) = 2 * rad(asin(x/4)) * 2 / pi;}
]
\addplot [only marks, samples=150, domain=-4:4] ({f(x) },{rand * ( a(f(x)) - b(f(x)) ) / 2} );
\draw (0,0) ellipse [x radius=2, y radius=1];
\end{axis}

\end{tikzpicture}

\end{document}