# How to draw this figure in Mathematica?

How can I draw such a figure in Mathematica?

Let, $\gamma=0.5$, $N=1\times10^{-12}$ and $F$ is exponentially distributed with mean 1.

I believe it is based on germ-grain coverage model $\left\{(X_i , C_i )\right\}$

• @HenrikSchumacher, what are the unspecified parameters here? Oct 18, 2017 at 7:55
• @HenrikSchumacher, Please see my EDIT. I have provided the value. Can you help me now? Oct 18, 2017 at 9:06
• @HenrikSchumacher, have you tried $\gamma=0$? You might have non-overlapping regions. Would you please post your code here? Oct 18, 2017 at 9:09
• I posted the code in the mean time, Now you can try yourself. Do not forget to post your result as an answer when you managed to do it correctly! Oct 18, 2017 at 9:25
• Why did you remove the images? The question doesn't make sense anymore. Oct 25, 2017 at 6:19

I do not understand the part about positioning the points X_i, so I just sample uniformly on some $(2 R) \times (2 R)$ square.

R = 2.5;
n = 20;
\[Lambda] = 10.;
X = RandomReal[{-R, R}, {n, 2}];
F = RandomVariate[ExponentialDistribution[1], n];

\[Gamma] = 2.0;
\[Tau] = 0.5;
\[Beta] = 3.;
ell = t \[Function] t^-\[Beta];
NN = 10^-12 8000^\[Beta];

r = Table[F[[i]] ell[Norm[{x, y} - X[[i]]]], {i, 1, n}];
s = Table[r[[i]]/(\[Gamma] Total[Delete[r, i]] + NN), {i, 1, n}];
Show[
Table[
RegionPlot[s[[i]] >= \[Tau], {x, -R, R}, {y, -R, R},
MaxRecursion -> 2,
PlotPoints -> 40,
PlotStyle -> {FaceForm[{Opacity[0.5], ColorData[97][i]}]},
BoundaryStyle -> ColorData[97][i]],
{i, 1, n}],
Graphics[Point[X]]
]


Note: A rearranged the terms a bit. In the OP's example, the $8000$ appearing within the power $-3$ is pretty nasty from a numerical point of view. That also explains why $N$ was chosen so tiny.

Edit: With $\gamma = \frac{1}{2}$, this did not work out well, but $\gamma = 2$ seems to do the job. Still, the cells are not as convex as in the original example, but at least they seem to be disjoint.

• Very beautiful！！！ Oct 26, 2017 at 9:48
• @partida Thank you! Oct 26, 2017 at 9:53