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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 $\left\{(X_i , C_i )\right\}$

enter image description here

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  • $\begingroup$ @HenrikSchumacher, what are the unspecified parameters here? $\endgroup$ Oct 18, 2017 at 7:55
  • $\begingroup$ @HenrikSchumacher, Please see my EDIT. I have provided the value. Can you help me now? $\endgroup$ Oct 18, 2017 at 9:06
  • $\begingroup$ @HenrikSchumacher, have you tried $\gamma=0$? You might have non-overlapping regions. Would you please post your code here? $\endgroup$ Oct 18, 2017 at 9:09
  • $\begingroup$ 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! $\endgroup$ Oct 18, 2017 at 9:25
  • $\begingroup$ Why did you remove the images? The question doesn't make sense anymore. $\endgroup$ Oct 25, 2017 at 6:19

1 Answer 1

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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]]
 ]

enter image description here

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.

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  • $\begingroup$ Very beautiful!!! $\endgroup$
    – partida
    Oct 26, 2017 at 9:48
  • $\begingroup$ @partida Thank you! $\endgroup$ Oct 26, 2017 at 9:53

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