Random non-overlapping disks in a square

Question

I've found a nice code snippet by @belisarius from this question, that I'll reproduce here for reference:

SeedRandom[5];

f := {RandomReal[{0, 10}, 2], RandomReal[{0.05, 3}]}

l = {f};

While[Length@l < 20, While[k = f;
    Not[And @@ ((# + k)[[2]] < EuclideanDistance[#[[1]], k[[1]]] & /@ 
   l)]];
AppendTo[l, k]];

Graphics[{Circle @@@ l, FaceForm[Transparent], EdgeForm[Red], 
    Polygon[{{0, 0}, {0, 10}, {10, 10}, {10, 0}, {0, 0}}]}]

Now, I would like to add two things to it :

1. All circles (disks) should be randomly colorized (from any color palette),

2. The parts outside the square should be removed, while leaving the disk's part inside.

Also, I would like a denser generator: all random sized disks should touch its neighbors.

How could we achieve this, using Mathematica version 7.0 ?

Answer 1

The trick is to place some random non-overlapping Disks in your square area, then use the DistanceTransform to find a point in your square area that is the farthest from its nearest disk. (Such a point will be equidistant from at least two disks--generally three or more disks.) Place a new disk centered at this point, with its radius equal to the distance to its nearest existing disk. Iterate this procedure (here 60 times) using Nest. Then merely plot colored Circles with those centers and radii.

What is nice about this algorithm is that it iteratively finds the largest possible circle consistent with the existing circles, and therefore the radii will never increase during iteration. Note too that it is possible that the algorithm will introduce circles whose centers are on the perimeter of your square area.

startingcenters = {{{5, 8}, 1}, {{1, 6}, 2}, {{9, 8}, .5}, 
                   {{8, 2}, .5}, {{1, 1}, 1}}; 
f[centerset_List] := 
 Module[{m = ImageData[
         DistanceTransform[
         Image[
         Graphics[
         Disk @@@ centerset,
         PlotRange -> {{0, 10}, {0, 10}},
         ImageSize -> {1000, 1000}]]]]},
  {{#[[2]]/100, 10 - #[[1]]/100}, Max[m]/100} &@ Position[m , Max[m]][[1]]
  ];
finalcenters = Nest[Union[#, {f[#]}] &, startingcenters, 60];
Graphics[{Hue[RandomReal[]], #} & /@ (Circle @@@ finalcenters),
  PlotRange -> {{-0.04, 10.04}, {-0.04, 10.04}},
  Epilog -> {Red, Line[{{0, 0}, {0, 10}, {10, 10}, {10, 0}, {0, 0}}]}]

Answer 2

The following code is faster, but has its own problems. That is, for some settings of the radius bounds it is impossible to fit a large number of disks into the square. In addition, there are fewer tangent points than in the answer by @DavidGStork; however, every disk is tangent to at least one other.

The radius of a new disk is the minimum of the distances to all disk centres minus the disk radii. If the minimum is negative then the new centre lies within an existing disk.

MinCircleRadius[newCentre_List, circles_List] :=
   Min[Map[Norm[newCentre - #] &, circles[[All, 1]]] - circles[[All, 2]]]

Find a new disk with radius between rlo and rhi, within a square of side 10, centred on the origin.

FindNewCircle[circles_, rlo_, rhi_] :=
    Block[{c = RandomReal[{-5., 5.}, 2], r},
       While[(r=MinCircleRadius[c, circles])<rlo || r>rhi, c=RandomReal[{-5.,5.},2]];
       {c, r}]

Place disks.

PackSomeCircles[n_, rlo_, rhi_] :=
   Nest[Flatten[{#, {FindNewCircle[#, rlo, rhi]}}, 1] &,
        {{RandomReal[{-4, 4}, 2], RandomReal[{rlo, rhi}]}}, n]

Begin with a small lower bound to avoid impossible fits.

Manipulate[
   Module[{c},
      SeedRandom[seed];
      c = PackSomeCircles[n, rlo, rhi];
      Graphics[{
         EdgeForm[{Thick, Black}],
         Map[{ColorData[cs, #[[2]]], Apply[Disk, #]} &, c],
         Thickness[0.02], Line[{{-5,-5}, {5,-5}, {5,5}, {-5,5}, {-5,-5}}]
      }, Background -> Gray, PlotRange -> 5 {{-1, 1}, {-1, 1}}]],
   {{n, 100, "Number of Disks"}, 2, 250, 1, Appearance -> "Labeled"},
   {{rlo, 0.02, "Radius Lower Bound"}, 0.02, 0.99*rhi, Appearance -> "Labeled"},
   {{rhi, 1.0, "Radius Upper Bound"}, 1.01*rlo, 2.0, Appearance -> "Labeled"},
   {{seed, 1, "Random Seed"}, 1, 2000, 1, Appearance -> "Labeled"},
   {{cs, "DarkRainbow", "Colour Scheme"}, ColorData["Gradients"]}]

Answer 3

With the solution above suggested by David G. Stork, I made this complete (?) working solution.

I'm now wondering if it could be improved in some way. Currently, it's a bit slow.

SeedRandom[];

f := {RandomReal[{0, 10}, 2], RandomReal[{0.5, 3}]}
l = {f};

While[Length@l < 10, While[k = f;
    Not[And @@ ((# + k)[[2]] < EuclideanDistance[#[[1]], k[[1]]] & /@l)]];
    AppendTo[l, k]];

Graphics[{Circle @@@ l, FaceForm[Transparent], EdgeForm[Black], Polygon[{{0, 0}, {0, 10}, {10, 10}, {10, 0}, {0, 0}}]}]

numCircles = 100;

g[centerset_List] := Module[{m = ImageData[DistanceTransform[Image[Graphics[Disk @@@ centerset, 
     PlotRange -> {{0, 10}, {0, 10}}, 
     ImageSize -> {1000, 1000}]]]]}, {{#[[2]]/100, 10 - #[[1]]/100}, Max[m]/100} &@Position[m, Max[m]][[1]]];

finalcenters = Nest[Union[#, {g[#]}] &, Circle @@@ l, numCircles];

Graphics[{ColorData["Aquamarine", RandomReal[]], #} & /@ (Disk @@@finalcenters), PlotRange -> {{-0.04, 10.04}, {-0.04, 10.04}}, Epilog -> {Black, Line[{{0, 0}, {0, 10}, {10, 10}, {10, 0}, {0, 0}}]}]

(I'm unable to upload a preview picture from my HD. The upload interface doesn't work with OS X ?)