Radial Random Walk

Question

I'm trying to generate a spherical distribution of radial random walk points in 3D space. The following code works, but the random walk lines aren't radial. Why ? Where is my mistake ?

MinSprite := 0.006; (* min radius of sprites *)
MaxSprite := 0.03; (* max radius of sprites *)
SpriteOverlap := 0.75; (* min separation between sprites *)
IterationStep := 0.1;
NumberOfSteps := 20;
thickness = 0.09;
pointsmean = 20;
pointssd = 12;

SpriteSize[p_] := MinSprite + (MaxSprite - MinSprite)Norm[p];

SeedRandom[];
RandomWalk = Flatten[Table[{x,y,z}={dist Sqrt[1 - cosinus^2]Cos[phi],dist Sqrt[1 - cosinus^2]Sin[phi],dist cosinus};
    {u,v, w}={0.0, 0.0, 0.0};
    dist = RandomReal[{5,10}];
    phi = RandomReal[{0,2Pi}];
    cosinus = RandomReal[{-1,1}];
    velocity = Abs[RandomReal[NormalDistribution[0,s]]];

Line[NestList[(
    u+=velocity Sqrt[1 - cosinus^2]Cos[phi];
    v+=velocity Sqrt[1 - cosinus^2]Sin[phi];
    w+=velocity cosinus;
    #+IterationStep{u,v, w})&,{x,y, z},NumberOfSteps]],{s,0.25,0.75,0.007}][[All,1]],1];

CloudsParticles = Flatten[Table[(#+RandomReal@LaplaceDistribution[0,thickness])&/@#,{Max[1,IntegerPart@RandomReal@NormalDistribution[pointsmean,pointssd]]}]&/@RandomWalk, 1];

max=Max[Norm/@CloudsParticles];
NormalizedParticles = CloudsParticles/max;

MinSeparation[p_] := SpriteOverlap SpriteSize[p];
KeepPoint[{p_,q_}] := Norm[p]<Norm[q]||Norm[p-q]>MinSeparation[p];
FilterOnce[pts_] := With[{nf=Nearest[pts]},Select[pts, KeepPoint[nf[#,2]]&]];
PointsCoords = FixedPoint[FilterOnce,NormalizedParticles];

ListPointPlot3D[PointsCoords,BoxRatios->{1,1,1},ImageSize->800,SphericalRegion->True,PlotStyle->{Blue,PointSize[Small]}]

Here's a sample of the output. As you can see, this isn't a radial distribution :

The mistake most probably lies in the RandomWalk declaration, but I can't see it. Anyone has an idea of what may be wrong ?

Take note that I'm using Mathematica 7.0 only.

EDIT :

I must admit that this method isn't a clever way of defining a random distribution of points around radial lines. I'll have to do it differently.

Answer 1

Put the line

{x,y,z}={dist Sqrt[1 - cosinus^2]Cos[phi],dist Sqrt[1 - cosinus^2]Sin[phi],dist cosinus};

Behind

velocity = Abs[RandomReal[NormalDistribution[0,s]]];

and the other expressions that set your variables, rather than before it. With

MinSprite := 0.006; (* min radius of sprites *)
MaxSprite := 0.03; (* max radius of sprites *)
SpriteOverlap := 0.75; (* min separation between sprites *)
IterationStep := 0.1;
NumberOfSteps := 20;
thickness = 0.09;
pointsmean = 20;
pointssd = 12;

You will see that setting

randomLines =
 Table[
  {u, v, w} = {0.0, 0.0, 0.0};
  dist = RandomReal[{5, 10}];
  phi = RandomReal[{0, 2 Pi}];
  cosinus = RandomReal[{-1, 1}];
  velocity = Abs[RandomReal[NormalDistribution[0, s]]];

  {x, y, z} =
   dist { Sqrt[1 - cosinus^2] Cos[phi], Sqrt[1 - cosinus^2] Sin[phi], 
     cosinus};

  Line[
   NestList[
    (u += velocity Sqrt[1 - cosinus^2] Cos[phi];
      v += velocity Sqrt[1 - cosinus^2] Sin[phi];
      w += velocity cosinus;
      # + IterationStep {u, v, w}) &,
    {x, y, z},
    NumberOfSteps
    ]
   ]

  ,
  {s, 0.25, 0.75, 0.007}

  ]

and then doing

Graphics3D@randomLines

yields a picture with radial random lines.

Remark

Note that

randomLines =
  Table[
   {u, v, w} = {0.0, 0.0, 0.0};
   dist = RandomReal[{5, 10}];
   velocity = Abs[RandomReal[NormalDistribution[0, s]]];

   {x, y, z} = dist RandomReal[{-1, 1}, 3];

   Line[
    NestList[
     ({u, v, w} = {u, v, w} + velocity {x, y, z};
       # + IterationStep {u, v, w}) &,
     {x, y, z},
     NumberOfSteps
     ]
    ]
   ,
   {s, 0.25, 0.75, 0.007}
   ];

also creates some radial random lines. Just as a side remark.