How to access distance of points from origin in Show with multiple ListPlot

Question

I would like to plot a graph of a random-generated list of points, so that they decay a little bit by moving from the origin. So far I wrote this piece of code, which seems to give me the desired output:

ClearAll[Evaluate[$Context <> "*"]]

MAXITER = 10;
Array[f, MAXITER, 0];


For[i = 0, i < MAXITER, i++, 

  ppXst = 1/(i + 1) * 1000;

  v = i/MAXITER;
  strip = 2/MAXITER;

    If[i == 0, strip == Abs[strip]];

  CircularDistribution[num_] :=

   Table[{Sqrt[RandomReal[{If[i == 0, .05, v], v + strip}]] {Cos[#], 
         Sin[#]} & //@ Random[Real, {0, 2 Pi}]}, {num}];

  f[i] = ListPlot[
    CircularDistribution[ppXst],
    AspectRatio -> 1, 
    ImageSize -> Medium,
    Axes -> False, 
    PlotStyle -> {{Gray, Opacity[.4], PointSize[.01]}}
    ];

  ];

Show[
 {Table[f[i], {i, 0, MAXITER - 1}],  
  Graphics[{Red, PointSize[.02],  Point[{0, 0}]}]}, 
 PlotRange -> All, 
 ImageSize -> Large
 ]

Now, what I really need is to color the points depending on the distance between them and the origin. The problem is that I did't find out a way to access the single point distance from the origin in the Show command, nor I can figure out how to color ListPlot by ListPlot in order to obtain an uniform coloured plot depending on the distance of the single point. I only get near what I want by doing so:

MAXITER = 10;
Array[f, MAXITER, 0];


For[i = 0, i < MAXITER, i++, 

  ppXst = 1/(i + 1) * 1000;
  v = i/MAXITER;
  strip = 2/MAXITER;
  If[i == 0, strip == Abs[strip]];

  CircularDistribution[num_] :=

   Table[{Sqrt[RandomReal[{If[i == 0, .05, v], v + strip}]] {Cos[#], 
         Sin[#]} & //@ Random[Real, {0, 2 Pi}]}, {num}];

  f[i] = ListPlot[
    CircularDistribution[ppXst],
    AspectRatio -> 1, 
    ImageSize -> Medium,
    Axes -> False, 
    PlotStyle -> {{Hue[v], Opacity[.4], PointSize[.01]}}
    ];

  ];

Show[
 {Table[f[i], {i, 0, MAXITER - 1}],  
  Graphics[{Red, PointSize[.02],  Point[{0, 0}]}]}, 
 PlotRange -> All, 
 ImageSize -> Large
 ]

which is not really what I want, as it's coloured per strip and not per single point distance. Also, if it's possible, I would like to superimpose to the final graph a Legend, similar to the color map of the ContourPlot which is easily achievable with PlotLegends-> Automatic, to show the distance associated to the color.

Answer 1

I define the f[i] as follows in your for loop to get all the points as a list.

f[i] = CircularDistribution[ppXst];

Then get all the points as a list.

pt = (Flatten /@ Flatten[Table[f[i], {i, 0, MAXITER - 1}], 1]);

Now you define a radial color function.

col = Function[{x, y}, ColorData["SolarColors"][Sqrt[x^2 + y^2]]];
Graphics[{{Red, PointSize[.02], 
   Point[{0, 0}]}, {Directive[col @@ #, [email protected]], PointSize[.01], 
     Point[#]} & /@ pt}]

To choose which color you want I ran over all available and made the plot. You can choose the color name and replace it in the above code which uses "SolarColors".

Answer 2

Graphics[
 {
  PointSize[Large],
  Red,
  Point[{0, 0}],
  PointSize[Medium],
  Table[
   {
    ColorData["Rainbow"][Rescale[r, {30, 0}]]
    , Point[
     CoordinateTransform[
      "Polar" -> "Cartesian", {r, RandomReal[{-π, π}]}]]
    }
   , {r, RandomVariate[GammaDistribution[10, 2], 3000]}
   ]
  }]

---

For the legend

Grid@Table[
  {ColorData["Rainbow"][Rescale[r, {30, 0}]], r}, {r, 0, 30, 5}
  ]

---

_{Code and plots done in Mathemathica 11.1.1 on Win7}