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plotting - How can I easily visualize density plots with singularities with the least loss of detail?


I'm trying to visualize electric fields. StreamPlot is helpful:


ef[q_, source_, at_] = (k q)/Norm[source - at]^3 (at - source)
myfield[x_, y_] =
   (ef[-1, {-1, 0}, {x, y}] + ef[+1, {+1, 0}, {x, y}]) /. k -> 8.99 10^9;
StreamPlot[myfield[x, y], {x, -3, 3}, {y, -3, 3}]


Output from <code>StreamPlot</code>, correctly showing the field lines


But when I want to see the strength of the field as well, StreamDensityPlot understandably chokes because $\lim_{(x,y) \to (\pm 1,0)} \lVert \texttt{myfield}[x,y] \rVert = \infty$. This is what I get:


StreamDensityPlot[myfield[x, y], {x, -3, 3}, {y, -3, 3}, 
 ColorFunction -> "TemperatureMap"]

Output from <code>StreamDensityPlot</code>: blue everywhere with a couple of bright dots at the poles


Now, I can adjust the scalar field by Mining the actual Norm with some fixed value:


StreamDensityPlot[{myfield[x, y], 
  Min[Norm[myfield[x, y]], 10^11]}, {x, -3, 3}, {y, -3, 3}, 

 ColorFunction -> "TemperatureMap"]

Output from <code>StreamDensityPlot</code> with a <code>Min</code> clamp; a bit nicer


But this requires some trial and error to find a good value, and it still doesn't look particularly great (there's noticeable clipping). More importantly, there's really only two regions: the poles (red) and the farfield (blue); I don't really gain much insight into the field strength, at, say, $(0, \frac{1}{2})$.


Throwing a Log in there gives you more contrast, but you still have to fiddle with the clamp:


StreamDensityPlot[{myfield[x, y], 
  Min[Log[Norm[myfield[x, y]]], 28]}, {x, -3, 3}, {y, -3, 3}, 
 ColorFunction -> "TemperatureMap"]

Output of <code>StreamDensityPlot</code> with a <code>Min</code> clamp of the <code>Log</code> transform



Hence my question. How can I get StreamDensityPlot to



  • yield nice output even when the domain contains singularities,

  • such that I can clearly see both the singularities and the farfield, and the regions in between,

  • while still retaining a reasonable degree of physical accuracy,

  • and as automatically as possible? (e.g., I don't very much like having to manually specify the scalar field)


I read the main StreamDensityPlot documentation and skimmed the "Options" section (that's how I found that you can manually specify the scalar field) but didn't see anything pertinent.



Answer



vals = Table[Norm[myfield[x, y]], {x, -3, 3, 6/100}, {y, -3, 3, 6/100}];

m = Mean@Log@Flatten@vals;
st = StandardDeviation@Log@Flatten@vals;

(* For some cases you may use these instead
   m = NIntegrate[Log@Norm[myfield[x, y]], {x, -3, 3}, {y, -3, 3}]/36;
   st = NIntegrate[(m - Log@Norm[myfield[x, y]])^2, {x, -3, 3}, {y, -3, 3}]/36 // Sqrt;
*)
Manipulate[
 sta = a st; 
 StreamDensityPlot[{

       myfield[x, y],  
       Rescale[Log@Norm[myfield[x, y]], {m - sta, m + sta}]}, 
       {x, -3, 3}, {y, -3, 3}, ColorFunction -> "TemperatureMap", 
       ColorFunctionScaling -> False], 
{a, 1, 5}]

Mathematica graphics


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