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I am trying to plot an Amoeba, e.g we consider the polynomial f(z, w) = 2 z - w + 1 and solve it explicitly , the solutions satisfy the equation w = 2z + 1, where $w,z\in \mathbb{C}$ under the map $$(Log|z|, Log|2z+1|)$$

So, I did the following:

ParametricPlot[{Log[Abs[r Exp[I s]]], Log[Abs[2 r Exp[I s] + 1]]}, {r, 0, 3}, {s, -π, π}]

I want to get the picture showed below, but the result has flaws -- I am not supposed to get those curves inside the region

enter image description here

The result should look like this:

enter image description here

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    $\begingroup$ When I run your code I don't get the green background or the lines inside the figure. $\endgroup$
    – C. E.
    Jun 3, 2015 at 16:20
  • $\begingroup$ Fine with me too $\endgroup$
    – Sektor
    Jun 3, 2015 at 16:29

2 Answers 2

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ParametricPlot[{Log[Abs[r*Exp[I*s]]], Log[Abs[2*r*Exp[I*s] + 1]]}, {r,
   0, 10}, {s, -Pi, Pi}, Frame -> False, Axes -> False, 
 PlotRange -> All, PlotStyle -> {Blue, Opacity[1]}, PlotPoints -> 60]

enter image description here

If you want to get the original pictue then:

ParametricPlot[{Log[Abs[r Exp[I s]]], Log[Abs[2 r Exp[I s] + 1]]}, {r,
   0, 3}, {s, -\[Pi], \[Pi]}, PlotTheme -> "Classic", 
 Background -> LightGreen, ImagePadding -> 15]

enter image description here

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Add a few options to get rid of the plot-junk:

ParametricPlot[{Log[Abs[r Exp[I s]]], Log[Abs[2 r Exp[I s] + 1]]}, 
  {r, 0, 3}, {s, -π, π}, Frame -> False, Axes -> False, PlotRange -> All]

enter image description here

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