# Plot: Construct an Amoeba

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

The result should look like this:

• When I run your code I don't get the green background or the lines inside the figure. Commented Jun 3, 2015 at 16:20
• Fine with me too Commented Jun 3, 2015 at 16:29

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]


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]


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]