### Trying to draw the Amoeba

With Mathematica, it's possible to graph $e^{-k x} + e^{-k y} = 1,e^{-k x} - e^{-k y} = 1$ and $e^{-k x} + e^{-k y} = -1$ to get the amoeba of **1 + x + y** when **k = 1**. Then by plotting when $k \to \infty$, these graphs converge to the tropical polynomial

However, I meant to draw **Min(1,x,y)** which I fixed by shifting my coordinates **x,y** by the vector **(1,1)**. This is a bit ad-hoc and I am probably not understanding how constant functions "tropicalize".

### Main question

I would like to "fatten" **Min[x, y, 1, x + y + 1]** into its amoeba, so I thought the right curve should be **1 + x + y + xy**. However, my "neck" is disappearing in the scaling limit. *How should I scale the coefficients correctly to get my amoeba?*

Following Zeb's suggestion (but a few second before he posted it) I came up with this imge

However, this "dequantization" procedure doesn't always produce the whole tropical curve. Here's the curve I drew to "requantize" **Min[1, x , y , x+ y + 1, -2 + 2x , 2y]**. A line has to be missing b/c of the zero tension condition, as in Tropical Mathematics by David Speyer and Bernd Sturmfels.

Here is the code. You have to draw 4 different versions of the curve to get all the absolute values. Maybe this should somehow involve complex phases as well.

```
q[x_] := E^( x)
f[a_, b_, c_] := c + a + b + c a b + (1/c^2) a^2 + b^2;
{x0, y0} = { -1, -3};
L = 5;
k = 8;
ContourPlot[
{ f[q[k x], q[k y], q[k ]] == 0, f[-q[k x], q[k y], q[k ]] == 0,
f[q[k x], -q[k y], q[k ]] == 0, f[-q[k x], -q[k y], q[k ]] == 0},
{x, x0, x0 + L}, {y, y0, y0 + L}
]
```

Ideally, I want to take any tropical curve and fatten it into its amoeba. Tropical conics and cubics seem the best starting point.

In anticipation of comments, by "amoeba" here I think I mean the boundary of the 2D region which is usually called "amoeba".