# Cannot remove ragged edges of TransformedRegion plot

I am working with two mappings of a region in the complex plane and I can't eliminate the ragged edges of the region in the second map and do not see anyway to add points to make the perimeter smooth and was wondering if there is an option I'm not aware of?. The maps are:

$$r1=\{z\in\mathbb{C} : |\log(z)\big|\leq 1\}$$

$$r2=\{z^{1/z}: z\in \text{r1}\}$$

I'm using the following code:

r1 = ImplicitRegion[
Abs[Log[r Exp[I t]]] <= 1, {{r, 0, 3}, {t, -Pi, Pi}}]
r1Plot = Region[r1, Axes -> True, PlotRange -> 3]

g[{r_, t_}] := ReIm@((r Exp[I t])^(1/(r Exp[I t])));
r2 = TransformedRegion[r1, g];
r2Plot = Region[r2, Axes -> True, PlotRange -> 3]


This produces (the perimeter edges are more noticeable in the notebook): • r1Plot = Style[Region[r1, Axes -> True, PlotRange -> 3, PerformanceGoal -> "Quality"], Antialiasing -> True]. I don't know how to avoid the mesh lines artifacts which shows up when using Antialiasing, however. See: mathematica.stackexchange.com/questions/381/…, mathematica.stackexchange.com/q/2629/55405 – WeavingBird1917 Jan 2 at 13:23
• Thanks. However, it's the second plot that has the polygon-like perimeter on part of the region I'm trying to smooth-out. And when I use those options on this plot, I still obtain the rough edges. Also, I know the region should be smooth by other methods. – Dominic Jan 2 at 13:36

If you use BoundaryDiscretizeRegion, you can control the resolution of the boundary:
BoundaryDiscretizeRegion[r2, MaxCellMeasure -> "Length" -> 0.005] 