Minkowski sum and product of 2D-regions

Question

Let $A_j$, $j=1,\dots,n$ be subsets of $\mathbb C$ defined by $$ A_j=\{z\,|\,f_j(z)\geq0\} $$ where $f_j:\mathbb C\to\mathbb R$ are given functions.

I am interested in the following subsets of $\mathbb C$ (maybe they could be called Minkowski sum and product respectiverly): $$ A_1+A_2+\dots+A_n := \{z_1+z_2+\dots+z_n\,|\,z_j\in A_j\,\forall j=1,\dots,n\} \,,$$ $$ A_1\cdot A_2\cdot\dots\cdot A_n := \{z_1 z_2\cdots z_n\,|\,z_j\in A_j\,\forall j=1,\dots,n\} \,.$$ How can I plot them in Mathematica?

Example Let's take the ball $$A_1=A_2=\{z\in\mathbb C\,|\, |z-1|\leq 1\} = \{1+r\,e^{it}\,|\,0\leq r\leq1,\,0\leq t<2\pi\}$$ then I would like to plot the region in $\mathbb C$ obtained by multiplying any two points in the ball, that is $$A_1\cdot A_2=\{z_1\cdot z_2 \,|\, |z_1-1|\leq 1,\, |z_2-1|\leq1\} =\{(1+r_1e^{it_1})(1+r_2e^{it_2}) \,|\, 0\leq r1,r_2\leq1,\,0\leq t_1,t_2<2\pi\} \,.$$ But I am not able to do it because ParametricPlot requires only 2 parameters, while I have 4 (that are $t_1,t_2,r_1,r_2$).

Answer 1

If

1. you have only two regions $A_1$ and $A_2$, and
2. every point in their sum/product can be attained as the sum/product of points on the boundaries of $A_1$ and $A_2$,

then you could do something like this:

ParametricPlot[
 Through@{Re, Im}@((1 + Exp[I t1]) (1 + Exp[I t2])), 
 {t1, -Pi, Pi}, {t2, -Pi, Pi}, Mesh -> 7]