# Combining several geometric regions by adding pointwise?

Given two regions $$D_1, D_2 \subseteq \mathbb{R}^2$$, I'm looking to combine them by adding pointwise: $$D_1 + D_2 := \left\{ (a+x, b+y) \mid (a,b) \in D_1, (x,y) \in D_2 \right\}.$$ Even more generally, given a map $$f:\mathbb{R}^l \times \mathbb{R}^m \to \mathbb{R}^n$$, combine the regions $$D_1\subseteq \mathbb{R}^l, D_2 \subseteq \mathbb{R}^m$$ using $$f$$ over all pointwise combinations: $$f(D_1 , D_2) := \left\{ f (\vec{a}, \vec{x} ) \mid \vec{a} \in D_1, \vec{x} \in D_2 \right\}.$$

Is there an elegant way to do this in Mathematica? Something like f[r1,r2] or Outer[f, r1,r2,1]? Here's a MWE and the closest I've been able to come (which works, but feels clunky). My approach is to explicitly construct the Cartesian product using RegionProduct and then apply a mapping f using TransformedRegion

r1=Rectangle[]; r2=Disk[]; f[{a_,b_,x_,y_}]:={a+x,b+y} Region@TransformedRegion[RegionProduct[r1, r2], f]

Surely there's a built-in function or piece of syntax I'm missing?