I'm trying to find the term for something I thought of but I assume someone else has done it before.
Let us say I have two stamps, a square one and a circle one. I first stamp down the square one, then, for every point that has been stamped by the square one I stamp the circle one down with the center of the circle exactly on point. This can also be done in reverse order where the circle is stamped first and the square is stamped on every point that the circle is stamped. What I will get as a result is commonly referred to as a 'rounded square' where a circle is split into four parts and their edges are connected with straight lines.
The same shape can be achieved by taking a function $S(x,y)$ that is 1 if the point $(x,y)$ is inside the square and otherwise 0, and $C(x,y)$ that is 1 if the point $(x,y)$ is inside the square and otherwise 0. Then we take the convolution of this function, call it $R(x,y)$, and then define the resulting shape as the set of points where $R(x,y)\neq0$.
What is something like this called?
This is called the Minkowski sum. For two sets $A, B \subseteq \mathbb{R}^n$ of vectors in $\mathbb{R}^n$ it is given by the set of all pairwise sums
$$A + B = \{ a + b : a \in A, b \in B \}.$$