# Visualising the Minkowski sum of two triangles

Consider the figure given below. I am told that the Minkowski sum of the two triangles $$A,B \subset \mathbb{R}^2$$ is the regular hexagon, as shown right next to them.

The Minkowski sum is defined as $$A+ B = \{a+b: a\in A, b\in B\}$$

Note that $$A$$ and $$B$$ are such that $$A$$ is the reflection of $$B$$ about the origin (and vice versa). So, knowing that the sum is a regular hexagon, I can somewhat make sense of it, but I'm not really able to see why it is a regular hexagon in the first place! Is there an intuitive way of looking at this? Finding $$A+ B = \{a+b: a\in A, b\in B\}$$ manually using every $$a\in A, b\in B$$ is obviously impossible, so there must be some special $$a_1,a_2,...,a_p \in A$$ and $$b_1, b_2,...,b_q \in B$$ we should use to construct the figure (extreme cases of some sort). Which ones are these?

As an addendum, I am wondering if there is some general intuition (at least in $$\mathbb{R}$$,$$\mathbb{R}^2$$, and $$\mathbb{R}^3$$) I can carry with me, when finding the Minkowski sum of two figures (sets)?

P.S.
I am not looking for algorithms to find Minkowski sums.

• Cut them out from paper. Pick the bottom left corner of the second triangle and traverse with it over the whole of the first triangle. What sort of figure gets covered by all of the copies of the second triangle as its bottom left corner moves across the first triangle?
– user700480
Commented Dec 23, 2020 at 13:26

In your notation, we're looking at translates of $$a$$ in the boundary of $$A$$ by $$b$$ in the boundary of $$B$$, where each frame shows a single value of $$b$$. When the shapes are convex, the Minkowski sum is given by taking the convex hull of these "extreme" sums. In fact, in the case of convex polygons (or polyhedra, in the 3-dimensional case), it suffices to take just the vertices of $$A$$ and $$B$$.