Minkowski addition

In mathematics, the sumset of two subsets A and B of an (additive) abelian group is formed by adding each element of A to each element of B: A + B = { a + b ∣ a ∈ A , b ∈ B } . {\displaystyle A+B=\{a+b\mid a\in A,\ b\in B\}.} In geometry, the Minkowski sum of two subsets A and B of a Euclidean space is the set of the points whose position vectors form the sumset of the position vectors of A and B. The Minkowski sum depends on the choice of an origin in the Euclidean space. As a change of origin amounts to translate the Minkowski sum, the Minkowski sum is defined up to a translation, and its shape and orientation are well defined. The Minkowski difference (also Minkowski subtraction, Minkowski decomposition, or geometric difference) is the corresponding inverse, where ( A − B ) {\textstyle (A-B)} produces a set that could be summed with B to recover A. This is defined as the complement of the Minkowski sum of the complement of A with the reflection of B about the origin. − B = { − b | b ∈ B } A − B = ( A ∁ + ( − B ) ) ∁ {\displaystyle {\begin{aligned}-B&=\{\mathbf {-b} \,|\,\mathbf {b} \in B\}\\A-B&=(A^{\complement }+(-B))^{\complement }\end{aligned}}} This definition allows a symmetrical relationship between the Minkowski sum and difference. Note that alternately taking the sum and difference with B is not necessarily equivalent. The sum can fill gaps which the difference may not re-open, and the difference can erase small islands which the sum cannot recreate from nothing. ( A − B ) + B ⊆ A ( A + B ) − B ⊇ A A − B = ( A ∁ + ( −...