Minkowski addition

In geometry, the Minkowski sum of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B: The Minkowski difference (also Minkowski subtraction, Minkowski decomposition, or geometric difference) is the corresponding inverse, where 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. 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. In 2D the Minkowski sum and difference are known as dilation and erosion. An alternative definition of the Minkowski difference is sometimes used for computing intersection of convex shapes. This is not equivalent to the previous definition, and is not an inverse of the sum operation. Instead it replaces the vector addition of the Minkowski sum with a vector subtraction. If the two convex shapes intersect, the resulting set will contain the origin. The concept is named for Hermann Minkowski. For example, if we have two sets A and B, each consisting of three position vectors (informally, three points), representing the vertices of two triangles in , with coordinates and then their Minkowski sum is which comprises the vertices of a hexagon. For Minkowski addition, the , containing only the zero vector, 0, is an identity element: for every subset S of a vector space, The empty set is important in Minkowski addition, because the empty set annihilates every other subset: for every subset S of a vector space, its sum with the empty set is empty: For another example, consider the Minkowski sums of open or closed balls in the field which is either the real numbers or complex numbers If is the closed ball of radius centered at in then for any and also will hold for any scalar such that the product is defined (which happens when or ).