Minkowski addition

The Minkowski sum (after Hermann Minkowski ) of two sets A and B with elements of a vector space is the resulting set of sums of all elements of A and all elements of B.

Definition

Be two subsets of a vector space. Then the Minkowski sum is defined by

Some of the Minkowski sum is also listed with the characters instead of the normal plus sign. In the field of linear algebra and functional analysis, however, this can lead to confusion with the direct sum.

Applications is the Minkowski sum, for example, in the 2D and 3D computer graphics and image processing ( specifically morphology, . Is there, however, usually called binary dilation or dilating The counterpart is the erosion), in the linear programming (for example, the Minkowski sum of a polytope and a Polyederkegels ), in the functional analysis and in the robot controller.

Properties

The Minkowski sum is associative, commutative and distributive with respect to the union of sets, that is.

For the power of the Minkowski sum, because each element is added with each and multiple sums are only once in the set.

The Minkowski sum of convex sets is a convex set again. For convex sets, the computation of the Minkowski sum can also be very easily done graphically: push a polytope on the edge of the other along and the covered area is the Minkowski sum.

Example

Where A and B elements:

Then the Minkowski sum of A and B to stubborn calculation:

The point (1,0) is three times before, that is,

A and B represent isosceles triangles ( convex) represents the Minkowski sum is a convex hexagon which you have been incurred by driving along B may be regarded on the edge of A, as the picture shows.