Minkowski addition

File:Сумма Минковского.svg

The red figure is the Minkowski sum of blue and green figures.

In geometry, the Minkowski sum (also known as dilation) 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, i.e., the set

$A + B = \{\mathbf{a}+\mathbf{b}\,|\,\mathbf{a}\in A,\ \mathbf{b}\in B\}.$

Analogously, the Minkowski difference is defined as

$A - B = \{\mathbf{a}-\mathbf{b}\,|\,\mathbf{a}\in A,\ \mathbf{b}\in B\}.$

File:Minkowski-sumex4.svg

Minkowski sum A + B

File:Minkowski-sumex2.svg

B

File:Minkowski-sumex1.svg

A

The concept is named for Hermann Minkowski.

Example

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 $\mathbb{R}^2$, with coordinates

A = {(1, 0), (0, 1), (0, −1)} 

and

B = {(0, 0), (1, 1), (1, −1)} ,

then the Minkowski sum is
A + B = {(1, 0), (2, 1), (2, −1), (0, 1), (1, 2), (1, 0), (0, −1), (1, 0), (1, −2)}, which looks like a hexagon, with three 'repeated' points at (1, 0).

For Minkowski addition, the zero set {0}, containing only the zero vector 0, is an identity element: For every subset S, of a vector space

S + {0} = S;

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: S + $\emptyset$ = $\emptyset$.

In the convex hull of the red set, each blue point is a convex combination of some red points.

Minkowski addition of sets. The sum of the squares Q₁=[0,1]² and Q₂=[1,2]² is the square Q₁+Q₂=[1,3]².

Convex hulls of Minkowski sums

Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition:

• For all subsets S₁ and S₂ of a real vector-space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls

Conv(S₁ + S₂) = Conv(S₁) + Conv(S₂).

This result holds more generally for each finite collection of non-empty sets

Conv(∑S_n) = ∑Conv(S_n).

In mathematical terminology, the operations of Minkowski summation and of forming convex hulls are commuting operations.

If S is a convex set then also $\mu S+\lambda S$ is a convex set; furthermore

$\mu S+\lambda S=(\mu+\lambda)S$ for every $\mu,\lambda \geq 0$.

Conversely, if this "distributive property" holds for all non-negative real numbers, $\mu, \lambda$, then the set is convex.

The figure shows an example of a non-convex set for which A + A ⊋ 2A.

File:Minkowskisum.svg

An example of a non-convex set such that A + A ≠ 2A

An example in 1 dimension is: B=[1,2]∪[4,5]. It can be easily calculated that 2B=[2,4]∪[8,10] but B+B=[2,4]∪[5,7]∪[8,10], hence again B+B ⊋ 2B.

Minkowski sums act linearly on the perimeter of two-dimensional convex bodies: the perimeter of the sum equals the sum of perimeters. Additionally, if K is (the interior of) a curve of constant width, then the Minkowski sum of K and of its 180° rotation is a disk. These two facts can be combined to give a short proof of Barbier's theorem on the perimeter of curves of constant width.

Applications

Minkowski addition plays a central role in mathematical morphology. It arises in the brush-and-stroke paradigm of 2D computer graphics (with various uses, notably by Donald E. Knuth in Metafont), and as the solid sweep operation of 3D computer graphics.

Motion planning

Minkowski sums are used in motion planning of an object among obstacles. They are used for the computation of the configuration space, which is the set of all admissible positions of the object. In the simple model of translational motion of an object in the plane, where the position of an object may be uniquely specified by the position of a fixed point of this object, the configuration space are the Minkowski sum of the set of obstacles and the movable object placed at the origin and rotated 180 degrees.

Numerical control (NC) machining

In numerical control machining, the programming of the NC tool exploits the fact that the Minkowski sum of the cutting piece with its trajectory gives the shape of the cut in the material.

3d Solid Modeling

In OpenSCAD Minkowski sums are used to outline a shape with another shape creating a composite of both shapes.

Aggregation Theory

Minkowski sums are also frequently used in aggregation theory when individual objects to be aggregated are characterized via sets.

Algorithms for computing Minkowski sums

File:Shapley–Folkman lemma.svg

Minkowski addition and convex hulls. The sixteen dark-red points (on the right) form the Minkowski sum of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus-signs (+): The right plus-sign is the sum of the left plus-signs.

Planar case

Two convex polygons in the plane

For two convex polygons P and Q in the plane with m and n vertices, their Minkowski sum is a convex polygon with at most m + n vertices and may be computed in time O (m + n) by a very simple procedure, which may be informally described as follows. Assume that the edges of a polygon are given and the direction, say, counterclockwise, along the polygon boundary. Then it is easily seen that these edges of the convex polygon are ordered by polar angle. Let us merge the ordered sequences of the directed edges from P and Q into a single ordered sequence S. Imagine that these edges are solid arrows which can be moved freely while keeping them parallel to their original direction. Assemble these arrows in the order of the sequence S by attaching the tail of the next arrow to the head of the previous arrow. It turns out that the resulting polygonal chain will in fact be a convex polygon which is the Minkowski sum of P and Q.

Other

If one polygon is convex and another one is not, the complexity of their Minkowski sum is O(nm). If both of them are nonconvex, their Minkowski sum complexity is O((mn)²).

Essential Minkowski sum

There is also a notion of the essential Minkowski sum +_e of two subsets of Euclidean space. Note that the usual Minkowski sum can be written as

$A + B = \{ z \in \mathbb{R}^{n} \,|\, A \cap (\{z\} - B) \neq \emptyset \}.$

Thus, the essential Minkowski sum is defined by

$A +_{\mathrm{e}} B = \{ z \in \mathbb{R}^{n} \,|\, \mu \left[A \cap (\{z\} - B)\right] > 0 \},$

where μ denotes the n-dimensional Lebesgue measure. The reason for the term "essential" is the following property of indicator functions: while

$1_{A \,+\, B} (z) = \sup_{x \,\in\, \mathbb{R}^{n}} 1_{A} (x) 1_{B} (z - x),$

it can be seen that

$1_{A \,+_{\mathrm{e}}\, B} (z) = \mathop{\mathrm{ess\,sup}}_{x \,\in\, \mathbb{R}^{n}} 1_{A} (x) 1_{B} (z - x),$

where "ess sup" denotes the essential supremum.

See also

• Dilation
• Erosion
• Interval arithmetic
• Mixed volume (a.k.a. Quermassintegral or intrinsic volume)
• Parallel curve
• Shapley–Folkman lemma
• Zonotope
• Convolution

References

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• Mayer, A. and Zelenyuk, V. (2014) "Aggregation of Malmquist productivity indexes allowing for reallocation of resources," European Journal of Operational Research, 238:3, pp 774-785
• Zelenyuk, V. (2015) "Aggregation of scale efficiency," European Journal of Operational Research, 240:1, pp 269-277.

External links

• Minkowski Sums, in Computational Geometry Algorithms Library
• The Minkowski Sum of Two Triangles and The Minkowski Sum of a Disk and a Polygon by George Beck, The Wolfram Demonstrations Project.
• Minkowski's addition of convex shapes by Alexander Bogomolny: an applet
• Wikibooks:OpenSCAD User Manual/Transformations#minkowski by Marius Kintel: Application