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Minkowski addition

Minkowski addition is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Minkowski addition rather than just read about it. In short: 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…

Minkowski addition — main illustration
Minkowski addition — illustration

Key takeaways

  • Minkowski addition belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Minkowski addition to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Minkowski addition from memory before moving on to harder problems.

Reference excerpt

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 ∁ + ( − B ) ) ∁ A + B = ( A ∁ − ( − B ) ) ∁ {\displaystyle {\begin{aligned}(A-B)+B&\subseteq A\\(A+B)-B&\supseteq A\\A-B&=(A^{\complement }+(-B))^{\complement }\\A+B&=(A^{\complement }-(-B))^{\complement }\\\end{aligned}}}

In 2D image processing 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.

A − B = { a − b | a ∈ A , b ∈ B } = A + ( − B ) {\displaystyle A-B=\{\mathbf {a} -\mathbf {b} \,|\,\mathbf {a} \in A,\ \mathbf {b} \in B\}=A+(-B)}

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

A = { ( 1 , 0 ) , ( 0 , 1 ) , ( 0 , − 1 ) } {\displaystyle A=\{(1,0),(0,1),(0,-1)\}}

and

B = { ( 0 , 0 ) , ( 1 , 1 ) , ( 1 , − 1 ) } {\displaystyle B=\{(0,0),(1,1),(1,-1)\}}

… excerpt ends here. Continue reading the full article.

Illustrations

Minkowski addition: The red figure is the Minkowski sum of blue and green figures.
The red figure is the Minkowski sum of blue and green figures.
Minkowski addition: Minkowski sum A + B
Minkowski sum A + B
Minkowski addition: An example of a non-convex set such that A + A ≠ 2 A . {\textstyle A+A\neq 2A.}
An example of a non-convex set such that A + A ≠ 2 A . {\textstyle A+A\neq 2A.}
Minkowski addition: 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.
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.

Worked examples

Example 1 — a first encounter with Minkowski addition

Start with the simplest possible case. Write down what Minkowski addition claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Minkowski addition before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Minkowski addition ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Minkowski addition

In research
Minkowski addition appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Minkowski addition in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Minkowski addition is common in secondary-school and first-year university syllabi. It links to neighbouring topics Abelian group theory, Affine geometry, Binary operations, so understanding it makes those chapters shorter.
In everyday life
Look for Minkowski addition outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Minkowski addition in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Minkowski addition means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Minkowski addition out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Minkowski addition in simple terms?

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 spac…

Why does Minkowski addition matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Minkowski addition?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Minkowski addition.

Tags

  • Abelian group theory
  • Affine geometry
  • Binary operations
  • Convex geometry
  • Digital geometry
  • Geometric algorithms
  • Hermann Minkowski
  • Sumsets
  • Theorems in convex geometry
  • Variational analysis

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