Talk:Minkowski addition

In Essential Minkowski sum

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

How can you substract a set Y from an element z?

dilation and minkowski addition should be separate pages

Dilation in binary morphology is essentially equal to the Minkowski addition; however, the morphological term dilation has evolved significantly since, and dilation in complete lattices today is very different.

A separate page for morphological dilation should be created. I would gladly edit it. —Preceding unsigned comment added by Renatokeshet (talk • contribs) 14:08, 24 May 2008 (UTC)

Well, I finally found out how to do it myself, so I did it. Let me know if you have any objections. Thanks. Renato (talk) 07:16, 8 July 2008 (UTC)

Image

In this image's current version

• what is a in B+a?
• Every SVG-capable program I tested renders this image differently (purple lines instead of blue-red lines). Firefox even does not want to render it.

Also, what is the Minkowski difference?

Thanks, --Abdull (talk) 07:38, 29 June 2008 (UTC)

Attention needed

• Explain - what is it in relation to the world? what is it used for?

This mathematical concept is f.ex. used in robotics (obstacle avoidance). Wikispaghetti (talk) 09:19, 11 May 2015 (UTC)

• Layperson explanation - the definition is too precise for the general layperson to undersatnd

Chaosdruid (talk) 00:14, 8 August 2010 (UTC)

Digressions removed

I removed these digressions ( Kiefer.Wolfowitz 21:59, 10 May 2011 (UTC))

This defines a binary operation called Minkowski addition, named after Hermann Minkowski. It occurs in a basic step in proving Minkowski's theorem, in the form

C + C = 2C

for a convex symmetric set containing 0, where the left-hand side is the Minkowski sum and the right-hand side the enlargement by a factor of 2.

This operation is sometimes called (somewhat inappropriately) the convolution of the two sets. The actual convolution of the indicator functions of the set will be a function with the same support as the Minkowski sum.

Minkowski addition is also called the binary dilation of A by B.

Minkowski, does it refer to Hermann Minkowski?

http://en.wikipedia.org/wiki/Hermann_Minkowski

If so, is there a pointer to Minkowski's original work that could be incorporated? — Preceding unsigned comment added by 138.72.131.78 (talk) 22:06, 26 June 2013 (UTC)

The explanation is a stub

In the Example section of the article,

A = {(1, 0), (0, 1), (0, -1)}

B = {(0, 0), (1, 1), (1, -1)}

This corresponds to the blue (A) and green (B) triangles in the hexagonal image in the sidebar.

The example further explains that the Minkowski sum of these two sets is

A + B = {(1, 0), (2, 1), (2, -1), (0, 1), (1, 2), (0, -1), (1, -2)}

This is seven points, with the "extra" point being the first, (1, 0), in the exact center of the hexagon. Why include these particular seven? Why not omit (1, 0)? Why not include all three repetitions of (1, 0) involved in finding the cartesian product:

[(1, 0), (2, 1), (2, -1), (0, 1), (1, 2), (1, 0), (0, -1), (1, 0), (1, -2)]

and *then* remove the duplicates because sets do not contain duplicates? Or at least mention that the Minkowski sum is the sum of all ordered pairs of points in the cartesian product of the two sets? Or some equivalent plain english interpretation of { a + b | a ∈ A, b ∈ B } like "A set created by adding each element a in A to each element b in B"

And since "Minkowski subtraction" redirects here, can we get an example of the difference as well? The equation flows very similarly, { a - b | a ∈ A, b ∈ B } - you're still taking the cartesian product, but subtracting each pair, to get

[(1, 0), (0, -1), (0, 1), (0, 1), (-1, 0), (-1, 2), (0, -1), (-1, -2), (-1, 0)]

with duplicates removed-

A - B = {(1, 0), (0, -1), (0, 1), (-1, 0), (-1, 2), (-1, -2)}

At any rate, we need much more elaborate explanation of the mechanism at work, characterization of the resulting shapes, and what it means to be added or subtracted in this manner. ◗●◖ falkreon (talk) 20:09, 21 August 2020 (UTC)

Why is the example so complicated?

The #Example section is the first contact for many people with what a Minkowski sum is. Yet, instead of showing something simple like {1,2} + {4,7} = {1+4, 1+7, 2+4, 2+7} = {5, 8, 6, 9}, it gives an example using *vectors*! The first sentence is practically unreadble

> 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 $\mathbb {R} ^{2}$ (...)

what the heck is all this? — Preceding unsigned comment added by 185.73.25.204 (talk) 09:00, 18 March 2022 (UTC)