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This is a very naive question about the Minkowski addition. I hope not to be off-topic.

I read in the Wikipedia's article dedicated to it :

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.

I don't understand the relationship "cutting piece + trajectory = the shape of the cut in the material" .

By example, if I look at the first image of the article (blue+green = red) enter image description here

... I don't get it : where would be the cutting piece ? what about the trajectory ?

Any help would be appreciated !

suizokukan
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    You might be confused, because that's not quite true. Let $A$ be the shape of the cutting tool, and $P$ the trajectory of the cut. The shape of the cut is the sum of the translations of $A$ along the curve $P$. Minkowski sum gives you something similar, but not exactly, for example if $A$ is a circle of radius $1\mathrm{unit}$ and $P$ consists of edges of a square of size $10\mathrm{units}$, then the cut would be a rounded-corner square of size $11\mathrm{units}$ with the thickness $1\mathrm{unit}$ while the Minkowski sum will get you thickness $2\mathrm{units}$ and size $12\mathrm{units}$. – dtldarek Nov 25 '14 at 14:14

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