Sums vector sets A and B by adding each vector in A to each vector in B
The red figure is the Minkowski sum of blue and green figures.
In geometry, the Minkowski sum 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:
The Minkowski difference (also Minkowski subtraction, Minkowski decomposition, or geometric difference)[1] is the corresponding inverse, where 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.[2]
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.
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.[3] 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.
The concept is named for Hermann Minkowski.
^Hadwiger, Hugo (1950), "Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt", Math. Z., 53 (3): 210–218, doi:10.1007/BF01175656, S2CID 121604732, retrieved 2023-01-12
^Li, Wei (Fall 2011). GPU-Based Computation of Voxelized Minkowski Sums with Applications (PhD). UC Berkeley. pp. 13–14. Retrieved 2023-01-10.
^Lozano-Pérez, Tomás (February 1983). "Spatial Planning: A Configuration Space Approach" (PDF). IEEE Transactions on Computers. C-32 (2): 111. doi:10.1109/TC.1983.1676196. hdl:1721.1/5684. S2CID 18978404. Retrieved 2023-01-10.
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