Editing Minkowski addition (section)

== 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]]. It has also been shown to be closely connected to the [[Earth mover's distance]], and by extension, [[Transportation theory (mathematics)|optimal transport]].<ref>{{cite journal | journal = Discrete Applied Mathematics | title = Properties of the d-dimensional earth mover's problem | last1 = Kline | first1 = Jeffery | volume = 265 | year = 2019 | pages = 128–141 | doi = 10.1016/j.dam.2019.02.042| s2cid = 127962240 | doi-access = free }}</ref>

=== Motion planning ===
Minkowski sums are used in [[motion planning]] of an object among obstacles. They are used for the computation of the [[Configuration space (physics)|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.<ref>{{cite journal | last1 = Zelenyuk | first1 = V. | year = 2015 | title = Aggregation of scale efficiency | url = https://ideas.repec.org/a/eee/ejores/v240y2015i1p269-277.html | journal = European Journal of Operational Research | volume = 240 | issue = 1| pages = 269–277 | doi=10.1016/j.ejor.2014.06.038}}</ref><ref>{{cite journal | last1 = Mayer | first1 = A. | last2 = Zelenyuk | first2 = V. | year = 2014 | title = Aggregation of Malmquist productivity indexes allowing for reallocation of resources | url = https://ideas.repec.org/a/eee/ejores/v238y2014i3p774-785.html | journal = European Journal of Operational Research | volume = 238 | issue = 3| pages = 774–785 | doi=10.1016/j.ejor.2014.04.003}}</ref>

=== Collision detection ===
Minkowski sums, specifically Minkowski differences, are often used alongside [[Gilbert–Johnson–Keerthi distance algorithm|GJK algorithms]] to compute [[collision detection]] for convex hulls in [[physics engines]].