An algebra of polygons through the notion of negative shapes
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The paper gives a new formulation of the Minkowski sum of polygons. In the conventional Minkowski sum, the inverse operation is not well defined unless the polygons are restricted to be convex. In the proposed formulation, on the other hand, the set of polygons is extended to the set of "hyperpolygons" and the Minkowski sum forms a commutative group. Consequently, every polygon has its unique inverse, and the sum and the inverse operations can be taken freely. An example of a physical interpretation of the hyperpolygon is also given.