Minkowski's addition of convex shapes

Fix a point O in the plane. Point O is called the origin. The directed segment OA from the origin to an arbitrary point A in the plane is known as the A's radius-vector. Radius-vectors of two points can be added according to the rule of parallelogram. Sometimes we forget to mention the origin and talk of the sum A + B of two points. We may even justify such an apparent sloppiness by expanding the operation to addition of sets and showing that the shape of the result does not depend on the selection of the origin. Minkowski's addition of two sets X and Y is defined as

In other words, to find Minkowski's sum of two sets one must consider the totality of all possible sums of a point from one set and a point from the other. If the origin is translated from point O to O', the sum of two sets is translated by the same distance, but in the opposite direction. Some other features of this operation may be surmised by toying with the applet below.

The applet demonstrates Minkowski's addition of two polygons with the number of sides between 3 and 9, inclusive. The labeling of the vertices of the sum reflects the fact that each such vertex is always related to one vertex of one polygon and one vertex of the other. The addents are conditionally dubbed Left and Right, and their labels are combined (in this order) to form the label of the related vertex of the sum.

Discussion

Convex Sets

• Helly's Theorem
• First Applications of Helly's Theorem
• Crossed-Lines Construction of Shapes of Constant Width
• Shapes of constant width (An Interactive Gizmo)
• Star Construction of Shapes of Constant Width
• Convex Polygon Is the Intersection of Half Planes
• Minkowski's addition of convex shapes
• Perimeters of Convex Polygons, One within the Other
• The Theorem of Barbier
• A. Soifer's Book, P. Erdos' Conjecture, B. Grunbaum's Counterexample
• Reuleaux's Triangle, Extended

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny