Zonogons: Minkowski Sums of Line Segments in 2D

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The Minkowski sum of two subsets in the plane, and , is the set of all sums , where and .


This Demonstration shows the Minkowski sum of line segments with an endpoint at the origin. The line segments joining the origin to all possible sums of the endpoints are also shown.

The Minkowski sum of two line segments is a parallelogram. With more segments, the resulting figure is called a zonogon, which, like a parallelogram, has opposite sides equal and parallel.

In 3D (not shown here), a zonohedron is again the Minkowski sum of such line segments. Its opposite faces are zonogons that are equal, parallel, and symmetric about the center of the zonohedron.


Contributed by: George Beck (March 2011)
Open content licensed under CC BY-NC-SA




Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.