What Is Linkage?

As often happens in mathematical sciences which occasionally borrows its terms from the common language, the mathematical object known as linkage has no direct linkage to the many instances of the word's vernacular usage, although it does relates to some physical objects. Mathematical linkage serves to model robot limbs, arms in particular; the mechanism on top of an electric train (tram, streetcar) designed to maintain the contact with the overhead wires; a crank mechanism in any kind of vehicles, windshield wipers, and many others.

Linkage is a structure that like graph consists of vertices and edges, but there are very essential differences. First, while a graph edge is just an indicator that two vertices are adjacent to each other and could be depicted as any line joining the two vertices, a linkage edge is a rigid shape which is usually but not necessarily is a straight line segment of a definite length. Second, the vertices are often considered as joints at which the edges are hinged. The edges may rotate about incident joints (their end points), in which motion the angles between the edges may be fixed preserved or left free.

The theory of linkage exploits tools from many areas of mathematics - geometry, algebra, topology, in particular.

References

J. O'Rourke, How To Fold It: The Mathematics of Linkages, Origami and Polyhedra, Cambridge University Press, 2011

Linkages

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