Linkage

Linkages are linked (or constrained) mechanical structures such as those used in robots, excavators, engines, etc. They are also extensively used for character animation in modeling systems.

Sylvester, Kempe and Cayley developed the geometry associated with the theory of linkages in the 1870s. Kempe proved that every finite segment of an algebraic curve can be generated by a linkage in the manner of Watt's curve.

Muirhead (1911-1912, pp. 69-71) gives a linkage construction for evaluating a real polynomial at a real input . Successive similar triangles multiply a signed displacement by , and adjustable sliding segments add the coefficients. Repeating these operations evaluates in the nested form of Horner's rule. The final signed displacement is , so a real root is found by varying until the endpoint coincides with the corresponding zero position. A particular mechanism may cover only a restricted input interval, with other real roots obtained by transforming the equation.