A transformation for plane coordinates is presented and is used to derive the equation, in plane coordinates, for the dual torus, which is the envelope of two-freedom motion of a plane. Five distinct forms of the dual torus are described. Two kinds of nodal points are found which relate to linkage motion. The principle of identical surface-enveloping by four cognate spatial dyad linkages is developed, and the existence of all forms of spatial ERRR linkages is confirmed. Finally, some comments are made about duality of the dual torus and the point torus.

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