In the broad category of couplings considered the two coupled bodies have either one or two relative degrees of freedom. Points, planes, and lines guided by such couplings trace a variety of spatial loci restricted by the requirement that they must all be algebraic. The order, class, or degree of such loci is related to the tracing element, and is proved to be always exactly identifiable with an invariant property of a particular coupling, here called the “feather” of the coupling. This property is central to two new theorems which unify many hitherto distinct facets of linkages. Some shortcuts are revealed to assist in more detailed study of mechanical movements.

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