Singular foci of planar linkages

The focal points of a curve traced by a planar linkage capture essential information about the curve. Knowledge of the singular foci can be helpful in the design of path-generating linkages and is essential to the determination of path cognates. This paper shows how to determine the singular foci of planar linkages built with rotational links. The method makes use of a general formulation of the tracing curve based on the Dixon determinant of loop equations written in isotropic coordinates. In simple cases, the singular foci can be read off directly from the diagonal of the Dixon matrix, while the worst case requires only the solution of an eigenvalue problem. The method is demonstrated for one inversion each of the Stephenson-3 six-bar and the Watt-1 six-bar.