The geometry of singular foci of planar linkages

The focal points of a curve traced by a planar linkage capture essential information about the curve. In a previous paper, we showed how to determine the singular foci of planar linkages from an expression for the tracing curve derived by use of the Dixon determinant. This paper gives an alternative approach to finding the singular foci, one that lends itself to simple geometric interpretations and does not require a derivation of the tracing curve equation. In many cases, singular foci can be determined from a simple graphical construction. The method is demonstrated for one inversion each of the Stephenson-3 six-bar and the Watt-1 six-bar. A by-product of the study is a technique for illustrating the non-real points on a tracing curve. Knowledge of the singular foci will be helpful in further study of path cognates.