The vertices A,B,w,z will always form a parallelogram, which suggests the relationship

w=z+B

where B is any complex number (other than 0). So in a sense, our linkage computes the sum of a given complex number with a constant complex number.

Take a look at the following linkage and see if you can determine what w is, as a function of z.



Now, keep in mind we'd like to keep the two long external edges moving as a single edge each, so the vertex drawn halfway along each of those edges is actually only a pivot for the two edges that meet at the bottom center vertex.

What is the function that determines w, based on the value of z, from the picture above?