Linkages and Polynomials

Occasionally we may have a linkage such that whenever you specify where one particular vertex (what we've called a hinge) is, another particular vertex has a unique location, which depends only on the location of the first. Here's an example:



Can you see that z can be moved in any direction, and that the position of w is uniquely decided by the position of z?

In these cases, since some specified vertex w is uniquely determined by the location of another specified vertex z, then we can consider the complex function, such that, given z as a complex valued input, the function produces w, a complex number.

What function would we get from considering the example above? Suppose A=0, and B is some arbitrary complex number.

Hint: what shape is formed by the vertices A, B, w, z?