  For a given graph whose edges are labeled with general real numbers, we
consider the set of functions from the vertex set into the Euclidean plane such
that the distance between the images of neighbouring vertices is equal to the
corresponding edge label. This set of functions can be expressed as the zero
set of quadratic polynomials and our main result characterizes the number of
complex irreducible components of this zero set in terms of combinatorial
properties of the graph. In case the complex components are three-dimensional,
then the graph is minimally rigid and the component number is a well-known
invariant from rigidity theory. If the components are four-dimensional, then
they correspond to one-dimensional coupler curves of flexible planar
mechanisms. As an application, we characterize the degree of irreducible
components of such coupler curves combinatorially.
