The configuration space of a crank rocker

Every point of the square is a pair of angles — the crank's and the rocker's — and the curve is where the coupler is exactly the right length to join them. That curve *is* the mechanism: one equation in two angles leaves one freedom, which is the mobility. It has 2 components, and that is the two assembly branches. A built linkage cannot cross between them, because there is no path in the set to cross by — and each goes all the way round, which is what makes the crank a crank. The square's left and right edges are the same line, and so are its top and bottom.