Play with the colored points in the leftmost picture.
This step is crucial for the method proposed in the article.
It generates a correspondence between 5 points on a circle (modulo Moebius trasnformation) to a Pentagon Linkage (modulo rotation and translation).
For that we take the five points on a circle to start with. A theorem of Boris Springborn implies that as long as no three of these points coincide there is a Moebius transformation that moves their centroid to the origin of the circle. If we furthermore require that one of theose points appears at a certain angle, this transformation is uniquely determined. This step is performed in the second image. In the above animation the red point is fixed to the point (1,0).
After this transformation we can easily reconstruct a linkage from the transformed points, by joining the unit vectors one to another in a given order.