Work on "Churning Polygons" Background of this question is that I recently stumbled over the problem of deforming polygons in area-preserving way, i.e. modifying the angles between adjacent edges while preserving edge-lengths, topological equivalence to a circle and size of enclosed area.  


Questions: 
  
  
*
  
*Has that problem been encountered and worked on before?
  I'am looking for existing work, because I encountered some interesting questions when investigating the problem, e.g. what the minimum number of edges of polygons allowing such deformations is (I conjecture that it must be six).  
  
*What to do with the further questions I encountered?


Any pointers to articles or blogs related to the problem would be of help to me.  
Remark:
I chose the preliminary term "churning" as an analogy to the approximately surface-area and volume preserving deformations of the stomach; I am however no native English speaker, so a more appropriate verb may exist.
 A: If I understand your question correctly, you're asking about (signed) area-preserving deformations of planar polygonal linkages. One place to start reading about polygonal linkages is this chapter by Connelly and Demaine. You might be interested about questions about the configuration space of such linkages, for that see "On the moduli space of polygons in the Euclidean plane" by Kapovich and Millson and the papers which cite this. 
Let me turn to your comment / question:

[What is] the minimum number of edges of polygons allowing such deformations [...] (I conjecture that it must be six).

The dimension of the moduli space of a planar $n$-gonal linkage (with plane isometries quotiented out) is $n-3$. Preserving the area function gives one more constraint on this space. Therefore, at a generic value of the area $A$, the dimension of the space of polygonal linkages with area $A$ (if nonempty) should be $n-4$. Thus (as I see Gerhard Paseman has already pointed out in the comments) nontrivial area-preserving deformations exist for $n\geq 5$.
The Morse-theoretic properties of the area function have been the subject of study. At certain special values of $A$, the area function is critical (i.e. the space of fixed-area polygonal linkages will have some singularity). It turns out that the "typical" critical configurations of the polygon are cyclic (i.e. vertices lie on a circle)! This is a result of Panina and Khimshiashvili. There's quite a bit of followup work on this result, to give one random recent example, see this recent preprint of Panina and Siersma.
