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. 


  




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> **Questions:** 
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> * Has that problem been encountered and worked on before?  
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>  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?
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> 


  



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.



Work on "Churning Polygons"
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](http://www.csun.edu/~ctoth/Handbook/chap9.pdf). 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](https://www.math.ucdavis.edu/~kapovich/EPR/plane.pdf) 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](https://en.wikipedia.org/wiki/Morse_theory) 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](https://link.springer.com/article/10.1007/s10958-009-9417-z). 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](https://arxiv.org/abs/1702.07468).


