I am aware that there have been many works on the problem of discretizing a surface into polygons, however, I wonder if in any work the problem of doing so to get polygons with edges of the same length has been discussed?
Edit: Just to be clear, I am interested not only in the case where the edges form polygons; all I want is the edges to be of the same length (and don't care if they enclose a 3D shape).
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You might look into the work of Helmut Pottmann in Vienna, e.g.,
Helmut Pottmann et al., "Geometric modeling with conical meshes and developable surfaces," 2006. (author link)
In particular, their PQ-strips are very much oriented toward "panelization," that is, the need to actually build an architectural model with physical panels, perhaps glass panels. The PQ-strip results in planar quadrilaterals of approximately the same shape. Various "shape optimization" techniques are used to optimize the parameters that minimize the manufacturing costs.
Another source is this PDF poster from the same TU Wein group.
An interesting question. My immediate thought on it is that the curvature of the surface that one can represent will be rather restricted. If you fix an edge length, you will often need to have discretized points that lie outside of the original surface.
Think about regular triangles, they have a limited set of curvatures. If you use anything other than these triangles, it is not given that all the points can lie in the plane.
Perhaps there is always a trivial solution where you have one big polygon with lots of edges, perhaps its not convex. But this seems to run counter to the original purpose of discretization.