Say I have a bendable 2D sheet, on which I've drawn a circle. The sheet can bend, but it cannot be stretched or shrunk in any direction within its tangent plane. I then drape the sheet over a surface with coordinates $(x,y,z(x,y))$. I want to know the shape of the circle as projected on the $xy$ plane.
I'm assuming the surface has no overhangs.
I'm setting up the problem by saying the sheet has internal coordinates $(u,v)$, and a displacement field then maps those points to points on the surface: $(x(u,v),y(u,v),z(x,y))$. But I'm not sure how to enforce that there is no stretching/shrinking in any direction, since it's more than enforcing constant area.
Short of a final answer, does this type of 2D deformation have a name? Can you point me to any references? I'm not a differential geometry person, so the lower level, the better.
If the answer is not well defined in the presence of Gaussian curvature, let’s assume there is none.
I believe this is an example of an isometry: https://en.wikipedia.org/wiki/Isometry