One tricky thing about curved folding origami is the interaction of the mesh edge directions and the ruling lines (paper stays very close to a developable surface, so the curvature in one direction will be zero).
If you know the ruling directions, you can model curved fold origami as thin planar quads. (See this paper: http://graphics.stanford.edu/~niloy/research/folding/folding_sig_08.html)
However, what makes it tricky is that the direction of these ruling lines often isn’t obvious from the flat crease pattern, and sometimes it even has to change during folding. If the mesh doesn’t match these directions, it won’t behave properly when you simulate the folding. If you make a real physical version first, sometimes you can use that to identify the ruling directions.
A different approach is to use a fairly fine unstructured triangular mesh with some bending resistance and allowing only a small amount of stretching of the edges. Like in this video https://vimeo.com/88649160 but also including mesh edges along the crease lines where you set the bending resistance to zero.
Yet another very recent interesting approach is described in this paper: http://igl.ethz.ch/projects/dog-space/
Here a special angle condition preserves developability, even for a coarse quad mesh that doesn’t have to match the ruling directions. I already tried a version of this from their previous paper:
http://igl.ethz.ch/projects/developable/
In kangaroo this didn’t even need any new goals, as there is already one for equal angles:
https://vimeo.com/273918494
I didn’t yet try and implement the newest paper which includes the creasing though. Would be interesting to test that out…