I find myself recalling the advanced CAD/CAM course I took. More specifically, surfacing and the inaccuracy of flattening a quilt. I find that my surface area of the formed surface and flattened surface are different. I was wondering if it had to do with the definition of the second fundamental form that the software is using. Or, if was other vector calculus theorem that the software was using.
The surface
It's because you are converting a surface with curvature to one that has no curvature. I recently saw a nice video on maths Youtube channel Numberphile as applied to why folding a pizza slice prevents it from drooping. Essentially, a single bend does not change curvature of the flat piece; but if the pizza tries to bend, that does add curvature, which requires stretching. Once there is stretching there is no limit on the difference in the areas.
https://www.youtube.com/watch?v=gi-TBlh44gY
I love that it includes Gauss and Cliff Stoll.
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