Are there cases were unrolling/flattening a surface ( not sure what the name of the process is, maybe area preserving?) does not equal the unflattened/unrolled form? The motivation for asking this question is as follows :
When trying to calculate length of a helix, the following two methods give the same answer
- using an arc length integral or
- unrolling the cylinder ( e.g. https://www.youtube.com/watch?v=Ce64D_yGDvY )
the problem with 2 is that the process is not mathematically justified to be area preserving, although intuitively it does make sense and happens to also match he correct answer by 1.
my question is under what conditions process 2 is justified, or an example of when no amount of flattening/unrolling will not work/correct? (I am looking for something similar to examples for every where continuous function being no where differentiable for example https://en.wikipedia.org/wiki/Weierstrass_function)