Due to this theory any mapping from the globe to a paper neccesairly have disortions. My question is there a theory which states the number of necessary map that gets this error down to a certain percentage? I.e. if I wanted to make an atlas with enough pages show that everything shows up with distances preserves or at most 20 miles off for all diatances?
2026-03-31 10:59:52.1774954792
Thoughts on Theorema Egregium
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In principle, if you take the usual parametrization of the sphere in spherical coordinates, then you can use its Jacobian to bound the error and subdivide the parameter domain into rectangles until the error is below a given threshold.
You can also subdivide the icosahedron into a triangle mesh until the error is below a given threshold:
(image from the OpenGL Programming Guide.)