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Chris asked, what if each of these nodes on Helga's map were the "South Pole" of a different sterographic projection. Good point. This leads us to consider the possibility of maps from a surface that is no longer a regular sphere. Imagine how the "shadows" cast by points on a sphere would be distorted if the glass were of irregular refractive index, or if the glass sphere were no longer a perfect sphere but some lumpy mass. Then we could have all sorts of non-regular lensing. Caustics.
To explore this further will take us into the realm of differential geometry, the mathematics of local coordinates, and how a set of local coordinates may be patched together to form a manifold.
xinwei@stanford.edu