>>13774753>It's impossible to do it to a full sphere but it should be possible to do it with one part. Only one axis has to have parallel lines.Even locally its still impossible. think about it this way - take two squares one on top of each other . for simplicity, work in the northern hemisphere though the same argument holds mutatis mutandis in the souther hemisphere. Npw, lets form these squares out of of a 2 longitude lines intersected by 3 latitude lines; the latitude lines are equal number of degrees apart. The top square will have smaller area on the sphere. Now, if we use a map to a square grid, the longitude lines and latitude lines become the lines on this grid meaning the squares now have the same area - aka we have area distortion. Even locally there doesn't exist isometry since sphere has nonzero curvature at every point.
