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Stereographic projection is one way to make a flat map of the earth.
Because the earth is spherical, any map must distort shapes or sizes to some degree. The rule for stereographic projection has a nice geometric description. Think of the earth as transparent sphere, sitting on the plane of the paper. Now imagine a light bulb at the north pole, which shines through the sphere. Each point on the sphere casts a shadow on the paper, and that is where it is drawn on the map.