Why stereographic projection preserves angles between curves but does not preserve area?
One way to think about it is that you're mapping to a plane from a sphere. If both angles and lengths were preserved, you'd have the same curvature on each. But clearly the curvature isn't the same on each (try adding angles in a "triangle" on the sphere!). Since angles are preserved, you have to lose the lengths to account for the change in geometry.
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