Error in a proof for the area of the circle?

[MegaMoh]

Do you know why Greenland is larger (relatively) on maps than on globes?

Yes, because of the curvature of a sphere, you can't just make that 3D object completely 2D(or so they say) so they had to change the maps by scaling some countries up and others down than others. But that isn't related to changing a 2D object to another 2D object.

 

[lev888]

Yes, because of the curvature of a sphere, you can't just make that 3D object completely 2D(or so they say) so they had to change the maps by scaling some countries up and others down than others. But that isn't related to changing a 2D object to another 2D object.

We don't change the maps by scaling countries. We project a 3d globe onto a 2d map. There are many ways to do it. E.g. put a point light source in the center of the globe and trace the resulting shadow of each country on a 2d surface to get a map. If you wrap the 2d surface around the equator countries close to the poles will increase in size. How can it be? We project each point on the globe onto the corresponding point on the map. Same infinite number of points should result in the same area, according to your approach.

 

[MegaMoh]

We don't change the maps by scaling countries. We project a 3d globe onto a 2d map. There are many ways to do it. E.g. put a point light source in the center of the globe and trace the resulting shadow of each country on a 2d surface to get a map. If you wrap the 2d surface around the equator countries close to the poles will increase in size. How can it be? We project each point on the globe onto the corresponding point on the map. Same infinite number of points should result in the same area, according to your approach.

I don't wanna turn this into a pointless argument but I just don't see how projecting points has to do with what I did.