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Icosahedron

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What is the relationship betwenn RP^2 and CP?

Espesicially, why are their stenographic representations different?

As far as I understand the stenographic representation for RP^2 goes like that:

a sphere with antipodal points identified is put above the R^2 plane, lines through the origin of the sphere cut the R^2 plane and project the sphere to the plane and so on.

Whereas, the stenographic representation for CP is: take a sphere, cut it in the middle with the R^2 plane and take lines starting from the north pole projecting from the sphere to the plane.

Espesicially, why are their stenographic representations different?

As far as I understand the stenographic representation for RP^2 goes like that:

a sphere with antipodal points identified is put above the R^2 plane, lines through the origin of the sphere cut the R^2 plane and project the sphere to the plane and so on.

Whereas, the stenographic representation for CP is: take a sphere, cut it in the middle with the R^2 plane and take lines starting from the north pole projecting from the sphere to the plane.

**And**take lines starting from the south pole projecting from the sphere to the*other side*of the plane. Plus, transistion functions connecting these two mappings.
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