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What is the geometric intuition of projective plane and space?

I can understand affine plane and 3 dimension affine space, for higher dimension, at least I can imagine it similarly as the 2,3 dimensional case.

Please feel freely on commenting and helping me. Thank for reading my question.

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Look at this answer. – Arturo Magidin Feb 22 '12 at 20:58
In "How not to be wrong", Ellenberg gives a beautiful description of the projective plane: . – L Spice Jun 5 at 21:19

The intuition is that of a landscape painter: the projective plane adds the points on the horizon to the usual plane when viewed, like here

from slightly above. From the point of view of an observer floating slightly above the plane, lines of sight come in two flavors: those that intersect the plane below her, and those that do not, corresponding to the points on the horizon ("at infinity" in the usual mathematical jargon). Each such line of sight corresponds to a point in the projective plane, and this connects with the usual mathematical description of the projective plane as the set of lines through the origin in 3-dimensional space.

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