projective plane

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projective plane

[prə′jek·tiv ′plān]
The topological space obtained from the two-dimensional sphere by identifying antipodal points; the space of all lines through the origin in Euclidean space.
More generally, a plane (in the sense of projective geometry) such that (1) every two points lie on exactly one line, (2) every two lines pass through exactly one point, and (3) there exists a four-point.

Projective Plane


in its original meaning, the Euclidean plane with the addition of the points and line at infinity. From the topological standpoint, the projective plane is a closed, non-orientable surface with Euler characteristic 1.

projective plane

The space of equivalence classes of vectors under non-zero scalar multiplication. Elements are sets of the form

kv: k != 0, k scalar, v != O, v a vector

where O is the origin. v is a representative member of this equivalence class.

The projective plane of a vector space is the collection of its 1-dimensional subspaces. The properties of the vector space induce a topology and notions of smoothness on the projective plane.

A projective plane is in no meaningful sense a plane and would therefore be (but isn't) better described as a "projective space".