Projective plane

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In mathematics, the projective plane is a geometric construction that extends the concept of a plane. In the ordinary plane, two lines typically intersect in a single point, but there are some pairs of lines — namely, parallel lines — that do not intersect. The projective plane is, in one view, the ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two lines in the projective plane intersect in one and only one point.

The projective plane has two common definitions. The first comes from linear algebra; it produces planes that are homogeneous spaces for some of the classical groups. Important examples include the real projective plane \mathbb{RP}^2 and the complex projective plane \mathbb{CP}^2. The second, more general definition comes from axiomatic geometry and finite geometry; it is suitable for study of the incidence properties of plane geometry.

The projective plane generalizes to higher-dimensional projective spaces; that is, a projective plane is a 2-dimensional projective space.


Linear-algebraic definition

In one view, the projective plane is the set of lines through the origin in 3-dimensional space, and a line in the projective plane arises from a plane through the origin in 3-dimensional space. This idea can be made precise as follows.[1]

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