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In mathematics, real projective space, denoted or (), is the topological space of lines passing through the origin 0 in the real space +. It is a compact , smooth manifold of dimension n , and is a special case G r ( 1 , R n + 1 ) {\displaystyle \mathbf {Gr} (1,\mathbb {R} ^{n+1})} of a Grassmannian space.
A projective space is a topological space, as endowed with the quotient topology of the topology of a finite dimensional real vector space. Let S be the unit sphere in a normed vector space V , and consider the function π : S → P ( V ) {\displaystyle \pi :S\to \mathbf {P} (V)} that maps a point of S to the vector line passing through it.
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism.
Real projective space RP n is a compactification of Euclidean space R n. For each possible "direction" in which points in R n can "escape", one new point at infinity is added (but each direction is identified with its opposite). The Alexandroff one-point compactification of R we constructed in the example above is in fact homeomorphic to RP 1.
In this construction, each "point" of the real projective plane is the one-dimensional subspace (a geometric line) through the origin in a 3-dimensional vector space, and a "line" in the projective plane arises from a (geometric) plane through the origin in the 3-space. This idea can be generalized and made more precise as follows.
Generally, a projective n-space is formed from antipodal pairs on a sphere in (n+1)-space; in this case the sphere is a circle in the plane. The real projective line is a complete projective range that is found in the real projective plane and in the complex projective line. Its structure is thus inherited from these superstructures.
For example, the Grassmannian () is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than . [ 1 ] [ 2 ] When V {\displaystyle V} is a real or complex vector space, Grassmannians are compact smooth manifolds , of dimension k ( n − k ) {\displaystyle k(n-k)} . [ 3 ]
The Lie group SO(3) is diffeomorphic to the real projective space (). [4] Consider the solid ball in of radius π (that is, all points of of distance π or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point ...