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Projective space

In [mathematics] a ** projective space ** is a set of elements similar to the set * P(V) * of lines through the origin of a [vector space] * V * . The cases when * V * = ** R ** 2 or * V * = ** R ** 3 are the [projective line] and the [projective plane] , respectively.

The idea of a projective space relates to [perspective] , more precisely to the way an eye or a camera projects a 3D scene to a 2D image. All points which lie on a projection line (i.e. a "line-of-sight"), intersecting with the [focal point of the camera] , are projected onto a common image point. In this case the vector space is ** R ** 3 with the camera focal point at the origin and the projective space corresponds to the image points.

Projective spaces can be studied as a separate field in mathematics, but are also used in various applied fields, [geometry] in particular. Geometric objects, such as points, lines, or planes, can be given a representation as elements in projective spaces based on [homogeneous coordinates] . As a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. For example, in the standard geometry for the plane two lines always intersect at a point except when the lines are parallel. In a projective representation of lines and points, however, such an intersection point exists even for parallel lines, and it can be computed in the same way as other intersection points.

Other mathematical fields where projective spaces play a significant role are [topology] , the theory of [Lie group] s and [algebraic group] s, and their representation theories.

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