Editing Projective space (section)

== Motivation ==
[[File:Affine space R3.png|thumb|Projective plane and [[central projection]]]]
As outlined above, projective spaces were introduced for formalizing statements like "two [[coplanar lines]] intersect in exactly one point, and this point is at infinity if the lines are [[parallel lines|parallel]]". Such statements are suggested by the study of [[perspective (graphical)|perspective]], which may be considered as a [[central projection]] of the [[three dimensional space]] onto a [[plane (geometry)|plane]] (see ''[[Pinhole camera model]]''). More precisely, the entrance pupil of a camera or of the eye of an observer is the ''center of projection'', and the image is formed on the ''projection plane''.

Mathematically, the center of projection is a point {{mvar|O}} of the space (the intersection of the axes in the figure); the projection plane ({{math|''P''{{sub|2}}}}, in blue on the figure) is a plane not passing through {{mvar|O}}, which is often chosen to be the plane of equation {{math|1=''z'' = 1}}, when [[Cartesian coordinates]] are considered. Then, the central projection maps a point {{mvar|P}} to the intersection of the line {{mvar|OP}} with the projection plane. Such an intersection exists if and only if the point {{mvar|P}} does not belong to the plane ({{math|''P''{{sub|1}}}}, in green on the figure) that passes through {{mvar|O}} and is parallel to {{math|''P''{{sub|2}}}}.

It follows that the lines passing through {{mvar|O}} split in two disjoint subsets: the lines that are not contained in {{math|''P''{{sub|1}}}}, which are in one to one correspondence with the points of {{math|''P''{{sub|2}}}}, and those contained in {{math|''P''{{sub|1}}}}, which are in one to one correspondence with the directions of parallel lines in {{math|''P''{{sub|2}}}}. This suggests to define the ''points'' (called here ''projective points'' for clarity) of the projective plane as the lines passing through {{mvar|O}}. A ''projective line'' in this plane consists of all projective points (which are lines) contained in a plane passing through {{mvar|O}}. As the intersection of two planes passing through {{mvar|O}} is a line passing through {{mvar|O}}, the intersection of two distinct projective lines consists of a single projective point. The plane {{math|''P''{{sub|1}}}}
defines a projective line which is called the ''line at infinity'' of {{math|''P''{{sub|2}}}}. By identifying each point of {{math|''P''{{sub|2}}}} with the corresponding projective point, one can thus say that the projective plane is the [[disjoint union]] of {{math|''P''{{sub|2}}}} and the (projective) line at infinity.

As an [[affine space]] with a distinguished point {{mvar|O}} may be identified with its associated [[vector space]] (see ''{{slink|Affine space|Vector spaces as affine spaces}}''), the preceding construction is generally done by starting from a vector space and is called [[projectivization]]. Also, the construction can be done by starting with a vector space of any positive dimension.

So, a projective space of dimension {{mvar|n}} can be defined as the set of [[vector line]]s (vector subspaces of dimension one) in a vector space of dimension {{math|''n'' + 1}}. A projective space can also be defined as the elements of any set that is in natural correspondence with this set of vector lines.

This set can be the set of [[equivalence class]]es under the equivalence relation between vectors defined by "one vector is the product of the other by a nonzero scalar". In other words, this amounts to defining a projective space as the set of vector lines in which the zero vector has been removed.

A third equivalent definition is to define a projective space of dimension {{mvar|n}} as the set of pairs of [[antipodal points]] in a sphere of dimension {{mvar|n}} (in a space of dimension {{math|''n'' + 1}}).