Editing Projective space (section)

{{short description|Completion of the usual space with "points at infinity"}}
[[File:Railroad-Tracks-Perspective.jpg|thumb|right|In [[perspective (graphical)|graphical perspective]], parallel (horizontal) lines in the plane intersect at a [[vanishing point]] (on the [[horizon]]).]]
In [[mathematics]], the concept of a '''projective space''' originated from the visual effect of [[perspective (graphical)|perspective]], where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a [[Euclidean space]], or, more generally, an [[affine space]] with [[points at infinity]], in such a way that there is one point at infinity of each [[affine space#direction|direction]] of [[parallel lines]].

This definition of a projective space has the disadvantage of not being [[isotropic]], having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In [[synthetic geometry]], ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the [[axioms of projective geometry]]. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the following definition, which is more often encountered in modern textbooks.

Using [[linear algebra]], a projective space of dimension {{mvar|n}} is defined as the set of the [[vector line]]s (that is, vector subspaces of dimension one) in a [[vector space]] {{mvar|V}} of dimension {{math|''n'' + 1}}. Equivalently, it is the [[quotient set]] of {{math|''V'' \ {{mset|0}}}} by the [[equivalence relation]] "being on the same vector line". As a vector line intersects the [[unit sphere]] of {{mvar|V}} in two [[antipodal points]], projective spaces can be equivalently defined as spheres in which antipodal points are identified. A projective space of dimension 1 is a [[projective line]], and a projective space of dimension 2 is a [[projective plane]].

Projective spaces are widely used in [[geometry]], allowing for simpler statements and simpler proofs. For example, in [[affine geometry]], two distinct lines in a plane intersect in at most one point, while, in [[projective geometry]], they intersect in exactly one point. Also, there is only one class of [[conic section]]s, which can be distinguished only by their intersections with the line at infinity: two intersection points for [[hyperbola]]s; one for the [[parabola]], which is tangent to the line at infinity; and no real intersection point of [[ellipse]]s.

In [[topology]], and more specifically in [[manifold theory]], projective spaces play a fundamental role, being typical examples of [[non-orientable manifold]]s.