Editing Projective geometry (section)

== Description ==
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Projective geometry is less restrictive than either [[Euclidean geometry]] or [[affine geometry]]. It is an intrinsically non-[[Metric (mathematics)|metrical]] geometry, meaning that facts are independent of any metric structure. Under the projective transformations, the [[incidence structure]] and the relation of [[projective harmonic conjugate]]s are preserved. A [[projective range]] is the one-dimensional foundation. Projective geometry formalizes one of the central principles of perspective art: that [[parallel (geometry)|parallel]] lines meet at [[infinity]], and therefore are drawn that way. In essence, a projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line". Thus, two parallel lines meet on a horizon line by virtue of their incorporating the same direction.

Idealized directions are referred to as points at infinity, while idealized horizons are referred to as lines at infinity. In turn, all these lines lie in the plane at infinity. However, infinity is a metric concept, so a purely projective geometry does not single out any points, lines or planes in this regard—those at infinity are treated just like any others.

Because a [[Euclidean geometry]] is contained within a projective geometry—with projective geometry having a simpler foundation—general results in Euclidean geometry may be derived in a more transparent manner, where separate but similar theorems of Euclidean geometry may be handled collectively within the framework of projective geometry. For example, parallel and nonparallel lines need not be treated as separate cases; rather an arbitrary projective plane is singled out as the ideal plane and located "at infinity" using [[homogeneous coordinates]].

Additional properties of fundamental importance include [[Desargues' Theorem]] and the [[Pappus's hexagon theorem|Theorem of Pappus]]. In projective spaces of dimension 3 or greater there is a construction that allows one to prove [[Desargues' Theorem]]. But for dimension 2, it must be separately postulated.

Using [[Desargues' Theorem]], combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically. The resulting operations satisfy the axioms of a field – except that the commutativity of multiplication requires [[Pappus's hexagon theorem]]. As a result, the points of each line are in one-to-one correspondence with a given field, {{mvar|F}}, supplemented by an additional element, ∞, such that {{math|1={{var|r}} ⋅ ∞ = ∞}}, {{math|1=−∞ = ∞}}, {{math|1={{var|r}} + ∞ = ∞}}, {{math|1={{var|r}} / 0 = ∞}}, {{math|1={{var|r}} / ∞ = 0}}, {{math|1=∞ − {{var|r}} = {{var|r}} − ∞ = ∞}}, except that {{math|0 / 0}}, {{math|∞ / ∞}}, {{math|∞ + ∞}}, {{math|∞ − ∞}}, {{math|0 ⋅ ∞}} and {{math|∞ ⋅ 0}} remain undefined.

Projective geometry also includes a full theory of [[conic sections]], a subject also extensively developed in Euclidean geometry. There are advantages to being able to think of a [[hyperbola]] and an [[ellipse]] as distinguished only by the way the hyperbola ''lies across the line at infinity''; and that a [[parabola]] is distinguished only by being tangent to the same line. The whole family of circles can be considered as ''conics passing through two given points on the line at infinity'' — at the cost of requiring [[complex number|complex]] coordinates. Since coordinates are not "synthetic", one replaces them by fixing a line and two points on it, and considering the ''linear system'' of all conics passing through those points as the basic object of study. This method proved very attractive to talented geometers, and the topic was studied thoroughly. An example of this method is the multi-volume treatise by [[H. F. Baker]].