Editing Projectively extended real line (section)

== As a projective range ==
{{Main|Projective range}}
When the [[real projective line]] is considered in the context of the [[real projective plane]], then the consequences of [[Desargues' theorem]] are implicit. In particular, the construction of the [[projective harmonic conjugate]] relation between points is part of the structure of the real projective line. For instance, given any pair of points, the [[point at infinity]] is the projective harmonic conjugate of their [[midpoint]].

As [[projectivity|projectivities]] preserve the harmonic relation, they form the [[automorphism]]s of the real projective line. The projectivities are described algebraically as [[homography|homographies]], since the real numbers form a [[ring (mathematics)|ring]], according to the general construction of a [[projective line over a ring]]. Collectively they form the group [[PGL(2,R)|PGL(2,&nbsp;'''R''')]].

The projectivities which are their own inverses are called [[involution (mathematics)#Projective geometry|involutions]]. A '''hyperbolic involution''' has two [[fixed point (mathematics)|fixed point]]s. Two of these correspond to elementary, arithmetic operations on the real projective line: [[additive inverse|negation]] and [[multiplicative inverse|reciprocation]]. Indeed, 0 and ∞ are fixed under negation, while 1 and −1 are fixed under reciprocation.