Editing Projectively extended real line (section)

== Geometry ==

Fundamental to the idea that {{math|∞}} is a point ''no different from any other'' is the way the real projective line is a [[homogeneous space]], in fact [[homeomorphic]] to a circle. For example the [[general linear group]] of 2&thinsp;×&thinsp;2 real [[invertible matrix|invertible]] [[matrix (mathematics)|matrices]] has a [[transitive action]] on it. The [[group action]] may be expressed by [[Möbius transformation]]s (also called linear fractional transformations), with the understanding that when the denominator of the linear fractional transformation is {{math|0}}, the image is {{math|∞}}.

The detailed analysis of the action shows that for any three distinct points ''P'', ''Q'' and ''R'', there is a linear fractional transformation taking ''P'' to 0, ''Q'' to 1, and ''R'' to {{math|∞}} that is, the [[group (mathematics)|group]] of linear fractional transformations is [[transitive action|triply transitive]] on the real projective line. This cannot be extended to 4-tuples of points, because the [[cross-ratio]] is invariant.

The terminology [[projective line]] is appropriate, because the points are in 1-to-1 correspondence with one-[[dimension (vector space)|dimensional]] [[linear subspace]]s of <math>\mathbb{R}^2</math>.