Editing Projective line over a ring (section)

== Chains ==
The [[real line#In real algebras|real line]] in the [[complex plane]] gets permuted with circles and other real lines under [[Möbius transformation]]s, which actually permute the canonical embedding of the [[real projective line]] in the [[complex projective line]]. Suppose ''A'' is an [[algebra over a field]] ''F'', generalizing the case where ''F'' is the real number field and ''A'' is the field of complex numbers. The canonical embedding of P<sup>1</sup>(''F'') into P<sup>1</sup>(''A'') is
: <math>U_F[x, 1] \mapsto U_A[x, 1] , \quad U_F[1, 0] \mapsto U_A[1, 0].</math>
A '''chain''' is the image of P<sup>1</sup>(''F'') under a homography on P<sup>1</sup>(''A''). Four points lie on a chain [[if and only if]] their cross-ratio is in ''F''. [[Karl von Staudt]] exploited this property in his theory of "real strokes" [reeler Zug].<ref>{{citation |last=von Staudt |first=Karl |authorlink=Karl von Staudt |date=1856 |title=Beträge zur Geometrie der Lage }}</ref>

=== Point-parallelism ===
Two points of P<sup>1</sup>(''A'') are '''parallel''' if there is ''no'' chain connecting them. The convention has been adopted that points are parallel to themselves. This relation is [[invariant (mathematics)|invariant]] under the action of a homography on the projective line. Given three pair-wise non-parallel points, there is a unique chain that connects the three.<ref>[[Walter Benz]], Hans-Joachim Samaga, & Helmut Scheaffer (1981) "Cross Ratios and a Unifying Treatment of von Staudt's Notion of Reeller Zug", pp. 127–150 in ''Geometry – von Staudt's Point of View'', Peter Plaumann & Karl Strambach editors, Proceedings of NATO Advanced Study Institute, Bad Windsheim, July/August 1980, [[D. Reidel]], {{isbn|90-277-1283-2}}, {{MathSciNet|id=0621313}}</ref>