Linked from

The 14 pages that link to Cross-ratio, each with the reason it gives.

• Projective planeRelated: It is a central quantity preserved by projective maps of lines in the plane.

• Projective transformationRelated: It supplies a numerical invariant preserved even when lengths and angles change.

• Möbius transformationRelated: It characterizes the transformations that preserve the relative geometry of four points.

• Pascal's theoremRelated: Cross-ratio preservation supplies tools for proving incidence results on a conic.

• Jakob SteinerRelated: This invariant captures a central structure in the projective geometry of Steiner’s era.

• Complete quadrilateralRelated: The configuration’s diagonal points yield a classical harmonic range on each diagonal line.

• August Ferdinand MöbiusRelated: Möbius used the cross-ratio to characterize projective relationships independently of coordinates.

• Jean-Victor PonceletRelated: It exemplifies a projective invariant, the kind of property central to his subject.

• Schwarz–Christoffel mappingRelated: It expresses the conformally meaningful placement of four prevertices.

• Directed angleRelated: Directed angles help express the cross ratio of four concyclic points through angle relations.

• Inversive geometryRelated: Its invariance captures a deeper structure shared by inversion and related transformations.

• Steiner conicRelated: Projective correspondence preserves cross-ratios among rays in each pencil.

• Thomsen's theoremRelated: Projectivities preserve this quantity, providing an algebraic way to track the mappings.