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- Cross-ratio abstract "In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line. Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere.The cross-ratio is preserved by the fractional linear transformations andit is essentially the only projective invariant of a quadruple of points, which underlies its importance for projective geometry. In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio. Cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century.".
- Cross-ratio thumbnail Pappusharmonic.svg?width=300.
- Cross-ratio wikiPageExternalLink v=onepage&q&f=false.
- Cross-ratio wikiPageExternalLink Cross-Ratio.shtml.
- Cross-ratio wikiPageExternalLink kmath543.htm.
- Cross-ratio wikiPageExternalLink p01527115762n730.
- Cross-ratio wikiPageExternalLink MoebApplet.html.
- Cross-ratio wikiPageID "401767".
- Cross-ratio wikiPageRevisionID "601431056".
- Cross-ratio hasPhotoCollection Cross-ratio.
- Cross-ratio subject Category:Projective_geometry.
- Cross-ratio subject Category:Ratios.
- Cross-ratio type Abstraction100002137.
- Cross-ratio type MagnitudeRelation113815152.
- Cross-ratio type Ratio113819207.
- Cross-ratio type Ratios.
- Cross-ratio type Relation100031921.
- Cross-ratio comment "In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line.".
- Cross-ratio label "Birapporto".
- Cross-ratio label "Cross-ratio".
- Cross-ratio label "Doppelverhältnis".
- Cross-ratio label "Dubbelverhouding".
- Cross-ratio label "Dwustosunek".
- Cross-ratio label "Rapport anharmonique".
- Cross-ratio label "Razão anarmônica".
- Cross-ratio label "Двойное отношение".
- Cross-ratio label "交比".
- Cross-ratio sameAs Doppelverhältnis.
- Cross-ratio sameAs Rapport_anharmonique.
- Cross-ratio sameAs Birapporto.
- Cross-ratio sameAs 비조화비.
- Cross-ratio sameAs Dubbelverhouding.
- Cross-ratio sameAs Dwustosunek.
- Cross-ratio sameAs Razão_anarmônica.
- Cross-ratio sameAs m.023yz8.
- Cross-ratio sameAs Q899539.
- Cross-ratio sameAs Q899539.
- Cross-ratio sameAs Cross-ratio.
- Cross-ratio wasDerivedFrom Cross-ratio?oldid=601431056.
- Cross-ratio depiction Pappusharmonic.svg.
- Cross-ratio isPrimaryTopicOf Cross-ratio.
