cross ratio

—crossratio, adj.Geom.a ratio relating four points in the real or complex plane. Also called anharmonic ratio.[188085]
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in projective geometry, ratio that is of fundamental importance in characterizing projections. In a projection of one line onto another from a central point (see Figure—>), the double ratio of lengths on the first line (AC/AD)/(BC/BD) is equal to the corresponding ratio on the other line. Such a ratio is significant because projections distort most metric relationships (i.e., those involving the measured quantities of length and angle), while the study of projective geometry centres on finding those properties that remain invariant. Although the cross ratio was used extensively by early 19thcentury projective geometers in formulating theorems, it was felt to be a somewhat unsatisfactory concept because its definition depended upon the Euclidean (Euclidean geometry) concept of length, a concept from which projective geometers wanted to free the subject altogether. In 1847 the German mathematician Karl G.C. von Staudt showed how to effect this separation by defining the cross ratio without reference to length. In 1873 the German mathematician Felix Klein (Klein, Felix) showed how the basic concepts in Euclidean geometry of length and angle magnitude could be defined solely in terms of von Staudt's abstract cross ratio, bringing the two geometries together again, this time with projective geometry occupying the more basic position.* * *
Universalium. 2010.
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Crossratio — D is the harmonic conjugate of C with respect to A and B 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… … Wikipedia
cross ratio — noun Etymology: cross (III) : an anharmonic ratio in mathematics * * * cross ratio, adj. Geom. a ratio relating four points in the real or complex plane. Also called anharmonic ratio. [1880 85] … Useful english dictionary
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