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In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance"〔A. Cayley (1859) "Sixth Memoir on Conics", p 82, §§209 to 229〕 where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and in his book ''Vorlesungen über Nicht-Euklidischen Geometrie''(1928). The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry. The field of non-Euclidean geometry rests largely on the footing provided by Cayley–Klein metrics. ==Definition== Suppose that ''Q'' is a fixed quadric in projective space. If ''p'' and ''q'' are 2 points then the line through ''p'' and ''q'' intersects the quadric ''Q'' in two further points ''a'' and ''b''. The Cayley–Klein distance ''d''(''p'',''q'') from ''p'' to ''q'' is proportional to the logarithm of the cross-ratio: : for some fixed constant ''C''. Cayley–Klein geometry is the study of the group of motions that leave the Cayley–Klein metric invariant. This group is obtained as the collineations for which the absolute is stable. Indeed, cross-ratio is invariant under any collineation, and the stable absolute enables the metric comparison, which will be equality. The extent of Cayley–Klein geometry was summarized by Horst and Rolf Struve in 2004:〔H & R Struve (2004) page 157〕 :There are three absolutes in the real projective line, seven in the real projective plane, and 18 in real projective space. All classical non-euclidean projective spaces as hyperbolic, elliptic, Galilean and Minkowskian and their duals can be defined this way. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cayley–Klein metric」の詳細全文を読む スポンサード リンク
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