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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cross-ratio ： ウィキペディア英語版 Cross-ratio In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C and D on a line, their cross ratio is defined as : $(A,B;C,D)\; =\; \backslash frac$ where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.) The quadruple of points (''A'', ''B'', ''C'', ''D'') divides the line harmonically precisely if the cross-ratio of the quadruple is 1. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from harmonic division; hence the name ''anharmonic ratio''. The cross-ratio is preserved by the fractional linear transformations and it is essentially the only projective invariant of a quadruple of collinear 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.〔A theorem on the anharmonic ratio of lines appeared in the work of Pappus, but Michel Chasles, who devoted considerable efforts to reconstructing lost works of Euclid, asserted that it had earlier appeared in his book ''Porisms''.〕 Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere. ==Definition== The cross-ratio of a 4-tuple of distinct points on the real line with coordinates ''z''1, ''z''2, ''z''3, ''z''4 is given by :$(z\_1,z\_2;z\_3,z\_4)\; =\; \backslash frac.$ It can also be written as a "double ratio" of two division ratios of triples of points: :$(z\_1,z\_2;z\_3,z\_4)\; =\; \backslash frac:\backslash frac.$ The same formulas can be applied to four different complex numbers or, more generally, to elements of any field and can also be extended to the case when one of them is the symbol ∞, by removing the corresponding two differences from the formula. The formula shows that cross-ratio is a function of four points, generally four numbers $z\_1,\; \backslash \; z\_2,\; \backslash \; z\_3,\; \backslash \; z\_4$ taken from a field. In geometry, if A, B, C and D are collinear points, then the cross ratio is defined similarly as :$(A,B;C,D)\; =\; \backslash frac$ where each of the distances is signed according to a fixed orientation of the line. 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「Cross-ratio」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.