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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 Linear fractional transformation ： ウィキペディア英語版 Linear fractional transformation In mathematics, the phrase linear fractional transformation usually refers to a Möbius transformation, which is a homography on the complex projective line P(C) where C is the field of complex numbers. More generally in mathematics, C may be replaced by another ring (''A'', +, ×).〔N. J. Young (1984) ("Linear fractional transformations in rings and modules" ), Linear Algebra and its Applications 56:251–90〕 Then a linear fractional transformation refers to a homography over P(''A''), the projective line over a ring. When ''A'' is a commutative ring, then the linear fractional transformation has the familiar form :$z\; \backslash mapsto\; \backslash frac\; ,\; \backslash quad\; z,\; a,\; b,\; c,\; d\; \backslash isin\; A\; .$ Otherwise homographies are expressed (''az'' + ''b'', ''cz'' + ''d'') with homogeneous coordinates. The equivalence of such coordinates is expressed :$U(az\; +\; b,\; cz\; +d)\; \backslash sim\; U((cz\; +\; d)^(az\; +\; b),\; 1)\; .$ ==Conformal property== The commutative rings of split-complex numbers and dual numbers join the ordinary complex numbers as rings that express angle. In each case the exponential map applied to the imaginary axis produces an isomorphism between one-parameter groups in (''A'', + ) and in the group of units (''U'', × ): :$\backslash exp(y\; j)\; =\; \backslash cosh\; y\; +\; j\; \backslash sinh\; y,\; \backslash quad\; j^2\; =\; +1\; ,$ :$\backslash exp(y\; \backslash epsilon)\; =\; 1\; +\; y\; \backslash epsilon,\; \backslash quad\; \backslash epsilon^2\; =\; 0\; ,$ :$\backslash exp(y\; i)\; =\; \backslash cos\; y\; +\; i\; \backslash sin\; y,\; \backslash quad\; i^2\; =\; -1\; .$ The "angle" ''y'' is hyperbolic angle, slope, or circular angle according to the host ring. A linear fractional transformation can be generated by multiplicative inversion ''z'' → 1/''z'' and affine transformations ''z'' → ''a z'' + ''b''. Conformality can be confirmed by showing the generators are all conformal. The translation ''z'' → ''z'' + ''b'' is a change of origin and makes no difference to angle. To see that ''z'' → ''az'' is conformal, consider the polar decomposition of ''a'' and ''z''. In each case the angle of ''a'' is added to that of ''z'' giving a conformal map. Finally, inversion is conformal since ''z'' → 1/''z'' sends $\backslash exp(y\; b)\; \backslash mapsto\; \backslash exp(-y\; b),\; \backslash quad\; b^2\; =\; 1,\; 0,\; -1\; .$ 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「Linear fractional transformation」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.