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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 Laguerre formula ： ウィキペディア英語版 Laguerre formula The Laguerre formula (named after Edmond Laguerre) provides the acute angle $\backslash phi$ between two proper real lines, as follows: :$\backslash phi=|\backslash frac1\backslash operatorname\; \backslash operatorname(I\_1,I\_2,P\_1,P\_2)|$ where: * $\backslash operatorname$ is the principal value of the complex logarithm * $\backslash operatorname$ is the cross-ratio of four collinear points * $P\_1$ and $P\_2$ are the points at infinity of the lines * $I\_1$ and $I\_2$ are the intersections of the absolute conic, having equations $x\_0=x\_1^2+x\_2^2+x\_3^2=0$, with the line joining $P\_1$ and $P\_2$. The expression between vertical bars is a real number. Laguerre formula can be useful in computer vision, since the absolute conic has an image on the retinal plane which is invariant under camera displacements, and the cross ratio of four collinear points is the same for their images on the retinal plane. == Derivation == It may be assumed that the lines go through the origin. Any isometry leaves the absolute conic invariant, this allows to take as the first line the ''x'' axis and the second line lying in the plane ''z''=0. The homogeneous coordinates of the above four points are :$(0,1,i,0),\backslash \; (0,1,-i,0),\backslash \; (0,1,0,0),\backslash \; (0,\backslash cos\backslash phi,\backslash pm\backslash sin\backslash phi,0),$ respectively. Their nonhomogeneous coordinates on the infinity line of the plane ''z''=0 are $i$, $-i$, 0, $\backslash pm\backslash sin\backslash phi/\backslash cos\backslash phi$. (Exchanging $I\_1$ and $I\_2$ changes the cross ratio into its inverse, so the formula for $\backslash phi$ gives the same result.) Now from the formula of the cross ratio we have $\backslash operatorname\; (I\_1,I\_2,P\_1,P\_2)=-\backslash frac=e^.$ 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「Laguerre formula」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.