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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 Quadric (projective geometry) ： ウィキペディア英語版 Quadric (projective geometry) In projective geometry, a quadric is the set of points of a projective space where a certain quadratic form on the homogeneous coordinates becomes zero. We shall restrict ourself to the case of ''finite-dimensional projective spaces''. ==Quadratic forms== Let $K$ be a field and $\backslash mathcal\; V(K)$ a vector space over $K$. A mapping $\backslash rho$ from $\backslash mathcal\; V(K)$ to $K$ such that : (Q1) $\backslash rho(\backslash lambda\backslash vec\; x)=\backslash lambda^2\backslash rho(\backslash vec\; x\; )$ for any $\backslash lambda\backslash in\; K$ and $\backslash vec\; x\; \backslash in\; \backslash mathcal\; V(K)$. : (Q2) $f(\backslash vec\; x,\backslash vec\; y\; ):=\backslash rho(\backslash vec\; x+\backslash vec\; y)-\backslash rho(\backslash vec\; x)-\backslash rho(\backslash vec\; y)$ is a bilinear form. is called quadratic form. The bilinear form $f$ is symmetric''.'' In case of $\backslash operatornameK\backslash ne2$ we have $f(\backslash vec\; x,\backslash vec\; x)=2\backslash rho(\backslash vec\; x)$, i.e. $f$ and $\backslash rho$ are mutually determined in a unique way. In case of $\backslash operatornameK=2$ we have always $f(\backslash vec\; x,\backslash vec\; x)=0$, i.e. $f$ is ''symplectic''. For $\backslash mathcal\; V(K)=K^n$ and $\backslash vec\; x=\backslash sum\_^x\_i\backslash vec\; e\_i$ ($\backslash$ is a base of $\backslash mathcal\; V(K)$) $\backslash rho$ has the form : $$ \rho(\vec x)=\sum_^ a_x_ix_k\texta_:= f(\vec e_i,\vec e_k)\texti\ne k\texta_:= \rho(\vec e_i)\texti=k and : $f(\backslash vec\; x,\backslash vec\; y)=\backslash sum\_^\; a\_(x\_iy\_k+x\_ky\_i)$. For example: : $n=3,\backslash \; \backslash rho(\backslash vec\; x)=x\_1x\_2-x^2\_3,\; \backslash \; f(\backslash vec\; x,\backslash vec\; y)=x\_1y\_2+x\_2y\_1-2x\_3y\_3.$ 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「Quadric (projective geometry)」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.