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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 Hesse pencil ： ウィキペディア英語版 Hesse pencil In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation :$\backslash lambda(x^3+y^3+z^3)\; +\; \backslash mu\; xyz\; =0.$ Each curve in the family is determined by a pair of parameter values ($\backslash lambda,\backslash mu$) (not both zero) and consists of the points in the plane whose homogeneous coordinates $(x,y,z)$ satisfy the equation for those parameters. Multiplying both $\backslash lambda$ and $\backslash mu$ by the same scalar does not change the curve, so there is only one degree of freedom in selecting a curve from the pencil, but the two-parameter form given above allows either $\backslash lambda$ or $\backslash mu$ (but not both) to be set to zero. Each curve in the pencil passes through the nine points of the complex projective plane whose homogeneous coordinates are some permutation of 0, –1, and a cube root of unity. There are three roots of unity, and six permutations per root, giving 18 choices for the homogeneous coordinates of each point, but they are equivalent in pairs giving only nine points. More generally, the Hesse pencil can be defined in the same way as the family of cubics through these nine points over any field containing a cube root of unity in place of the complex numbers. The nine common points of the Hesse pencil are the inflection points of each of the cubics in the pencil. Any line that passes through at least two of these nine points passes through exactly three of them; the nine points and twelve lines through triples of points form the Hesse configuration. ==References== * * 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「Hesse pencil」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.