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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 Twisted cubic ： ウィキペディア英語版 Twisted cubic In mathematics, a twisted cubic is a smooth, rational curve ''C'' of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (''the'' twisted cubic, therefore). It is generally considered to be the simplest example of a projective variety that isn't linear or a hypersurface, and is given as such in most textbooks on algebraic geometry. It is the three-dimensional case of the rational normal curve, and is the image of a Veronese map of degree three on the projective line. ==Definition== It is most easily given parametrically as the image of the map :$\backslash nu:\backslash mathbf^1\backslash to\backslash mathbf^3$ which assigns to the homogeneous coordinate $()$ the value :$\backslash nu:()\; \backslash mapsto\; ().$ In one coordinate patch of projective space, the map is simply the moment curve :$\backslash nu:x\; \backslash mapsto\; (x,x^2,x^3)$ That is, it is the closure by a single point at infinity of the affine curve $(x,x^2,x^3)$. Equivalently, it is a projective variety, defined as the zero locus of three smooth quadrics. Given the homogeneous coordinates () on P3, it is the zero locus of the three homogeneous polynomials :$F\_0\; =\; XZ\; -\; Y^2$ :$F\_1\; =\; YW\; -\; Z^2$ :$F\_2\; =\; XW\; -\; YZ.$ It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above; that is, substituting ''x''3 for ''X'', and so on. In fact, the homogeneous ideal of the twisted cubic ''C'' is generated by three algebraic forms of degree two on P3. The generators of the ideal are :$\backslash .$ 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「Twisted cubic」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.