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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 ： ウィキペディア英語版 Quadric In mathematics, a quadric, or quadric surface, is any ''D''-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates , the general quadric is defined by the algebraic equation〔Silvio Levy (Quadrics ) in "Geometry Formulas and Facts", excerpted from 30th Edition of ''CRC Standard Mathematical Tables and Formulas'', CRC Press, from The Geometry Center at University of Minnesota〕 :$$ \sum_^ x_i Q_ x_j + \sum_^ P_i x_i + R = 0 which may be compactly written in vector and matrix notation as: :$$ x Q x^\mathrm + P x^\mathrm + R = 0\, where is a row vector, ''x''T is the transpose of ''x'' (a column vector), ''Q'' is a matrix and ''P'' is a -dimensional row vector and ''R'' a scalar constant. The values ''Q'', ''P'' and ''R'' are often taken to be over real numbers or complex numbers, but a quadric may be defined over any ring. In general, the locus of zeros of a set of polynomials is known as an algebraic set, and is studied in the branch of algebraic geometry. A quadric is thus an example of an algebraic set. For the projective theory see Quadric (projective geometry). == Euclidean plane and space == Quadrics in the Euclidean plane are those of dimension ''D'' = 1, which is to say that they are curves. Such quadrics are the same as conic sections, and are typically known as conics rather than quadrics. In Euclidean space, quadrics have dimension ''D'' = 2, and are known as quadric surfaces. By making a suitable Euclidean change of variables, any quadric in Euclidean space can be put into a certain normal form by choosing as the coordinate directions the principal axes of the quadric. In three-dimensional Euclidean space there are 16 such normal forms.〔(Sameen Ahmed Khan ),(Quadratic Surfaces in Science and Engineering ), Bulletin of the IAPT, 2(11), 327–330 (November 2010). (Publication of the Indian Association of Physics Teachers). (Sameen Ahmed Khan ), (Coordinate Geometric Generalization of the Spherometer and Cylindrometer ), (arXiv:1311.3602 ) 〕 Of these 16 forms, five are nondegenerate, and the remaining are degenerate forms. Degenerate forms include planes, lines, points or even no points at all.〔Stewart Venit and Wayne Bishop, ''Elementary Linear Algebra (fourth edition)'', International Thompson Publishing, 1996.〕 + = 1 \, | |- |     Spheroid (special case of ellipsoid) | $+\; +\; =\; 1\; \backslash ,$ | |- |     Sphere (special case of spheroid) | $+\; +\; =\; 1\; \backslash ,$ | |- |     Elliptic paraboloid | $+\; -\; z\; =\; 0\; \backslash ,$ | |- |     Circular paraboloid (special case of elliptic paraboloid) | $+\; -\; z\; =\; 0\; \backslash ,$ | |- |     Hyperbolic paraboloid | $-\; -\; z\; =\; 0\; \backslash ,$ | |- |    Elliptic hyperboloid of one sheet | $+\; -\; =\; 1\; \backslash ,$ | |- |    Elliptic hyperboloid of two sheets | $+\; -\; =\; -\; 1\; \backslash ,$ | |- ! colspan="3" style="background-color: white;" | Degenerate quadric surfaces |- |     Elliptic cone | $+\; -\; =\; 0\; \backslash ,$ | |- |     Circular cone (special case of cone) | $+\; -\; =\; 0\; \backslash ,$ | |- |     Elliptic cylinder | $+\; =\; 1\; \backslash ,$ | |- |     Circular cylinder (special case of elliptic cylinder) | $+\; =\; 1\; \backslash ,$ | |- |     Hyperbolic cylinder | $-\; =\; 1\; \backslash ,$ | |- |     Parabolic cylinder | $x^2\; +\; 2ay\; =\; 0\; \backslash ,$ | |} 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「Quadric」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.