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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 Ellipsoid ： ウィキペディア英語版 Ellipsoid An ellipsoid is a closed quadric surface that is a three-dimensional analogue of an ellipse. The standard equation of an ellipsoid centered at the origin of a Cartesian coordinate system and aligned with the axes is :$++=1,$ The points (''a'',0,0), (0,''b'',0) and (0,0,''c'') lie on the surface and the line segments from the origin to these points are called the semi-principal axes of length ''a'', ''b'', ''c''. They correspond to the semi-major axis and semi-minor axis of the appropriate ellipses. There are four distinct cases of which one is degenerate: *$a>b>c$ — tri-axial or (rarely) scalene ellipsoid; *$a=b>c$ — oblate ellipsoid of revolution (oblate spheroid); * *$a=b=c$ — the degenerate case of a sphere; Mathematical literature often uses 'ellipsoid' in place of 'tri-axial ellipsoid'. Scientific literature (particularly geodesy) often uses 'ellipsoid' in place of 'ellipsoid of revolution' and only applies the adjective 'tri-axial' when treating the general case. Older literature uses 'spheroid' in place of 'ellipsoid of revolution'. Any planar cross section passing through the center of an ellipsoid forms an ellipse on its surface: this degenerates to a circle for sections normal to the symmetry axis of an ellipsoid of revolution (or all sections when the ellipsoid degenerates to a sphere.) ==Generalised equations== More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the solutions x to the equation :$(\backslash mathbf)^\backslash mathrm\backslash !\; A\backslash ,\; (\backslash mathbf)\; =\; 1,$ where ''A'' is a positive definite matrix and x, v are vectors. The eigenvectors of ''A'' define the principal axes of the ellipsoid and the eigenvalues of A are the reciprocals of the squares of the semi-axes: $a^$, $b^$ and $c^$.〔http://see.stanford.edu/materials/lsoeldsee263/15-symm.pdf〕 An invertible linear transformation applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable rotation, a consequence of the polar decomposition (also, see spectral theorem). If the linear transformation is represented by a symmetric 3-by-3 matrix, then the eigenvectors of the matrix are orthogonal (due to the spectral theorem) and represent the directions of the axes of the ellipsoid: the lengths of the semiaxes are given by the eigenvalues. The singular value decomposition and polar decomposition are matrix decompositions closely related to these geometric observations. 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「Ellipsoid」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.