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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 Willmore energy ： ウィキペディア英語版 Willmore energy In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is defined to be the integral of the square of the mean curvature minus the Gaussian curvature. It is named after the English geometer Thomas Willmore. ==Definition== Expressed symbolically, the Willmore energy of ''S'' is: :$\backslash mathcal\; =\; \backslash int\_S\; H^2\; \backslash ,\; dA\; -\; \backslash int\_S\; K\; \backslash ,\; dA$ where $H$ is the mean curvature, $K$ is the Gaussian curvature, and ''dA'' is the area form of ''S''. For a closed surface, by the Gauss–Bonnet theorem, the integral of the Gaussian curvature may be computed in terms of the Euler characteristic $\backslash chi(S)$ of the surface, so :$\backslash int\_S\; K\; \backslash ,\; dA\; =\; 2\; \backslash pi\; \backslash chi(S),$ which is a topological invariant and thus independent of the particular embedding in $\backslash mathbb^3$ that was chosen. Thus the Willmore energy can be expressed as :$\backslash mathcal\; =\; \backslash int\_S\; H^2\; \backslash ,\; dA\; -\; 2\; \backslash pi\; \backslash chi(S)$ An alternative, but equivalent, formula is :$\backslash mathcal\; =\; \backslash int\_S\; (k\_1\; -\; k\_2)^2\; \backslash ,\; dA$ where $k\_1$ and $k\_2$ are the principal curvatures of the surface. 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「Willmore energy」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.