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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 Barycentric coordinate system ： ウィキペディア英語版 Barycentric coordinate system In geometry, the barycentric coordinate system is a coordinate system in which the location of a point of a simplex (a triangle, tetrahedron, etc.) is specified as the center of mass, or barycenter, of usually unequal masses placed at its vertices. Coordinates also extend outside the simplex, where one or more coordinates become negative. The system was introduced (1827) by August Ferdinand Möbius. ==Definition== Let $\backslash mathbf\_1,\; \backslash ldots,\; \backslash mathbf\_n$ be the vertices of a simplex in an affine space ''A''. If, for some point $\backslash mathbf$ in ''A'', :$(\; a\_1\; +\; \backslash cdots\; +\; a\_n\; )\; \backslash mathbf\; =\; a\_1\; \backslash ,\; \backslash mathbf\_1\; +\; \backslash cdots\; +\; a\_n\; \backslash ,\; \backslash mathbf\_n$ and at least one of $a\_1,\; \backslash ldots,\; a\_n$ does not vanish then we say that the coefficients ($a\_1,\; \backslash ldots,\; a\_n$) are ''barycentric coordinates'' of $\backslash mathbf$ with respect to $\backslash mathbf\_1,\; \backslash ldots,\; \backslash mathbf\_n$. The vertices themselves have the coordinates $\backslash mathbf\_1=(1,\; 0,\; 0,\; \backslash ldots,\; 0),\; \backslash mathbf\_2=(0,\; 1,\; 0,\; \backslash ldots,\; 0),\; \backslash ldots,\; \backslash mathbf\_n=(0,\; 0,\; 0,\; \backslash ldots,\; 1)$. Barycentric coordinates are not unique: for any ''b'' not equal to zero, ($b\; a\_1,\; \backslash ldots,\; b\; a\_n$) are also barycentric coordinates of ''p''. When the coordinates are not negative, the point $\backslash mathbf$ lies in the convex hull of $\backslash mathbf\_1,\; \backslash ldots,\; \backslash mathbf\_n$, that is, in the simplex which has those points as its vertices. Barycentric coordinates, as defined above, are a form of homogeneous coordinates. Sometimes values of coordinates are restricted with a condition :$\backslash sum\; a\_i\; =\; 1$ which makes them unique; then, they are affine coordinates. 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「Barycentric coordinate system」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.