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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 Menger curvature ： ウィキペディア英語版 Menger curvature In mathematics, the Menger curvature of a triple of points in ''n''-dimensional Euclidean space R''n'' is the reciprocal of the radius of the circle that passes through the three points. It is named after the Austrian-American mathematician Karl Menger. ==Definition== Let ''x'', ''y'' and ''z'' be three points in R''n''; for simplicity, assume for the moment that all three points are distinct and do not lie on a single straight line. Let Π ⊆ R''n'' be the Euclidean plane spanned by ''x'', ''y'' and ''z'' and let ''C'' ⊆ Π be the unique Euclidean circle in Π that passes through ''x'', ''y'' and ''z'' (the circumcircle of ''x'', ''y'' and ''z''). Let ''R'' be the radius of ''C''. Then the Menger curvature ''c''(''x'', ''y'', ''z'') of ''x'', ''y'' and ''z'' is defined by :$c\; (x,\; y,\; z)\; =\; \backslash frac1.$ If the three points are collinear, ''R'' can be informally considered to be +∞, and it makes rigorous sense to define ''c''(''x'', ''y'', ''z'') = 0. If any of the points ''x'', ''y'' and ''z'' are coincident, again define ''c''(''x'', ''y'', ''z'') = 0. Using the well-known formula relating the side lengths of a triangle to its area, it follows that :$c\; (x,\; y,\; z)\; =\; \backslash frac1\; =\; \backslash frac,$ where ''A'' denotes the area of the triangle spanned by ''x'', ''y'' and ''z''. Another way of computing Menger curvature is the identity :$c(x,y,z)=\backslash frac$ where $\backslash angle\; xyz$ is the angle made at the ''y''-corner of the triangle spanned by ''x'',''y'',''z''. Menger curvature may also be defined on a general metric space. If ''X'' is a metric space and ''x'',''y'', and ''z'' are distinct points, let ''f'' be an isometry from $\backslash$ into $\backslash mathbb^$. Define the Menger curvature of these points to be :$c\_\; (x,y,z)=c(f(x),f(y),f(z)).$ Note that ''f'' need not be defined on all of ''X'', just on ', and the value ''c''''X'' ''(x,y,z)'' is independent of the choice of ''f''. 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「Menger curvature」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.