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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 Envelope (mathematics) ： ウィキペディア英語版 Envelope (mathematics) In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions. ==Envelope of a family of curves== Let each curve ''C''''t'' in the family be given as the solution of an equation ''f''''t''(''x'', ''y'')=0 (see implicit curve), where ''t'' is a parameter. Write ''F''(''t'', ''x'', ''y'')=''f''''t''(''x'', ''y'') and assume ''F'' is differentiable. The envelope of the family ''C''''t'' is then defined as the set of points for which :$F(t,\; x,\; y)\; =\; (t,\; x,\; y)\; =\; 0$ for some value of ''t'', where $\backslash partial\; F/\backslash partial\; t$ is the partial derivative of ''F'' with respect to ''t''. Note that if ''t'' and ''u'', ''t''≠''u'' are two values of the parameter then the intersection of the curves ''C''''t'' and ''C''''u'' is given by :$F(t,\; x,\; y)\; =\; F(u,\; x,\; y)\; =\; 0\backslash ,$ or equivalently :$F(t,\; x,\; y)\; =\; \backslash frac\; =\; 0.$ Letting u→t gives the definition above. An important special case is when ''F''(''t'', ''x'', ''y'') is a polynomial in ''t''. This includes, by clearing denominators, the case where ''F''(''t'', ''x'', ''y'') is a rational function in ''t''. In this case, the definition amounts to ''t'' being a double root of ''F''(''t'', ''x'', ''y''), so the equation of the envelope can be found by setting the discriminant of ''F'' to 0. For example, let ''C''''t'' be the line whose ''x'' and ''y'' intercepts are ''t'' and 1−''t'', this is shown in the animation above. The equation of ''C''''t'' is :$\backslash frac+\backslash frac=1$ or, clearing fractions, :$x(1-t)+yt-t(1-t)=t^2+(-x+y-1)t+x=0.\backslash ,$ The equation of the envelope is then :$(-x+y-1)^2-4x=(x-y)^2-2(x+y)+1=0.\backslash ,$ Often when ''F'' is not a rational function of the parameter it may be reduced to this case by an appropriate substitution. For example if the family is given by ''C''θ with an equation of the form ''u''(''x'', ''y'')cosθ+''v''(''x'', ''y'')sinθ=''w''(''x'', ''y''), then putting ''t''=''e''''i''θ, cosθ=(''t''+1/''t'')/2, sinθ=(''t''-1/''t'')/2''i'' changes the equation of the curve to :$u(t+)+v(t-)=w$ or :$(u-iv)t^2-2wt+(u+iv)=0.\backslash ,$ The equation of the envelope is then given by setting the discriminant to 0: :$(u-iv)(u+iv)-w^2=0\backslash ,$ or :$u^2+v^2=w^2.\backslash ,$ 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「Envelope (mathematics)」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.