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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 Dual curve ： ウィキペディア英語版 Dual curve In projective geometry, a dual curve of a given plane curve ''C'' is a curve in the dual projective plane consisting of the set of lines tangent to ''C''. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If ''C'' is algebraic then so is its dual and the degree of the dual is known as the ''class'' of the original curve. The equation of the dual of ''C'', given in line coordinates, is known as the ''tangential equation'' of ''C''. The construction of the dual curve is the geometrical underpinning for the Legendre transformation in the context of Hamiltonian mechanics.〔See 〕 ==Equations== Let ''f''(''x'', ''y'', ''z'')=0 be the equation of a curve in homogeneous coordinates. Let ''Xx''+''Yy''+''Zz''=0 be the equation of a line, with (''X'', ''Y'', ''Z'') being designated its line coordinates. The condition that the line is tangent to the curve can be expressed in the form ''F''(''X'', ''Y'', ''Z'')=0 which is the tangential equation of the curve. Let (''p'', ''q'', ''r'') be the point on the curve, then the equation of the tangent at this point is given by :$x\backslash frac(p,\; q,\; r)+y\backslash frac(p,\; q,\; r)+z\backslash frac(p,\; q,\; r)=0.$ So ''Xx''+''Yy''+''Zz''=0 is a tangent to the curve if :$X=\backslash lambda\; \backslash frac(p,\; q,\; r),\backslash ,Y=\backslash lambda\; \backslash frac(p,\; q,\; r),\backslash ,Z=\backslash lambda\; \backslash frac(p,\; q,\; r).$ Eliminating ''p'', ''q'', ''r'', and λ from these equations, along with ''Xp''+''Yq''+''Zr''=0, gives the equation in ''X'', ''Y'' and ''Z'' of the dual curve. For example, let ''C'' be the conic ''ax''2+''by''2+''cz''2=0. Then dual is found by eliminating ''p'', ''q'', ''r'', and λ from the equations :$X=\; 2\backslash lambda\; ap,\backslash ,Y=2\backslash lambda\; bq,\backslash ,Z=2\backslash lambda\; cr,\backslash ,Xp+Yq+Zr=0.$ The first three equations are easily solved for ''p'', ''q'', ''r'', and substituting in the last equation produces :$\backslash frac+\backslash frac+\backslash frac=0.$ Clearing 2λ from the denominators, the equation of the dual is :$\backslash frac+\backslash frac+\backslash frac=0.$ For a parametrically defined curve its dual curve is defined by the following parametric equations: :$X=\backslash frac$ :$Y=\backslash frac$ The dual of an inflection point will give a cusp and two points sharing the same tangent line will give a self intersection point on the dual. 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「Dual curve」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.