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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 kappa curve ： ウィキペディア英語版 kappa curve In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek letter ϰ (kappa). The kappa curve was first studied by Gérard van Gutschoven around 1662. In the history of mathematics, it is remembered as one of the first examples of Isaac Barrow's application of rudimentary calculus methods to determine the tangent of a curve. Isaac Newton and Johann Bernoulli continued the studies of this curve subsequently. Using the Cartesian coordinate system it can be expressed as: :$x^2(x^2\; +\; y^2)=a^2y^2$ or, using parametric equations: :$$ \begin x&=&a\sin t\\ y&=&a\sin t\tan t \end In polar coordinates its equation is even simpler: :$r=a\backslash tan\backslash theta$ It has two vertical asymptotes at $x=\backslash pm\; a$, shown as dashed blue lines in the figure at right. The kappa curve's curvature: :$\backslash kappa(\backslash theta)=\backslash sin(2\backslash theta)\backslash right"\; TITLE="\backslash sin(2\backslash theta)\backslash right">)$ ==Tangents via infinitessimals== The tangent lines of the kappa curve can also be determined geometrically using differentials and the elementary rules of infinitessimal arithmetic. Suppose x and y are variables, while a is taken as a constant. From the definition of the kappa curve, :$x^2(x^2\; +\; y^2)-a^2y^2\; =\; 0$ Now, an infinitessimal change in our location must also change the value of the left hand side, so :$d\; (x^2(x^2\; +\; y^2)-a^2y^2)\; =\; 0$ Distributing the differential and applying appropriate rules, :$d\; (x^2(x^2\; +\; y^2))-d\; (a^2y^2)\; =\; 0$ :$(2\; x\; dx\; )\; (x^2+y^2)\; +\; x^2\; (2xdx\; +\; 2y\; dy)\; -\; a^2\; 2y\; dy\; =\; 0$ :$(\; 4\; x^3\; +\; 2\; x\; y^2)\; dx\; +\; (\; 2\; y\; x^2\; -\; 2\; a^2\; y\; )\; dy\; =\; 0$ :$x\; (\; 2\; x^2\; +\; y^2\; )\; dx\; +\; y\; (x^2\; -\; a^2)\; dy\; =\; 0$ :$\backslash frac\; =\; \backslash frac$ 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「kappa 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.