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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 Implicit curve ： ウィキペディア英語版 Implicit curve In mathematics an implicit curve is a plane curve which is defined by an equation : $F(x,y)=0.$ Hence an implicit curve can be considered as the set of zeros of a function of two variables. ''Implicit'' means, that the equation is not solved for either x or y. If $F(x,y)$ is a polynomial in two variables, the corresponding curve is called an ''algebraic curve'', and specific methods are available for studying it. The graph of a function is usually described by an equation $y=f(x)$ and is called an ''explicit'' representation. The third essential description of a curve is the ''parametric'' one: , where the ''x''- and ''y''-coordinates of curve points are represented by two functions $x(t)\backslash ,\; ,\; y(t)$ dependent on a common parameter $t.$ The change of representations is unusually simple only, when the explicit representation $y=f(x)$ is given: $y-f(x)=0$ (implicit), $(t,f(t))$ (parametric). ''Examples'' of implicit curves: # a line: $x+2y-3=0\backslash \; ,$ # a circle: $x^2+y^2-4=0\; \backslash \; ,$ # the Semicubical parabola: $x^3-y^2=0\; \backslash \; ,$ # Cassini ovals $(x^2+y^2)^2-2c^2(x^2-y^2)-(a^4-c^4)=0$ (see picture), # $\backslash sin(x+y)-\backslash cos(xy)+1=0$ (see picture). The first four examples are algebraic curves, but the last one is not algebraic. The first three examples possess simple parametric representations, which is not true for the 4th and 5th example. Especially the 5th example shows the possible complicated geometric structure of an implicit curve. The implicit function theorem describes conditions, under which an equation $F(x,y)=0$ can be solved (theoretically) for x and/or y. But in general the solution may be not conducted. This theorem is the key for the computation of essential geometric features of the curve: tangents, normals, curvature (see below). In praxis implicit curves have an essential drawback: their visualization is difficult (see below). But there are computer programs, enabling to display an implicit curve (see weblinks). Special properties of implicit curves make them essential tools in geometry and computer graphic (see below). An implicit curve with an equation $F(x,y)=0$ can be considered as the level curve of level 0 of the surface $z=F(x,y)$ (see third picture). == Formulas == For the following formulas the implicit curve will be defined by an equation $F(x,y)=0$ , where function $F$ meets the needed mathematical requirements. The partial derivatives of $F$ are $F\_x$, $F\_y$, $F\_$. 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「Implicit 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.