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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 Parametric surface ： ウィキペディア英語版 Parametric surface A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters $\backslash quad\; \backslash vec\; r:\; \backslash Bbb^2\; \backslash rightarrow\; \backslash Bbb^3.$ Parametric representation is a very general way to specify a surface, although implicit equations are even more general. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization. == Examples == * The simplest type of parametric surfaces is given by the graphs of functions of two variables: ::$z\; =\; f(x,y),\; \backslash quad\; \backslash vec\; r(x,y)\; =\; (x,\; y,\; f(x,y)).$ * Rational surfaces are surface that admits parameterizations by rational functions. A rational surface is an algebraic surface. Given an algebraic surface, it is commonly easier to decide if it is rational than to compute its rational parameterization, if it exists. * Surfaces of revolution give another important class of surfaces that can be easily parametrized. If the graph ''z'' = ''f''(''x''), ''a'' ≤ ''x'' ≤ ''b'' is rotated about the ''z''-axis then the resulting surface has a parametrization :: $\backslash vec\; r(u,\backslash phi)\; =\; (u\backslash cos\backslash phi,\; u\backslash sin\backslash phi,\; f(u)),$ \quad a\leq u\leq b, 0\leq\phi < 2\pi. :It may also be parameterized :: $\backslash vec\; r(u,v)\; =\; (u\backslash frac,\; u\backslash frac,\; f(u)),$ \quad a\leq u\leq b, :showing that, if the function is rational, then the surface is rational. * The straight circular cylinder of radius ''R'' about ''x''-axis has the following parametric representation: ::$\backslash vec\; r(x,\; \backslash phi)\; =\; (x,\; R\backslash cos\backslash phi,\; R\backslash sin\backslash phi).$ * Using the spherical coordinates, the unit sphere can be parameterized by :: : This parametrization breaks down at the north and south poles where the azimuth angle ''θ'' is not determined uniquely. The sphere is a rational surface. The same surface admits many different parametrizations. For example, the coordinate ''z''-plane can be parametrized as :$\backslash vec\; r(u,v)=(au+bv,cu+dv,\; 0)$ for any constants ''a'', ''b'', ''c'', ''d'' such that ''ad'' − ''bc'' ≠ 0, i.e. the matrix is invertible. 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「Parametric surface」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.