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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 Enneper's minimal surface ： ウィキペディア英語版 Enneper surface In mathematics, in the fields of differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: : $x\; =\; u(1\; -\; u^2/3\; +\; v^2)/3,\backslash$ : $y\; =\; -v(1\; -\; v^2/3\; +\; u^2)/3,\backslash$ : $z\; =\; (u^2\; -\; v^2)/3.\backslash$ It was introduced by Alfred Enneper 1864 in connection with minimal surface theory.〔J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975)〕〔(Francisco J. López, Francisco Martín, Complete minimal surfaces in R3 )〕〔Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny (2010). Minimal Surfaces. Berlin Heidelberg: Springer. ISBN 978-3-642-11697-1.〕 The Weierstrass–Enneper parameterization is very simple, $f(z)=1,\; g(z)=z$, and the real parametric form can easily be calculated from it. The surface is conjugate to itself. Implicitization methods of algebraic geometry can be used to find out that the points in the Enneper surface given above satisfy the degree-9 polynomial equation : $64\; z^9\; -\; 128\; z^7\; +\; 64\; z^5\; -\; 702\; x^2\; y^2\; z^3\; -\; 18\; x^2\; y^2\; z\; +\; 144\; (y^2\; z^6\; -\; x^2\; z^6)\backslash$ : $+\; 4\; (b^4\; c^2\; +\; b^2\; c^4)\; -\; 3\; (b^4\; d^2\; +\; c^4\; d^2)\; +\; 36\; (a\; b^2\; d^3\; -\; a\; c^2\; d^3)\; =\; 0.\backslash$ The Jacobian, Gaussian curvature and mean curvature are : $J\; =\; (1\; +\; u^2\; +\; v^2)^4/81,\backslash$ : $K\; =\; -(4/9)/J,\backslash$ : $H\; =\; 0.\backslash$ The total curvature is $-4\backslash pi$. Osserman proved that a complete minimal surface in $\backslash R^3$ with total curvature $-4\backslash pi$ is either the catenoid or the Enneper surface.〔R. Osserman, A survey of Minimal Surfaces. Vol. 1, Cambridge Univ. Press, New York (1989).〕 Another property is that all bicubical minimal Bézier surfaces are, up to an affine transformation, pieces of the surface.〔Cosín, C., Monterde, Bézier surfaces of minimal area. In Computational Science — ICCS 2002, eds. J., Sloot, Peter, Hoekstra, Alfons, Tan, C., Dongarra, Jack. Lecture Notes in Computer Science 2330, Springer Berlin / Heidelberg, 2002. pp. 72-81 ISBN 978-3-540-43593-8〕 It can be generalized to higher order rotational symmetries by using the Weierstrass–Enneper parameterization $f(z)=1,\; g(z)=z^k$ for integer k>1.〔 It can also be generalized to higher dimensions; Enneper-like surfaces are known to exist in $\backslash R^n$ for n up to 7.〔Jaigyoung Choe, On the existence of higher dimensional Enneper's surface, Commentarii Mathematici Helvetici 1996, Volume 71, Issue 1, pp 556-569〕 ==References== 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「Enneper 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.