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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 Björling problem ： ウィキペディア英語版 Björling problem In differential geometry, the Björling problem is the problem of finding a minimal surface passing through a given curve with prescribed normal (or tangent planes). The problem was posed and solved by Swedish mathematician Emanuel Gabriel Björling,〔E.G. Björling, Arch. Grunert , IV (1844) pp. 290〕 with further refinement by Hermann Schwarz.〔H.A. Schwarz, J. reine angew. Math. 80 280-300 1875〕 The problem can be solved by extending the surface from the curve using complex analytic continuation. If $c(s)$ is a real analytic curve in ℝ3 defined over an interval ''I'', with $c\text{'}(s)\backslash neq\; 0$ and a vector field $n(s)$ along ''c'' such that $||n(t)||=1$ and $c\text{'}(t)\backslash cdot\; n(t)=0$, then the following surface is minimal: :$X(u,v)\; =\; \backslash Re\; \backslash left\; (\; c(w)\; -\; i\; \backslash int\_^w\; n(w)\; \backslash wedge\; c\text{'}(w)\; \backslash ,\; dw\; \backslash right)$ where $w\; =\; u+iv\; \backslash in\; \backslash Omega$, $u\_0\backslash in\; I$, and $I\; \backslash subset\; \backslash Omega$ is a simply connected domain where the interval is included and the power series expansions of $c(s)$ and $n(s)$ are convergent.〔Kai-Wing Fung, Minimal Surfaces as Isotropic Curves in C3: Associated minimal surfaces and the Björling's problem. MIT BA Thesis. 2004 http://ocw.mit.edu/courses/mathematics/18-994-seminar-in-geometry-fall-2004/projects/main1.pdf〕 A classic example is Catalan's minimal surface, which passes through a cycloid curve. Applying the method to a semicubical parabola produces the Henneberg surface, and to a circle (with a suitably twisted normal field) a minimal Möbius strip. A unique solution always exists. It can be viewed as a Cauchy problem for minimal surfaces, allowing one to find a surface if a geodesic, asymptote or lines of curvature is known. In particular, if the curve is planar and geodesic, then the plane of the curve will be a symmetry plane of the surface.〔Björling problem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bj%C3%B6rling_problem&oldid=23196〕 ==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「Björling problem」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.