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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 Schild's ladder ： ウィキペディア英語版 Schild's ladder In the theory of general relativity, and differential geometry more generally, Schild's ladder is a first-order method for ''approximating'' parallel transport of a vector along a curve using only affinely parametrized geodesics. The method is named for Alfred Schild, who introduced the method during lectures at Princeton University. == Construction == The idea is to identify a tangent vector ''x'' at a point $A\_0$ with a geodesic segment of unit length $A\_0X\_0$, and to construct an approximate parallelogram with approximately parallel sides $A\_0X\_0$ and $A\_1X\_1$ as an approximation of the Levi-Civita parallelogramoid; the new segment $A\_1X\_1$ thus corresponds to an approximately parallel translated tangent vector at $A\_1.$ Formally, consider a curve γ through a point ''A''0 in a Riemannian manifold ''M'', and let ''x'' be a tangent vector at ''A''0. Then ''x'' can be identified with a geodesic segment ''A''0''X''0 via the exponential map. This geodesic σ satisfies :$\backslash sigma(0)=A\_0\backslash ,$ :$\backslash sigma\text{'}(0)\; =\; x.\backslash ,$ The steps of the Schild's ladder construction are: * Let ''X''0 = σ(1), so the geodesic segment $A\_0X\_0$ has unit length. * Now let ''A''1 be a point on γ close to ''A''0, and construct the geodesic ''X''0''A''1. * Let ''P''1 be the midpoint of ''X''0''A''1 in the sense that the segments ''X''0''P''1 and ''P''1''A''1 take an equal affine parameter to traverse. * Construct the geodesic ''A''0''P''1, and extend it to a point ''X''1 so that the parameter length of ''A''0''X''1 is double that of ''A''0''P''1. * Finally construct the geodesic ''A''1''X''1. The tangent to this geodesic ''x''1 is then the parallel transport of ''X''0 to ''A''1, at least to first order. 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「Schild's ladder」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.