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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 geodesic curvature ： ウィキペディア英語版 geodesic curvature In Riemannian geometry, the geodesic curvature $k\_g$ of a curve $\backslash gamma$ measures how far the curve is from being a geodesic. In a given manifold $\backslash bar$, the geodesic curvature is just the usual curvature of $\backslash gamma$ (see below), but when $\backslash gamma$ is restricted to lie on a submanifold $M$ of $\backslash bar$ (e.g. for curves on surfaces), geodesic curvature refers to the curvature of $\backslash gamma$ in $M$ and it is different in general from the curvature of $\backslash gamma$ in the ambient manifold $\backslash bar$. The (ambient) curvature $k$ of $\backslash gamma$ depends on two factors: the curvature of the submanifold $M$ in the direction of $\backslash gamma$ (the normal curvature $k\_n$), which depends only from the direction of the curve, and the curvature of $\backslash gamma$ seen in $M$ (the geodesic curvature $k\_g$), which is a second order quantity. The relation between these is $k\; =\; \backslash sqrt$. In particular geodesics on $M$ have zero geodesic curvature (they are "straight"), so that $k=k\_n$, which explains why they appear to be curved in ambient space whenever the submanifold is. ==Definition== Consider a curve $\backslash gamma$ in a manifold $\backslash bar$, parametrized by arclength, with unit tangent vector $T=d\backslash gamma/ds$. Its curvature is the norm of the covariant derivative of $T$: $k\; =\; \backslash |DT/ds\; \backslash |$. If $\backslash gamma$ lies on $M$, the geodesic curvature is the norm of the projection of the covariant derivative $DT/ds$ on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of $DT/ds$ on the normal bundle to the submanifold at the point considered. If the ambient manifold is the euclidean space $\backslash mathbb^n$, then the covariant derivative $DT/ds$ is just the usual derivative $dT/ds$. 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「geodesic curvature」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.