测地曲率

测地曲率：设P是曲线（C）上一点， $\alpha$ 是（C）在P点的单位切矢量， $\beta$ 是主法矢量， $\gamma$ 是副法矢量。再设n是曲面S在P点的单位法矢量。命 $\varepsilon = n \times \alpha$ 。

曲线（C）在P点的曲率矢量 $\ddot{r}=k\beta$ 在 $\varepsilon$ 上的投影（也就是在S上P点的切平面上的投影）

$k_{g}=\ddot{r}\cdot\varepsilon$

称为曲线（C）在P点的测地曲率。

单词	Geodesic curvature
释义	Geodesic curvature 中文百科测地曲率测地曲率：设P是曲线（C）上一点， $\alpha$ 是（C）在P点的单位切矢量， $\beta$ 是主法矢量， $\gamma$ 是副法矢量。再设n是曲面S在P点的单位法矢量。命 $\varepsilon = n \times \alpha$ 。曲线（C）在P点的曲率矢量 $\ddot{r}=k\beta$ 在 $\varepsilon$ 上的投影（也就是在S上P点的切平面上的投影） $k_{g}=\ddot{r}\cdot\varepsilon$ 称为曲线（C）在P点的测地曲率。英语百科 Geodesic curvature 测地曲率 In Riemannian geometry, the geodesic curvature $k_g$ of a curve $\gamma$ measures how far the curve is from being a geodesic. In a given manifold $\bar{M}$ , the geodesic curvature is just the usual curvature of $\gamma$ (see below), but when $\gamma$ is restricted to lie on a submanifold $M$ of $\bar{M}$ (e.g. for curves on surfaces), geodesic curvature refers to the curvature of $\gamma$ in $M$ and it is different in general from the curvature of $\gamma$ in the ambient manifold $\bar{M}$ . The (ambient) curvature $k$ of $\gamma$ depends on two factors: the curvature of the submanifold $M$ in the direction of $\gamma$ (the normal curvature $k_n$ ), which depends only from the direction of the curve, and the curvature of $\gamma$ seen in $M$ (the geodesic curvature $k_g$ ), which is a second order quantity. The relation between these is $k = \sqrt{k_g^2+k_n^2}$ . 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.
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Geodesic curvature

测地曲率

Geodesic curvature 测地曲率