曲面正交网下测地曲率计算公式的推导方法

Derivation of Calculation Formula of Geodesic Curvature Under the Coordinate Grid of the Orthogonal Curve

考虑曲面上曲线测地曲率计算公式的推导方法问题,在曲面正交坐标网下,给出曲面上曲线测地曲率计算公式的参数方程形式,并由此得出测地曲率计算Liouville公式的一种推导方法。充分利用坐标曲线网的正交性条件,介绍了一种推导Liouville公式的直接方法,由此发现两种推导过程的内在联系。在曲面上一般参数坐标网下,直接给出了测地线的参数方程所满足的微分方程组的形式,由此导出在曲面正交坐标网下测地线的微分方程组。

The derivation method of calculation formula of curve geodesic curvature on the curved surface was considered in the paper. Based on the coordinate grid of the orthogonal curve,the parametric equation form of the calculation formula of geodesic curvature on the surface was put forward,and thus a derivation method of Liouville formula to calculate the geodesic curvature was obtained. Taking good use of the orthogonal condition of curvilinear coordinate grid,a direct method for deriving the Liouville formula was introduced,which can figure out the internal links of these two derivation methods. Under the common parametric coordinate grid of curved surface,the form of differential equation set that parameter equation of geodesic curve meets was put forward directly,with which the geodesic curve differential equation set under coordinate grid of the orthogonal curve was derived.