测地坐标系中大地线的微分方程及微分关系式

Differential Equations and Differential Relationship of Geodesic Lines in Geodesic Coordinate System

应用微分几何和大地测量理论 ,提出并推证了在地球椭球面上的局部区域内以测地坐标为坐标参数的大地线二阶微分方程和一阶微分关系式 ,其间定义了在测地坐标系中大地线的方向角 ,并得出该方向角与大地方位角的关系式 .由此获得的大地线必要条件式与大地测量中的克莱劳定理相一致 .利用微分几何学中的Liouville公式也能证得完全相同的微分关系式 .这就为进行测地主题正反解并用以进行三维GIS建模及空间量度、分析奠定了理论基础 .

Based on the theory of differential geometry and geodesy,the second order differential equation and the first differential relationship are derived on the regional earth ellipsoid in this paper.The relationship between the azimuth and the defined direction angle in the geodesic coordinate system is therein obtained.The given necessary condition expression for geodesic line is the same as that one by the Clairaut theorem in geodetic coordinate system.The same first order differential relationship can also be derived from the Liouville formula on the differential geometry.It is useful for the direct and inverse solution of geodesic problem and applying in digital elevation model(DEM) and 3D GIS model.