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

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.