大地主题问题常用幂级数的第三扁率展开

The Third Flattening Expansion of Power Se-ries Commonly Used in Geodetic Problem

摘要: 针对传统方法大地主题问题解算用第一偏心率e和K为参数的幂级数展开收敛速度慢、形式复杂效率低下的问题，以第三扁率n代替以往参数将其级数展开式进行重新推导改化。结果表明，基于n的大地线问题幂级数展开式更为简洁，形式上更加简单，更加便于分析与应用；在中小距离的大地主题反解上，新的表达式精度最高达到0.0001 m，满足高精度的要求；在长距离大地主题反解上，对传统人工推导的结果进行了新的计算，依然保持精度相对一致性。同时结果发现：经过转换后，在保持了上述精度的同时，展开式从原来的10阶降到了3阶，形式上更为简单。与传统的贝塞尔大地反解问题相比，在长距离大地线的解算上也存在着一定优势。

Abstract: In view of the problems of slow convergence and low efficiency of the power series expansion with the first flattening e and K as the parameters of the traditional method for solving the geodetic problem, the third flattening n was used to replace the previous parameters and the series expan-sion was reduced and modified. The results show that the power series expansion of the geodetic problem based on n is more concise, simpler in form, and more convenient for analysis and applica-tion. For the inverse solution of the earth theme in small and medium distance, the accuracy of the new expression is up to 0.0001 m, which meets the requirement of high precision. In the inverse solution of long distance geodetic theme, a new calculation is made to the result of traditional artifi-cial derivation, and the accuracy is still consistent. At the same time, it is found that after the con-version, while maintaining the above accuracy, the expansion formula is reduced from the original order 10 to the order 3, which is simpler in form. Compared with the traditional Bessel geodetic in-verse problem, it also has some advantages in the calculation of long distance geodetic line.