空间直角坐标与大地坐标转换的拉格朗日反演方法

Transformation from Cartesian to Geodetic Coordinates Using Lagrange Inversion Theorem

以拉格朗日反演理论为基础,导出空间直角坐标向大地坐标转换的一种新的直接解法。该方法将归化纬度的正弦函数sinμ表达为以空间直角坐标(X,Y,Z)为基础的相关变量的多项式。为验证公式的精度水平和实用性,将WGS-84椭球参数代入验算,结果表明,新解法在-2×106 m≤H≤+1010 m范围内,展开至b4项纬度反解误差不超过9.71×10-6″,展开至b5项的误差不超过9.67×10-8″,有效范围比Bowring公式广,在H≥+105 m的区域精度明显优于Bowring公式。与迭代算法相比,新方法在H≥-2×106 m的区域精度与迭代算法精度相当,但计算效率明显优于迭代算法,展开至b4项的CPU执行时间分别约为迭代4次的1/2、展开至b5项的CPU执行时间约为迭代5次的1/10。兼顾精度和效率,本文算法优于Bowring公式和迭代算法。

According to Lagrange Inversion Theorem,a Taylor-series expansion method for transforming Cartesian to geodetic coordinates is obtained,which expresses the sine function of reduce latitude sinμas a convergent polynomial of geodetic coordinates(X,Y,Z).Comparative computation of the new method and Bowring's formula show that the new method is sufficiently precise in the sense that the maximum error of the latitude is less than9.71×10-6″,9.67×10-8″for the range of-2×106≤H ≤+1010,truncating at b4andb5 respectively,while Bowring's formula only works well for the range of-105≤H ≤ +105.Meanwhile,new algorithm is around50%~90%faster than the iterative algorithm with the approximate accuracy.