纬度正反解问题的傅里叶级数解算方法

讨论了一种在一般闭区间上连续的无穷级数展开方法,针对测量和地图学中常用纬度间正反解等一类问题,给出了一种傅里叶级数解算方法,并进行了相应的理论分析和计算。基于Mathematica计算机代数系统,推导出常用纬度与大地纬度间正反解的级数展开式,表明运用傅里叶级数解法可以简单表示和计算一类复杂奇函数的正反解无穷展开式。算例分析验证表明,该方法推导出的纬度正反解展开式是正确的。傅里叶级数方法为分析一类正反解问题无穷展开式的一般系数和形式提供了理论分析依据,具有通用性,丰富了纬度正反解变换的相关理论。

A Fourier Series Method for Forward and Inverse Solutions of Latitudes

An infinite series expansion method in general continuous closed interval is discussed.For calculating the forward and inverse solutions between the commonly used latitudes in surveying and cartography,a Fourier series method is given and then the corresponding theoretical analysis and calculations are conducted.Based on the computer algebra system Mathematica,the series expansions of the forward and inverse solutions between the commonly used latitudes and geodetic latitudes are derived,showing the series expansions of complex odd functions can be conveniently represented and calculated by using the Fourier series method.The example analysis shows the method and its expansions are correct.The method can be used to supply the theoretical basis for analyzing the general coefficients and forms of the forward and inverse expansions,which is universal and enrich the related theory of latitude transformations.