兰勃脱等角圆锥投影反解不同算法的解析

Analysis of Different Algorithms for Reverse Solution of Lambert Conic Conformal Projection

摘要: 在测量与地图制图中，等量纬度求解大地纬度是一种常见的投影反解计算，就该反解问题的几种不同算法进行研究，包括迭代法、等量纬差求解大地纬度的级数展开式及等量纬度求解大地纬度的直接算法。利用Mathematica对后两种算法的计算公式进行了详细推导，给出了其高阶系数展开式，同时对现有算法中存在的问题进行了解析。兰勃脱等角投影算例表明，所推导的公式其计算精度可达(1×10^-7)″~(1×10^-8)″，完全满足测量与地图投影高精度的要求。

Abstract:
Objectives In surveying and cartography, solving geodetic latitude is a common reverse calculation by conformal latitude.
Methods Several different algorithms of reverse solution are studied, including iterative method, series expansion of solving geodetic latitude by difference of conformal latitude and direct algorithm of inverse solution of geodetic latitude with conformal latitude. The calculation formulas of the latter two algorithms are derived by Mathematica. The higher-order coefficient expansion formula are given. At the same time, problems in existing algorithms are analyzed.
Results and Conclusions E xamples results of Lambert conic conformal projection show that the accuracy of the formula derived in this paper can reaches (1×10^-7)″-(1×10^-8)″. It fully meets the requirements of high precision in surveying and map projection.