复数高斯投影非迭代优化变换

Non-Iterative Optimization Transformation of Complex Gaussian Projection

摘要: 现有复数高斯投影非迭代变换，基于以偏心率$e$ 为参数的Taylor级数展开式，表达式较为冗长。以第3扁率$n$ 为参数的Taylor级数展开式，级数幂次降低一半，系数表达式得到简化，$c$ 及$c_f$ 系数的精简尤为显著。本文在现有高斯投影非迭代变换的基础上，选择适当的优化因子，使高斯投影椭圆积分的级数系数$k$ 及归一化系数$k_A$ 得到特别显著的精简优化。高斯投影变换的Taylor级数展开式，取级数幂次$je=12$ 或$jn=6$ ，其截断误差 < 1E−15，误差足以满足实际用于需求。变换实例表明，反算结果可精准回溯正算结果，回溯误差 < 1E−15。

Abstract: The existing non-iterative transformation of complex Gaussian projection is based on the Taylor series expansion with eccentricity $e$ as a parameter, and the coefficient expression is relatively lengthy. The Taylor series expansion with the third flattening rate $n$ as the parameter reduces the power of the series by half, simplifying the expression, especially the coefficients $c$ and $c_f$ . Based on the existing Gaussian projection non-iterative transformation, this paper selects appropriate optimization factors to achieve particularly significant simplification and optimization of the series coefficient $k$ and normalization coefficient $k_A$ in the Taylor expansion of the Gaussian projection elliptic integral. The Taylor series expansion of the Gauss projection transformation, with the series power $je=12$ or $jn=6$ , has a truncation error of < 1E−15, which is sufficient to meet the requirements of practical applications. The transformation example shows that the inverse calculation results can accurately trace back to the forward calculation results, with a traceback error less than 1E−15.