复数高斯投影变换非迭代算法的2条路径

Two Paths of Non-Iterative Algorithm for Complex Gaussian Projection Transformation

摘要: 高斯投影非迭代符号变换式，不需要迭代计算，更方便直接。将现有高斯投影变换以偏心率参数$e$ 表达的级数系数，用第3扁率$n$ 表达，级数幂次可降低一半，收敛更快，系数更为简洁。本文讨论了显函数及隐函数的Taylor级数展开的导数法及Series函数法。以复数高斯投影变换的非迭代算法为基础，利用计算机代数系统，对角度变换式，全新导出了以第3扁率$n$ 为参数的7次幂符号级数表达式。事实上，当级数幂次取至5次，高斯投影的变换精度为直角坐标 < 10⁻⁶ m，角度 < 10⁻¹² rad (10⁻⁷″)。复数高斯投影变换的核心是实数经纬度$\left(B,l\right)$ 与复数等距离纬度${\psi }_c$ 之间的角度变换。对复数等角纬度${\phi }_c$ 与复数等距离纬度${\psi }_c$ 之间的角度变换，给出了2条实现路径：间接变换路径(以复数纬度$B_c$ 为间接变量)及直接变换路径。

Abstract: Gaussian projection is a non-iterative symbol transformation that does not require iterative calculations and is more convenient and direct. Expressing the series coefficients of the existing Gaussian projection transform as eccentricity parameter $e$ and using the third flattening parameter $n$ can reduce the series power by half, converge faster, and simplify the coefficients. This article discusses the derivative method and Series function method for Taylor series expansion of explicit and implicit functions. Based on the non-iterative algorithm of complex Gaussian projection transformation, using computer algebra system, a new expression of the 7th power signed series with the third flattening $n$ as the parameter is derived for the angle transformation formula. In fact, when the power of the series is set to 5, the accuracy of Gaussian projection transformation is less than 10⁻⁶ m for Cartesian coordinates and less than 10⁻¹² rad (10⁻⁷″) for angles. The core of complex Gaussian projection transformation is the angle transformation between real latitude and longitude $\left(B,l\right)$ and complex equidistant latitude ${\psi }_c$ . Two implementation paths are proposed for the angle transformation between complex equiangular latitude ${\phi }_c$ and complex equidistant latitude ${\psi }_c$ : indirect transformation path (with complex latitude $B_c$ as the indirect variable) and direct transformation path.