Lee’s exact method was developed to enable the Gauss–Krüger (GK) projection to be implemented without iterative procedures via the expansion of the intermediate mapping (Thompson projection) into series approximations in terms of isothermal coordinates $$\left( \psi , \lambda \right) $$ ψ , λ for the forward mapping and GK coordinates $$\left( x, y \right) $$ x , y for the reverse mapping. The straightforward procedures expressed by new formulas for both forward and reverse mapping of the GK projection were composed by three sequential steps which essentially reveal the mapping procedures and intrinsic properties of the GK projection: The first step of deriving the isothermal coordinates $$\left( \psi , \lambda \right) $$ ψ , λ from the geodetic coordinates $$\left( \varphi , \lambda \right) $$ φ , λ specifies a conformal mapping (i.e., the Normal Mercator projection) of the Earth ellipsoid surface into the Euclidean plane excluding the South and North poles, and the subsequent two steps allow the GK projection to be expressed analytically via the elliptic functions and integrals. Based on the three-step procedure, the conformality and singularities over the entire ellipsoid of the Normal Mercator, Thompson and GK projections were analyzed and the fundamental domains of them were determined. With respect to the precision and efficiency, it was verified that the new algorithm and the complex latitude method had equivalent precision levels for the same orders of the third flatting n with the Krüger- n series from $$n^2$$ n 2 to $$n^{12}$$ n 12 but slower about 0.19 to $${0.21}\,\upmu \hbox {s}$$ 0.21 μ s than the Krüger- n series for a GK coordinates calculating. However, the new formulas provide series approximations for the forward mapping of the Thompson projection and projective transformations for the Normal Mercator, Thompson and GK projections.

中文翻译：

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Lee 的 Gauss-Krüger 投影精确方法的发展

Lee 的精确方法旨在通过将中间映射（汤普森投影）扩展为等温坐标 $$\left(\psi,\lambda) 的级数近似，无需迭代程序即可实现高斯-克吕格 (GK) 投影\right) $$ ψ , λ 用于正向映射，GK 坐标 $$\left( x, y \right) $$ x , y 用于反向映射。GK 投影的前向和反向映射的新公式表示的简单程序由三个连续步骤组成，这些步骤基本上揭示了 GK 投影的映射程序和内在属性： 导出等温坐标 $$\left( \psi , \lambda \right) $$ ψ , λ 来自大地坐标 $$\left( \varphi , \lambda \right) $$ φ , λ 指定了一个共形映射（即，地球椭球面的法向墨卡托投影）到欧几里得平面（不包括南极和北极），随后的两个步骤允许通过椭圆函数和积分解析表示 GK 投影。基于三步法，分析了法线墨卡托、汤普森和GK投影在整个椭球上的共形和奇异性，并确定了它们的基本域。在精度和效率方面，验证了新算法和复纬度方法对于与 Krüger-n 系列从 $$n^2$$n 2 到第三次平展 n 的相同阶数具有相同的精度水平。 $$n^{12}$$ n 12 但比 Krüger-n 系列计算 GK 坐标慢 0.19 到 $${0.21}\,\upmu \hbox {s}$$ 0.21 μ s。然而，