极区不分带高斯投影的正反解表达式

Forward and Inverse Expressions of Polar Gauss Projection without Zoning Limitations

针对传统高斯投影公式在极区难以应用的问题,通过引入等角余纬度及等量纬度的表达式,推导出严密的复数等角余纬度公式,进而得到严密的极区高斯投影正解表达式;借助符号迭代法及指数函数与三角函数间的关系式,推导出对应的极区高斯投影反解表达式;基于极区高斯投影正解表达式,推导出可用于极区的长度比、子午线收敛角公式;最后,以CGCS2000椭球为例,与实数型幂级数高斯投影公式计算的结果进行对比,验证了本文推导公式的正确性。由于本文推导公式不受带宽限制,且可用于整个极区的表示,对于编制极区地图及极区导航具有重要的参考价值。

As traditional formulae of Gauss projection could not be used in polar regions, strict equation of complex conformal colatitude was derived with relationship between conformal colatitude and isometric latitude introduced, and then strict forward expressions of Gauss projection suit for polar regions were carried out.Based on relationship between exponential and trigonometric functions, inverse expressions of polar Gauss projection were derived by means of symbol iteration method.With reference to the forward expressions, corresponding equations of length ratio and meridian convergence for polar Gauss projection were achieved.Finally, Taking CGCS2000 ellipsoid for example, by comparing with results calculated by formulae of Gauss projection in power series forms, correctness of the proposed expressions was verified.Expressions in this paper are all free from bandwidth, and can be used in the entire poles, which could provide important references for polar mapping and navigation.