球面高斯投影及其变形的闭合公式

摘要

将地球视为球体时，高斯投影与横墨卡托投影是同一种投影，因此高斯投影公式可表示为形式紧凑的闭合形式。根据相关定义，以高斯投影的闭合公式为基础，通过一系列数学变换，推导出高斯投影反解公式、经纬线投影方程、投影的长度比及子午线收敛角公式的严格解析式，并借助算例验证了推导结果的可靠性与正确性。与以往根据高斯投影的幂级数公式推得的结果进行了对比分析，结果表明：所推导出的公式逻辑上更严密、形式上更紧凑，既丰富了高斯投影理论，又完善了地图投影数学基础。%With the earth viewed as a sphere,Gauss projection is identified with transverse Mercator projection,so the formula of Gauss projection can be expressed in a compact closed form.According to relative definitions and based on the closed formula of Gauss projection,a series of mathematic transforms can lead to strict analytic expressions of inverse solution,meridian and parallel lines, length ratio and meridian convergence.They are tested and verified by examples.The results demon-strate that these derivations are logically strict and formally compact,compared with the existing re-sults obtained from the power series expansion of Gauss projection.Achievements made are of great value to enriching theories of Gauss projection and perfecting the mathematical foundation of map pro-jection.