非正轴墨卡托投影数学分析及适用性研究

As a common kind of conformal projection, with different location of projected surface relative to the earth ellipsoid, Mercator projection can be divided into normal and non-normal (transverse or oblique) Mercator projection. Normal Mercator projection is widely used as the mathematical foundation of nonpolar chart because of its mature theory, while applicability of non-normal Mercator projection is limited by its theory to be researched. Firstly, the transverse Mercator projection could not show entirely high-latitude area with its zoning and singularity in polar regions. Secondly, in traditional oblique Mercator projection, the earth is always approximatively viewed as a sphere, or needs suitable initial values with the earth’s flattening considered.In order to expand application of Mercator projection system，and improve shortages above in current Mercator projection theories, the project will conduct nonzonal transverse Mercator projection by means of complex functions, which could provide reference for polar scientific investigation and navigation. Combined with long-linear engineering, oblique ellipsoid will be realized by ellipsoid transformation methods, which could provide one alternative for high precision mapping or charting. On the basis of theoretical researches above, applicability study of non-normal Mercator projection will be constructed. Research contents in this project could enrich theory of Mercator projection, and develop its applicability, with theoretical innovation and broad application prospects.

墨卡托投影是一类常用的等角投影，因投影面与地球椭球相对位置的不同，可将其分为正轴和非正轴（横轴或斜轴）墨卡托投影。正轴墨卡托投影因理论研究的成熟性，已广泛用作非极区海图的数学基础，而非正轴墨卡托投影理论却仍待完善，致其适用性受限：主要表现在（1）横轴墨卡托投影在极区存在分带与奇异现象，难以完整地表示高纬度地区；（2）现有斜轴墨卡托投影公式多采用球近似的方法，而考虑地球扁率的算法需事先选定合适的初始值。为拓展墨卡托投影体系的适用性，针对目前墨卡托投影理论存在的不足进行改进，本项目将借助复变函数理论，实现极区无分带的横轴墨卡托投影，为极区科考及航海提供参考。结合长线型工程，借助椭球变换等方法实现斜轴墨卡托投影，为编制高精度地图或海图提供一种新思路。在以上理论研究的基础上，对非正轴墨卡托投影的适用性进行研究。本项目的研究内容可丰富墨卡托投影理论，拓展其适用性，具有理论创新与较好的应用前景。

项目以非正轴墨卡托投影为研究对象，深入开展了极区无分带横轴墨卡托投影数学分析、斜轴墨卡托投影数学分析、非正轴墨卡托投影适用性研究，构建了一系列严密的非正轴墨卡托投影理论算法模型，探讨了非正轴墨卡托投影在航海、地图表达中的应用，取得主要成果有：.(1)极区无分带高斯投影表达式消除了带宽的限制，解决了传统高斯投影公式难以在极区应用的问题，对于极区地图编制、航海导航具有参考价值；另，改进后的公式避免了极点奇异性，对于完善高斯投影的数学体系具有重要意义。.(2)根据航线所在位置，利用最小二乘法构建斜轴参考圆球，满足航线位斜轴参考圆球的 纬线圈附近；当航线沿非东西方向延伸时，与正轴墨卡托投影相比，斜轴墨卡托投影可有效减小航线附近海域投影长度变形；类似地，该投影在长线型海底管道、路上交通线路等非南北方向或非东西方向延伸的工程设计方面有较好应用前景，可用于重要航道或长线型工程高精度海图或地形图绘制。.(3)建立非正轴墨卡托投影与常用投影间的变换关系、研究常用曲线在非正轴墨卡托投影平面上的表示方法，进一步为海图转换、航线设计等实际应用提供支持。.