椭球几何学与任意高斯投影数学分析

Ellipsoid geometry is the foundation of geosciences, especially of geodesy. As the basic projection of topographic maps, GIS and various kinds of engineering projects, Gauss projection is widely used. However, the south-north direction and zoning greatly limit its application. These two types of problems are not solved perfectly due to the limitations of the artificial deduction in the past. Computer algebra provides strong support for further improvement and development of these two types of issues. This project intends to use the computer algebra with strong symbolic computing power to innovate the theory of ellipsoid geometry and Gauss projection mathematical analysis, to establish a new algorithm for calculating the earth's theme by means of spatial analytic geometry and differential geometry theory, to make the ellipsoid more suitable for regional control network by ellipsoidal transformation method, to build the expressions of arbitrary Gauss projection by complex function, which can be used for any long-line engineering with normal section line and large ellipse line studied. What’s more, by combining with engineering construction, such as China's high-speed railway construction, the deformation control program for Gaussian projection with different directions will be studied. Expected research results of the project will innovate and enrich the traditional theory of both geometric geodesy and Gauss projection, derive a number of simpler in form, theoretically more rigorous and accurate symbolic expressions, which has broad application prospect.

椭球几何学是地球科学尤其是大地测量学的基础。高斯投影是地形图、GIS和各类工程建设的基础投影方式，应用极其广泛，但正南北走向、分带极大地限制了其应用场合。这两类问题，由于受过去人工推导的限制，解决的绝非完美。计算机代数为这两类问题的进一步完善和发展，提供了强大的支持。本项目拟以具有强大符号运算能力的计算机代数为工具，革新椭球几何学和高斯投影数学分析理论；广泛利用空间解析几何和微分几何理论，建立大地主题解算新算法；通过椭球的平移、膨胀、变形，使椭球更适用于地区控制网；通过研究法截线、大椭圆线，建立非南北走向的任意高斯投影复变函数表示，使之更适用于任意走向长线型工程；结合工程建设，如我国高速铁路建设，研究任意方向高斯投影的变形控制方案。项目预期研究成果将革新和丰富传统的几何大地测量和高斯投影理论，导出一些理论上更严密，形式上更简单，精度上更精确的符号化表达式，具有广泛的应用前景。

本项目引入计算机代数分析方法，借助计算机代数系统的强大数学分析和符号推导能力研究椭球几何及其投影理论，提高了推导效率，消除了近似处理，得到形式更为简单、理论上更为严密、精度上更为精确的算法。革新了椭球几何学和高斯投影数学分析理论，并对新椭球上的几何实体元素建立算法模型和进行性质分析，同时也借助计算机代数系统实现椭球实体要素的可视化；广泛利用空间解析几何和微分几何理论，充分发掘微分方程间的规律性和内在的本质联系，建立大地主题解算新算法，同时也讨论了特殊情况下的正反解的计算问题。通过椭球变换得到新的椭球并进行任意方向高斯投影，可以较好的解决长线工程建设中的南北走向、分带带来的投影变形问题，深入的研究了法截线椭球高斯投影、大椭圆线椭球高斯投影，产出了一系列成果，丰富了斜轴高斯投影理论，同时建立非南北走向的任意高斯投影复变函数表示，使之更适用于任意走向长线型工程；结合具体的工程项目，利用椭球变换理论、独立坐标系和抵偿高程面等对任意高斯投影变换算法进行变形分析和控制，同时也对分带高斯投影间的转换关系进行了正确性检验，并编制了相应的计算机程序。通过实际的应用证明本项目的研究成果应用到具体工程建设中，可以为整个建设过程提供精密的坐标框架，提升测绘任务和工程建设的质量。