斜轴高斯投影计算机代数分析及其在长线工程中的应用理论研究

Gauss projection, as one kind of map projection, has found its wide application in geodesy, cartography and engineering surveying. Conventionally speaking, Gauss projection is very applicable to a region in the south to north directions. When Gauss projection is applied to a region in the east to west directions, it has to be divided into different projection zones. In this case, different coordinates transformations bring a lot of unconvenience and may introduce artificcel errors. In order to overcome these shortages and improve the practical effiency of Gauss projection, the oblique Gauss projection is proposed in the project. This will make projection zone fit the region to be projeted well and effectively reduce the distortoon errors. The project will make full use of computer algebra systems. Based on the analytical geometry, the oblique Gauss projection theory will be systematically investigated for the transformation, drift and deformation of ellipsoids. The main contents include three relatively independent parts. First, the oblique Gauss projection algorithm based on the normal section of ellipsoids is introduced and researched in detail. And then, the oblique Gauss projection algorithm based on the great ellipse of ellipsoids is introduced and investigated. The third part is the Gauss projection theory and algorithm based on the deformation of ellipsoids. Finally, the applications of the oblique Gauss projection in high-speed railway construction are discussed. The research achievements can be applied to the accurate surveying in high-speed railway, highway, oil pipeline, drainage area plan and economical zone constructions.

高斯投影是在大地测量学、地图学、工程测量学等领域得到极其广泛应用的一种地图投影。传统上高斯投影为限制东西方向变形需要分带，应用于东西跨度较大的长线工程时，频繁分带繁琐，信息要素表达零散，而且拼接又极易产生人为误差。为克服分带缺点，提高高斯投影的使用效率，有效降低高斯投影的长度变形，项目提出使用斜轴高斯投影。本项目拟以计算机代数为工具，运用空间解析几何理论，通过椭球的旋转、平移、收缩，系统地研究斜轴高斯投影理论，并建立一整套系统的斜轴高斯投影工程应用算法体系。主要内容包括分别以长线工程法截线和大椭圆线为基础的斜轴高斯投影理论和算法研究、以椭球变形为基础的斜轴高斯投影理论和算法研究、结合典型长线工程的斜轴高斯投影变形分析。项目的研究成果可应用于对变形具有较高要求的高铁线路、高速公路、输油管线和流域规划等长线工程的施工建设，也可广泛应用于我国自然水系、区域整治和经济带精确表述。

为提高高速铁路、客运专线、大型河流等长线工程平面控制测量精度和克服传统高斯投影频繁换带问题，本项目以计算机代数技术为基本的分析工具，深入研究坐标系统设计和投影变形，建立了法截面子午线椭球高斯投影、大椭圆线高斯投影以及椭球变换及其对长度变形等理论和算法体系，取得的研究成果主要有：（1）提出了一种新的建立工程独立坐标系统的方法——法截面子午线椭球高斯投影方法，系统分析了法截面子午线椭球高斯投影的性质和法截面子午线椭球扩展变换理论；（2）基于最小二乘估计、非线性规划最优理论等相关数学方法，提出了一种处理线路测区的控制点数据的方法——大椭圆线椭球高斯投影方法，系统建立了其理论体系并进行了实例验证；（3）系统研究了三种椭球变换方法的理论与变形规律，并将三种方法统一为一个广义的椭球面族；（4）进行了法截面子午线椭球工程应用研究，包括法截面子午线椭球的概要设计、数值计算，并对工程实例进行了验证；（5）深入研究了高斯投影复变函数表示的实数解理论、等距离球面高斯投影理论、极区不分带高斯投影理论以及辅助纬度与大地纬度正反解理论，扩展了高斯投影基础理论及其适用范围。项目相关成果可应用于高速铁路与客运专线建设、大型流域规划、特高压输电线与输油输气管线规划以及航线规划等领域。