以北极地区为中心的地图和海图投影数学分析研究

Along with the increase of the Arctic strategic position and the Arctic interests expansion of our country, it is absolutely necessary to compile Arctic-centered maps and charts. Traditional algorithms of map and chart projections have mainly problems used in the Arctic: (1)Some algorithms are derived based on the sphere, and their accuracies are not high because of the approximate disposal; (2)Some algorithms are in numerical form and not convenient for the promotion; (3) The rigorous analytical formulas are not available for the direct transformations between different projection coordinates, which makes the transformations tedious; (4)Some algorithms are singular and inconsistent because of the zonation. Based on the ellipsoid model, the forward and inverse solutions of Arctic-centered map projections with their distortion analysis, the direct transformations of Arctic-centered map projections, Arctic-centered chart projections and transformations, and mathematical analysis of generally used courses with their plotting methods are deeply researched with the help of computer algebra system which has powerful function of symbolical operation in this project. The new algorithms and mathematical models in symbolic form are established, which have stricter theory basis, more concise structure and higher accuracy compared to traditional ones. The research will further enrich the mathematical basis of modern map and chart projections. The research results could be used to compile Arctic-centered maps and charts, which could more accurately describe the whole Arctic, show the geopolitical relations between China and the Arctic with its neighboring countries, and ensure safety sailing in the Arctic Ocean.

随着北极战略地位的日益提升和我国北极利益的拓展，迫切需要编制以北极地区为中心的地图和海图。传统地图和海图投影算法在北极地区应用时存在的问题主要有：（1）基于球体模型导出，存在近似，精度不高；（2）以具体的数值形式给出，不便于推广应用；（3）未能建立投影坐标直接变换的严密解析式，计算繁琐；（4）存在奇异问题或分带现象，表达式不统一。本项目以地球椭球模型为基础，借助具有强大符号运算功能的计算机代数系统，深入研究以北极地区为中心的地图投影正反解及变形分析、地图投影直接变换、海图投影及其变换、常用航线的数学分析和绘算方法，推导和建立理论更为严密、结构更为简单、精度更为精确、符号化的新算法和新模型，进一步丰富当代地图和海图投影的数理基础。研究成果可用于以北极地区为中心的地图和海图编制，从而更准确地描绘出一个整体的北极，展现出中国与北极地区及其周边国家的地缘关系，为极区安全航行提供保障。

针对新形势下北极战略地位的提升和海图保障的迫切需求，本项目以地球椭球模型为基础，以各类投影数学模型为研究对象，深入开展了以北极地区为中心的地图和海图投影数学分析研究，采用计算机代数分析方法，借助计算机代数系统强大的数学分析功能，推导和建立了一系列理论上更为严密、结构上更为简单、精度上更为精确的符号形式的极区地图海图投影新算法和新模型，取得的主要研究成果有：.（1）构建了极区地图海图投影椭球偏心率幂级数函数模型，推导出了椭球常用纬度正反解和差异极值的符号表达式，显著提升了极区地图海图投影数学模型的严密性，提高了计算公式的精度和效率。.（2）通过对多种海图投影极区可用性的系统分析，给出了墨卡托投影75°以上纬区拓展应用的范围和极区航海图投影选择的理论依据，提出了合理的极区航海图投影选取方案。.（3）构建了极区球体横墨卡托投影和椭球高斯投影复变函数表示非奇异模型，给出了正轴与斜轴地理坐标正反解复数换算公式，推导出了极区斜轴等角投影及其变换的复数公式，为极区地图海图编制提供了理论基础。.（4）建立了极区地图海图投影转换的直接解模型，避免了过去间接变换导致的误差，显著地提高了投影变换的计算精度和计算效率。.（5）对常用航线在极区海图投影平面上的表象进行了系统的分析和研究，推导出了它们在极区海图投影平面上的方程、曲率及代曲直距公式，为极区航海导航提供了参考。.本项目实现了以北极地区为中心的地图和海图投影数学分析特定领域的一些突破和创新，进一步革新和完善了地图和海图投影的理论体系，丰富了地图和海图投影的数理基础。研究成果可广泛应用于地图制图、地理信息系统、海洋测绘、航海导航等领域。