基于流形映射原理的地图投影概念分析及其应用  

Concept Analysis of Map Projection and Its Applications Based on Manifold Mapping Principle

作　　者：过家春 刘熠 申文斌[3] 施贵刚 GUO Jiachun;LIU Yi;SHEN Wenbin;SHI Guigang(College of Civil Engineering,Anhui Jianzhu University,Hefei 230601,China;Yanqi Lake Beijing Institute of Mathematical Sciences and Applications,Beijing 101408,China;School of Geodesy and Geomatics,Wuhan University,Wuhan 430079,China)

机构地区：[1]安徽建筑大学土木工程学院,安徽合肥230601 [2]北京雁栖湖应用数学研究院,北京101408 [3]武汉大学测绘学院,湖北武汉430079

出　　处：《武汉大学学报（信息科学版）》2024年第12期2313-2322,共10页Geomatics and Information Science of Wuhan University

基　　金：国家自然科学基金(42274053,42122025,42030105);安徽省教育厅青年骨干教师境外访学研修项目(JWFX2023023);安徽建筑大学人才引进项目(2023QDZ30);国家重大科技基础设施项目(2023-EL-PT-000197);安徽省高等学校省级质量工程项目(2021jyxm0266,2022zygzts018)。

摘　　要：从拓扑流形和高斯投影的历史渊源着手,基于流形映射原理阐述辨析了地图投影的概念原理。从黎曼流形的角度重新认识地球椭球面或球面,分析了椭球面或球面的非欧几何特征、与平面的拓扑关系(不可展性和不同胚性)及其对地图投影的影响。以地图投影的流形映射原理为基础,地图投影的基本矛盾(地球曲面与地图平面之间的矛盾)应该包括不可展和不同胚两个方面,其中不可展性是任何地图投影方式都不可避免地产生投影变形的数学原理所在,不同胚性则对投影函数的定义域和值域、奇异点特征产生影响。同时,流形映射原理在等角投影的定义、等角投影的充分必要条件分析等方面的应用研究进一步验证了相关论断的正确性和可行性。此项工作为从黎曼流形映射的角度研究地图投影学拓展了研究思路。Objectives:After a concise statement of the historical origin of topological manifold and Gauss-Krüger projection,this paper expoundes and analyzes the principle of manifold mapping of cartographic pro-jection.Methods:The Earth ellipsoid or sphere is redefined from the perspective of Riemannian manifold and the non-Euclidean geometric characteristics,and the topological relationship with plane and the influence on map projection are analyzed.Results:Based on the principle of manifold mapping,this paper considers that the basic contradiction of map projection(i.e.the contradiction between the earth surface and the map plane)should include two aspects,namely,un-developability and un-homeomorphism,which have im-pacts on distortions,domains and singular points of map projection,etc.Meanwhile,the correctness and feasibility of the authorsassertions about the principle of manifold mapping are further verified in the defini-tion and the necessary and sufficient conditions of conformal map,etc.Conclusions:This work expands the research idea for studying map projection from the perspective of Riemannian manifold mapping.

关 键 词：地图投影 黎曼流形 流形映射 地图投影基本矛盾 等角投影 

分 类 号：P282[天文地球—地图制图学与地理信息工程]