摘要
By combining geographic information system (GIS), a new conception of “group” in which covers the complex connected region is introduced in the paper. Based on the conception of group, a generalized Euler formula and its properties are deduced and proved. The paper also describes the mathematical principles of generating topological information of polygon in GIS maps and the methods for checking up the veracity of topological relations of the map with Euler formula and general Euler formula. We have also obtained the quantitative relations among real nodes, chains, islands and groups with the formulas. At the same time, the paper introduces Whole Sphere Stereographic (WSS) projection into the GIS, and defines a new conception of “sea”. The deduction of general Euler formula and the introduction of WSS projection to GIS have developed new ways of delineating GIS topological models from plane to sphere and even constructing three dimensional (3-D) topological models.
By combining geographic information system (GIS), a new conception of 揼roup?in which covers the complex connected region is introduced in the paper. Based on the concep-tion of group, a generalized Euler formula and its properties are deduced and proved. The paper also describes the mathematical principles of generating topological information of polygon in GIS maps and the methods for checking up the veracity of topological relations of the map with Euler formula and general Euler formula. We have also obtained the quantitative relations among real nodes, chains, islands and groups with the formulas. At the same time, the paper introduces Whole Sphere Stereographic (WSS) projection into the GIS, and defines a new conception of 搒ea? The deduction of general Euler formula and the introduction of WSS projection to GIS have developed new ways of delineating GIS topological models from plane to sphere and even con-structing three dimensional (3-D) topological models.
