Spatial Analysis in curved spaces with Non-Euclidean Geometry

The ultimate goal of spatial information, both as part of technology and as science, is to answer questions and issues related to space, place, and location. Therefore, geometry is widely used for description, storage, and analysis. Undoubtedly, one of the most essential features of spatial information is geometric features, and one of the most obvious types of analysis is the geometric type and quantitative measures on such data. Most of the geometric analyzes and measurements used in different sections are based on Euclidean geometry. In other words, most of the known geometric analyzes are based on the assumption that only one line parallel to another line can be drawn from a point outside the line. Therefore, for example, the sum of the internal angles of a triangle should be 180 degrees, and regular tessellation in the plane is possible with only three types of regular polygons. In non-Euclidean geometries, the mentioned assumption and the results of following it are violated and no longer valid. The purpose of this research is to explain the need to use Non-Euclidean geometry. In this research, it is practically shown that location-based social networks or sensor networks can be addressed in the context of non-Euclidean geometry. This research also shows that the geometry governing the location-based social network is a hyperbolic geometry with negative curvature. This fact can be very effective to solve the problems such as routing and clustering. Moreover, the use of Non-Euclidean tessellations is a suitable tool for providing the user's current location service on the map in mobile and ubiquitous GIS.