Neighborhood relationship plays an important role in spatial analysis, map generalization, co-location data mining and other applications. From the perspective of computation, the formal model of neighborhood representation is a challenging question. This study presents a formal spatial data model for representing the planar spatial field with the support of Delaunay triangulation. Based on the three geometric elements in a triangle of the vertex, edge, and triangle area, the constructed data model describes the spatial objects of a point, line, and region respectively, as well as the neighborhood relationships among them. Three types of operators based on the model are formally defined, expanding, compressing and skeletonizing. For practical applications, three complex operators are extended by continuous and conditional operation. Through the application example of urban building generalization, this study illustrates the analysis of a neighborhood relationship and the detection of spatial conflicts, which is a crucial pre-process during map generalization. With the support of the proposed formal model of neighborhood representation, the generalization method uses the three basic operations of grouping, displacement and aggregation to perform decision making and detailed operation. The generalized result can retain the balance of built-up area better than that of other similar building generalization methods.