How to Get 3-D for the Price of 2-D—Topology and Consistency of 3-D Urban GIS

This article deals with topological concepts and models which are necessary to represent three-dimensional urban objects in a geographical information system (GIS). Depending on the shape and the representation of features, several classes with increasing topological complexity are identified and described. This complexity has strong impacts on the models and tools which are required to represent, manage and edit the data. One specific model we call ‘2.8-D map’ is identified, which covers many 3-D applications in GIS. It is a slight extension of a 2-D or 2.5-D model and preserves the algorithmic and conceptual simplicity of the 2-D case as much as possible. The model is described in a formal way. Integrity axioms are given, which detect errors in corresponding data sets safely and guarantee the consistency of 2.8-D maps in a mathematically sound and provable way. These axioms are effectively and efficiently checkable by automatic procedures. The model extends digital terrain models (2.5-D) by allowing for vertical walls and projections like balconies or ledges. The conceptual simplicity is due to the two-dimensional topology of the model. Thus bridges and tunnels are special cases; it is shown how to detect and handle these cases efficiently. Based on this model, thematic objects and their aggregation structures are defined in a consistent way.