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How to achieve consistency for 3D city models

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Abstract

Consistency is a crucial prerequisite for a large number of relevant applications of 3D city models, which have become more and more important in GIS. Users need efficient and reliable consistency checking tools in order to be able to assess the suitability of spatial data for their applications. In this paper we provide the theoretical foundations for such tools by defining an axiomatic characterization of 3D city models. These axioms are effective and efficiently supported by recent spatial database management systems and methods of Computational Geometry or Computer Graphics. They are equivalent to the topological concept of the 3D city model presented in this paper, thereby guaranteeing the reliability of the method. Hence, each error is detected by the axioms, and each violation of the axioms is in fact an error. This property, which is proven formally, is not guaranteed by existing approaches. The efficiency of the method stems from its locality: in most cases, consistency checks can safely be restricted to single components, which are defined topologically. We show how a 3D city model can be decomposed into such components which are either topologically equivalent to a disk, a sphere, or a torus, enabling the modeling of the terrain, of buildings and other constructions, and of bridges and tunnels, which are handles from a mathematical point of view. This enables a modular design of the axioms by defining axioms for each topological component and for the aggregation of the components. Finally, a sound, consistent concept for aggregating features, i.e. semantical objects like buildings or rooms, to complex features is presented.

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Notes

  1. Geometrically, a surface embedded in \( {\mathbb{R}^3} \) is a continuous, differentiable mapping from \( {\mathbb{R}^2} \) to \( {\mathbb{R}^3} \).

  2. The notion of “closed” as used throughout this paper [1, 24] should not be mixed up with the other point set topological notion of “closed” meaning that the boundary points of a point set belong to that set [1, 2].

  3. The focus of the concepts in [20] was the tessellation property; hence these axioms differ slightly from the axioms presented in this paper. Particularly, the umbrella axiom (axiom 3 in Table 1) ensuring the 2-manifold-property was not discussed in [20]. The 2.8D maps in [20] are topologically equivalent to a disk. In this paper, a more general notion of 2.8D maps is presented, which may be generalized to allow handles. Orientability (axiom 14 in Table 1)—in combination with other axioms—is a simple method to check whether the outer boundary of the map is homeomorphic to the boundary of a disk. Axiom 11 (uniqueness of the unbounded face OUT) is a clarification, which is implied by axiom 9 in [20].

  4. Below-surface solids are unique and bounded partially by 2.8D maps; they differ from solids representing below earth surface features [10], which are bounded completely.

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Acknowledgements

We thank Gerrit Kowatsch for assistance in preparing the illustrations and Jan Prinz for proof-reading.

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  1. Institute of Geodesy and Geoinformation, University of Bonn, Meckenheimer Allee 172, 53115, Bonn, Germany

    Gerhard Gröger & Lutz Plümer

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Correspondence to Gerhard Gröger.

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Gröger, G., Plümer, L. How to achieve consistency for 3D city models. Geoinformatica 15, 137–165 (2011). https://doi.org/10.1007/s10707-009-0091-6

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