Provably correct and complete transaction rules for updating 3D city models

Abstract

The shapes of our cities change very frequently. These changes have to be reflected in data sets representing urban objects. However, it must be assured that frequent updates do not affect geometric-topological consistency. This important aspect of spatial data quality guarantees essential assumptions on which users and applications of 3D city models rely: viz. that objects do not intersect, overlap or penetrate mutually, or completely cover one another. This raises the question how to guarantee that geometric-topological consistency is preserved when data sets are updated. Hence, there is a certain risk that plans and decisions which are based on these data sets are erroneous and that the tremendous efforts spent for data acquisition and updates become vain. In this paper, we solve this problem by presenting efficient transaction rules for updating 3D city models. These rules guarantee that geometric-topological consistency is preserved (Safety) and allow for the generation of arbitrary consistent 3D city models (Completeness). Safety as well as completeness is proven with mathematical rigor, guaranteeing the reliability of our method. Our method is applicable to 3D city models, which define—besides the terrain surface—complex spatial objects like buildings with rooms and storeys as interior structures, as well as bridges and tunnels. Those objects are represented as aggregations of solids, and their surfaces are complex from a topology point of view. 3D GIS models like CityGML, which are widely used to represent cities, provide the means to define semantics, geometry and topology, but do not address the problem of maintaining consistency. Hence, our approach complements CityGML.

Appendix A: Basic notions

In this paper we use standard notions from graph theory [19] to mathematical topology [1, 2, 20, 28]. In the following, we recapitulate the notions from mathematical topology which are necessary for defining the concepts presented in the paper.

1.1 A.1 Topology: 2-manifolds, boundaries, and genus

A 2-manifold surface, which plays a decisive role in geometrical and topological modelling [7, 21, 28–30], is a topological space where each point has a neighbourhood which is topologically equivalent to an open two-dimensional disk [20]. Examples for 2-manifolds are the open disk, the open cylinder surface, the hull of a cuboid, or the sphere. Figure 18 depicts the open disk (a) and the hull of a saddle roof building (b) as examples for 2-manifolds, as well as three counterexamples: three rectangles meeting in a common edge (c), the hull of two cuboids meeting in a common edge (d), and the hull of two saddle roof buildings, where the roof of one buildings penetrates the wall of the other (e).

Fig. 18

Examples for 2-manifold surfaces (a, b) and non 2-manifold surfaces (c–e)

Another property of surfaces, which is also a topological invariant, is the number of boundaries. An open disk has one boundary, whereas an open cylinder surface has two and a sphere has zero boundaries (c.f. Fig. 19). Surfaces without boundaries are closed. ⁵ They enclose a volume completely and hence are used in geometrical modelling to represent the boundary of solids [28, 30]. Some examples for a closed surface are the sphere, the hull of a cuboid, or the hull of a saddle roof building (c.f. Figs. 18b and 19b).

Fig. 19

Surfaces with different numbers of boundaries: a disk, one boundary, b sphere: zero boundaries, c cylinder surface: 2 boundaries

The number of handles is another topological invariant of surfaces. Handles in surfaces are used to model bridges, tunnels, arcades or similar phenomena. The number of handles of a surface is given by its genus. ⁶ The genus is defined as the maximal number of closed, continuous, non self intersecting and mutually non-intersecting curves, the cutting of which preserves the connectivity of the surface. A sphere, for example, has genus zero and hence no handles, while the genus of a torus is one. It has one handle (see Fig. 20).

Fig. 20

The sphere a has no handle (genus zero), the torus b has one handle (genus one). Each cycle on the surface in (a) separates the surface, while the cycle on the surface in (b) does not. Each additional, non-intersecting cycle separates the surface

1.2 A.2 Cell complexes

The theory of cell complexes [20], a branch of Algebraic Topology, defines concepts for aggregating primitive objects to more complex ones in a topologically clean manner. Hence, cell complexes are the base of many topological data models in GIS, CAD and Computer Graphics [7, 21, 29, 33]. Primitives are nodes, edges, faces and solids, which are also called 0-cells, 1-cells, 2-cells and 3-cells. Each n-cell is bounded by (n-1)-cells, which are called the boundary of the n-cell. A cell complex is an aggregation of cells where the following condition holds true: The intersection of two cells in the cell complex is either empty or is a cell which is part of the boundaries of both cells. Thus, cell complexes avoid overlapping cells as well as penetrations of cells, and touching of cells is defined explicitly by commonly shared boundaries. To exemplify the concept of cell complexes, Fig. 21 depicts a cell complex consisting of two 2-cells (a) and one consisting of two 3-cells (b). The touching of both cells is represented explicitly by a 1-cell and two 0-cells in a) and by a 2-cell (depicted in dark colour), four 1-cells and four 0-cells (each depicted in light colour) in b).

Fig. 21

Cell complexes consisting of two 2-cells (a) and of two 3-cells (b)