required. A valid TEN meets the criteria described in the definition of a TEN as a simplicial complex (as given earlier), including the positive orientation. Secondly, the volume feature boundary should be watertight (which is the case if the zero homomorphism holds). The last check is to see whether the interior of the volume feature is face-connected. An overall validation of topographic features in a simplicial complex-based structure requires therefore three steps: first the TEN has to be valid, second all individual features have to be valid and third and last the set of all features has to be consistent, meaning that there are no gaps or overlays between features. An editable data structure
A critical factor in the creation of data structure feasible for topographic data sets, is the possibility to perform incremental updates. Complete rebuilds of the TEN struc- ture will be time-consuming due to the expected data volume and should therefore be avoided. Furthermore, edits are usually related to local changes, so edit opera- tions are required that affect the TEN structure as locally as possible, thus trying to avoid ‘domino’ effects of alterations. Four steps can be distinguished in the feature insertion process: first a surface boundary triangulation of the feature is created. The resulting edges are then inserted as constrained edges into the TEN. Next one has to ensure the presence of constrained faces and as the fourth and last step the feature’s interior is modelled, including reclassifying the tetrahedrons involved. In order to demonstrate the usability of the simplicial complex-based approach for 3D topographic purposes, one need to guarantee successful update operations, since the frequent updates are a key characteristic of topography. Chapter 6 presented a new approach to the insertion of feature edges that guarantees successful insertion of these edges. Nine exhaustive and mutually exclusive cases were distinguished, depending on the type of intersection between the two nodes of a feature edge and the existing TEN structure. The following annotation is used: Iij, where i and j indicate the
dimension of the simplexes in which interior the two nodes of the constrained edge lie. As a result, the nine cases I01, I02, I03, I11, I12, I13, I22, I23 and I33 should be
interpret as (in case of I01) inserting an edge of which one node is already present
and the other lies on an existing edge, as (in case of I02) inserting an edge of which
one node is already present and the other lies on an existing triangle and so on. In each case the edge is inserted with a pure local impact on the TEN structure.
8.2
Main conclusions
The previous section summarised the main results of this research. Based on these results, the following conclusions can be drawn:
• This dissertation presents a new topological approach for 3D topography, based on a tetrahedral network. Operators and definitions from the field of simplicial homology are used to define and handle this structure of tetrahedrons. Ap- plying simplicial homology offers full control over orientation of simplexes and
140 Chapter 8. Conclusions
enables one to derive substantial parts of the TEN structure efficiently, instead of explicitly storing all primitives.
• Operators and definitions from simplicial homology provide a solid mathemati- cal foundation for the data structure.
• The simplicial complex-based approach and the vertex encoding including fea- ture identifiers eliminate the need for explicit storage of triangles, edges, nodes (in case of coordinate concatenation), constrained triangles, and constrained edges. As a result, it is a relatively compact data structure.
• The prevailing view that tetrahedrons, compared to polyhedrons, are more ex- pensive in terms of storage, proved to be outdated. Storage requirements for 3D objects in tetrahedronised form (excluding the space in between these objects) and 3D objects stored as polyhedrons, are in the same order of magnitude. Switching to a binary representation and storing coordinate differences instead of coordinates will lead to a significant storage reduction (about factor six) for the coordinate concatenation approach (including air and earth), thus justifying the full decomposition approach.
• A TEN has favourable characteristics from a computational point of view. All elements of the tetrahedral network consist of flat faces (important for clear inside/outside decisions), all elements are convex and they are well defined, thus allowing relatively easy implementation of operations, such as validation of 3D objects.
• DBMS characteristics as the usage of views, functions and function-based in- dexes are extensively used and contribute to the compactness and versatility of the data structure. Furthermore a database is capable of managing large data volumes, which is an essential characteristic in handling large scale 3D data. • A full volumetric approach contributes not only to improved analytical and val-
idation capabilities, but also enables future integration of topography and other 3D data within the same volume partition, like geological layers, polluted regions or air traffic and telecommunication corridors. Although the price of this ap- proach in terms of storage space is high (about 75% of the tetrahedrons models air or earth), this approach is likely to become justifiable due to current devel- opments like the trend towards sustainable urban development and increased focus on environmental issues. The impact of these factors on GI-developments is significant, as illustrated by for instance the environmentally-driven INSPIRE project (INSPIRE 2007).
• Constrained tetrahedronisation algorithm implementations are scarce and usu- ally not developed with GI or DBMS applications in mind. TetGen is used in this research and the TetGen website (2007) defines TetGen’s main goal as ‘to generate suitable meshes for solving partial differential equations by finite element or finite volume methods’. As a result these algorithms are usually not tested with large numbers of features and large numbers of incremental changes.