A data structure for 3D topography

A solid mathematical foundation

One of the key characteristics of the presented TEN data structure is that it is based on Poincar´e simplicial homology. The operators and definitions from this field, as presented in chapter 4, offer a solid mathematical foundation for the data structure. Simplexes are well defined, ordered and constructed of simplexes of lower dimension. The boundary operator can be used to derive these less dimensional simplexes. Based on the ordering of simplexes one can determine orientation. One might consider ori- entation as not a real mathematical characteristic of simplexes, since the concept of orientation is based on arbitrary conventions; nevertheless it enables useful operations within a GIS context, for instance in defining inside and outside. Within this context, simplicial homology offers some favourable characteristics. First, the zero homomor- phism results in consistent orientation of faces of a simplex. For instance, the normal vectors of the four boundary triangles of a tetrahedron will either all point inwards or all point outwards. Second, obtaining a simplex with opposite orientation requires only a transposition of its nodes.

A very important concept from simplicial homology is the simplicial complex, since such a set of connected simplexes will be used to model 3D topography. In terms of simplicial homology, a TEN can be defined as a simplicial complex of homogeneous dimension of three that consists of face-connected 3-simplexes. In addition to the requirements following from this definition (no self-intersections, no dangling edges, faces or tetrahedrons), a TEN is also supposed to consist of positively oriented 3- simplexes within the scope of this dissertation. In other words, the normal vectors of the boundary faces of each 3-simplex point outwards, which is common practice in computer graphics and GIS. Since topographic features will be represented as set of tetrahedrons, it is important that the boundary operator can also be applied to simplicial complexes, thus able to derive feature boundaries.

138 Chapter 8. Conclusions

A DBMS data structure

Applying definitions and operators from simplicial homology enables one to store a constrained TEN in a relatively compact way. The new simplicial complex-based method (as introduced in chapter 5) requires only explicit storage of tetrahedrons, while simplexes of lower dimensions (triangles, edges, nodes), constraints and topo- logical relationships can be derived in views. In this implementation, simplexes are encoded by their vertices. The boundary operator is used to derive views with the simplexes of lower dimension. In order to avoid explicit storage of constrained trian- gles and edges, simplex encoding is extended with a feature identifier. In other words, a tetrahedron is encoded as S3 =< v0, v1, v2, v3, f id >. The boundary operator is

implemented such that oriented triangles inherit the feature id from the tetrahedron they partially bound. Since the implementation uses triangles and opposite triangles (‘halftriangles’), neighbouring tetrahedrons are separated by a triangle and its op- posite. In case both tetrahedrons represent the same topographic feature, both the triangle and its opposite have the same inherited feature id. However, if the triangle is a constrained triangle, the tetrahedrons will represent different features and thus the triangle and its opposite will have different inherited feature identifiers. As a result, non-constrained triangles cancel out in pairs (since the triangle and its opposite will have identical triangle codes apart from the sign), while constrained triangles remain present. Topological relationships can also be derived, both at feature and at TEN level, for instance the neighbour relationship between tetrahedrons. This is achieved by creating a view with triangle opposites and defining the triangle view such that the coboundary is known. Neighbouring tetrahedrons can be identified by deriving the boundary triangles and querying their opposites first and their coboundaries af- terwards, thus resulting in the four neighbouring tetrahedrons.

Two variants in simplex encoding have been developed. The coordinate concate- nation is the most rigorous attempt to eliminate redundancy. By concatenating the x, y and z coordinate as node identifier, the node geometry in itself becomes an unique identifier. Simplexes of higher dimension are described by a concatenation of node codes. The alternative approach, identifier concatenation, requires an additional node table. Separate (meaningless) node identifiers are used to encode simplexes. This approach reduces the number of coordinate repetitions, since the coordinates of a specific node will be repeated in multiple tetrahedrons. Despite the substantially reduced storage requirements of identifier concatenation, this is not automatically the best solution. Applying coordinate concatenation yields the potential to sort bitwise- interleaved tetrahedron codes, such that this ordering acts as a spatial location code, thus eliminating the need for additional spatial index creation. Not only would this affect the overall storage requirements, but it will also affect performance. A clean comparison, based on multiple realistic data sets, is needed to determine the most suitable concatenation approach.

Validation can be performed at TEN level as well as at feature level. To validate a volume feature, three checks need to be performed. First of all a valid TEN is