3.2.3
Disadvantages of the TEN approach
The drawback of TENs, most often mentioned in literature, is the required amount of storage space. Amongst others, Zlatanova et al. (2004) state that TENs have a much larger database size in comparison with other 3D approaches. This line of reasoning is based on a comparison for a single building as shown in table 3.1, in which the number of tetrahedrons, triangles, edges and nodes is compared to the number of volumes, faces, edges and points in a polyhedron approach.
Building as polyhedron Building as explicit TEN
(1 volume) 8 tetrahedrons
7 faces 24 triangles
(15 edges) 25 edges
10 points 10 nodes
Table 3.1: Comparing storage requirements of a polyhedron and a TEN approach for the building in figure 4.7. The brackets indicate implicit presence (as opposite of explicit storage). In an explicit TEN, all elements are stored explicitly, although more elements might be modelled implicitly in an actual TEN implementation.
In order to reach acceptable performance, it has to be decided which primitives and which relationships between TEN elements will be stored explicitly. The perfor- mance requirements do not tolerate full storage of all possible relationships. Several approaches exist in 2D to reduce storage requirements of TINs by either working with an edge or a triangle based approach, in which not both triangles, edges and nodes are stored explicitly. Unfortunately, in the 3D situation and especially in case of a constrained TEN this might become very difficult. Nevertheless this drawback should be tackled at design level.
In addition to the (supposed) drawback of TEN storage requirements, one should realise that the described TEN approach includes a full tetrahedronisation of 3D space, thus adding air and earth features to the model. The addition of these features will influence storage requirements significantly.
Although both factors relate to storage requirements, a clear distinction has to be made between increasing storage requirements, triggered by the selection of a triangular data structure and increasing storage requirements, triggered by the full volumetric modelling approach. Especially the increases caused by the triangular data structure might be limited by a proper implementation of a TEN structure in a database. The upcoming chapters will present such an implementation.
3.3
The choice for the full 3D approach
This chapter first introduced a very pragmatic approach to 3D modelling, as it aims at modelling as much as possible in (less complicated) 2.5D, whereas full 3D mod- elling will be applied only in exceptional cases. Triangulations were selected (in the
Part II
A Data structure
for 3D Topography
Chapter 4
Theoretical foundations:
Poincar´e simplicial homology∗
The previous chapter answered the question how to develop a conceptual model de- scribing topographic features: a full 3D approach based on tetrahedrons is the most appropriate. As shown in the chapter 3, this network of tetrahedrons consists of tetra- hedrons, triangles, edges and nodes. These simplexes are constructed recursively: a tetrahedron is defined by four triangles, a triangle by three edges and an edge by two nodes. Intuitively, these relationships offer computational advantages in operations since they are known in advance.
Figure 4.1: Henri Poincar´e, founder of simplicial homology research
∗This chapter is partially based on Penninga and van Oosterom (2008a), A Simplicial Complex-
based DBMS Approach To 3D Topographic Data Modelling. Accepted for International Journal of Geographical Information Science: August 18, 2007. Publication scheduled for 2008.
58 Chapter 4. Theoretical foundations: Poincar´e simplicial homology
Although these favourable characteristics are recognised by others (Guibas and Stolfi 1985, Pilouk 1996, Wei et al. 1998), they are rarely explored in a structured fashion, except for Frank and Kuhn (1986), Carlson (1987), Egenhofer et al. (1989) and Paoluzzi et al. (1993) (see section 2.4.3). Nevertheless, in this research the solid mathematical foundations of simplexes will be applied. Simplexes, their characteris- tics and their mutual relationships are extensively studied in the late 19th century by Henri Poincar´e (see figure 4.1). His results still form the basis of the field of simplicial homology.
Applying simplicial homology offers significant advantages. Before introducing the simplicial homology-based data structure in the next chapter, this chapter will provide all relevant backgrounds of simplicial homology. Simplicial homology offers a mathematical description of simplexes and of their mutual relationships (section 4.1). A very important characteristic in handling geometries is orientation, as it is the basis for determining crucial relationships as left/right and inside/outside. The new data structure greatly benefits from the fact that simplicial homology enables full control over simplex orientation in each dimension. The underlying theory will be introduced in section 4.2. Another important aspect of the new approach is the representation of topographic features. These features are composed of several tetra- hedrons. In mathematical terms these combined simplexes are known as simplicial complexes (section 4.3). The last section (section 4.4) will show how the use of tetra- hedrons as basic building blocks for features will simplify operations on these features. In this chapter the following annotations will be used: Sn for a simplex of dimension
n, ∂ for the boundary and vi for a node. In mathematical terms vertex is the appro-
priate term, but in a GIS context node is more often used (especially in TINs and TENs). As a compromise this chapter will use the term node and the annotation vi.
4.1
Mathematical description of simplexes
Simplexes and the relationships between simplexes of different dimensions were stud- ied by mathematicians in the late 19th and early 20th century. This field of math- ematics was known as simplicial homology and is today considered part of the field of Algebraic Topology (Hatcher 2002). The foundations of simplicial homology are described by Jules Henri Poincar´e (1854-1912) in (Poincar´e 1895). Some relevant corrections and additions can be found in (Poincar´e 1899). Simplicial homology is the part of mathematics that deals with topological constructions of simplexes. Intu- itively a n-simplex can be described as the simplest geometry of dimension n, where simplest refers to minimising the number of points required to define such a simplex. For instance, one needs at least three points to define a 2D shape (a triangle) and these three points should not lie on the same line (since that would result in a 1D edge). A simplex can be seen as an elementary building block of its dimension; they are used to construct simplexes of higher dimension.
The previously introduced volumetric approach uses tetrahedrons to model the real world. These tetrahedrons in the TEN structure consist of nodes, edges and
4.1. Mathematical description of simplexes 59
triangles. All four data types are simplexes. A formal definition (Hatcher 2002) of a n-simplex Sn is given below:
Definition 1 A n-simplex Sn is the smallest convex set in Euclidian space (denoted
IRm) with n ≤ m, containing n + 1 points v
0, . . . , vn that do not lie in a hyperplane
of dimension less than n. As the n-dimensional simplex is defined by n + 1 nodes, it has the following notation: Sn=< v0, . . . , vn >.
Equivalent conditions to the hyperplane condition would be that the difference vectors v1− v0, . . . , vn− v0 are linearly independent or, if one considers v0, . . . , vn as set of
vectors, that these vectors are affinely independent. Figure 4.2 shows simplexes with dimension n = 0 . . . 3: respectively a node, an edge, a triangle and a tetrahedron.
Figure 4.2: Simplexes with dimension n = 0 . . . 3: node, edge, triangle, tetrahedron and their ‘orientation’ by (GIS) convention
Some observations on simplexes:
1. It is assumed that all simplexes are ordered. With any n-simplex, (n + 1)! distinct ordered simplexes are associated. Within GIScience, this ordering can be used to orient these simplexes according to specific conventions. Orienta- tion is 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 of geometries. As a result, one should interpret the term ‘orientation’ within the scope of this dissertation according to its common meaning within GIScience. Within this framework, one can say that all even numbers of permutations of an arbitrary ordered simplex Sn =< v0, . . . , vn >have similar orientation and
all odd numbers of permutations an opposite one. So for instance the following four statements are true:
54 Chapter 3. Two triangular data models for 3D topography
previous chapter) as data structure due to their strong computational capabilities. The triangulation and tetrahedronisations can be integrated at a conceptual level, as both TIN and TEN use nodes, edges and triangles. However, the actual connection at design and implementation level appeared to be very difficult. Another consequence of selecting a hybrid 2.5D/3D approach is inconsistent feature modelling: the viaduct example illustrated that a highway might be modelled sometimes as a volume and sometimes as a face. This inconsistency is hard to accept.
To overcome these drawbacks, a more rigid approach was designed. The full vol- umetric approach models everything in a TEN, including air and earth features, thus explicitly including space that is usually perceived as ‘empty’. Despite its potential substantial data storage requirements, the full 3D approach is selected as starting point for further research. The next part of this thesis (part II) will focus on a data structure, based on this conceptual model. As data storage is often considered to be a weakness of a TEN data structure (Zlatanova et al. 2004), this part will also focus on methods to reduce explicit data storage.
60 Chapter 4. Theoretical foundations: Poincar´e simplicial homology S0= < v0> S1= < v0, v1>= − < v1, v0> S2= < v0, v1, v2>= − < v0, v2, v1>=< v1, v2, v0>= − < v1, v0, v2>=< v2, v0, v1>= − < v2, v1, v0> S3= < v0, v1, v2, v3>= − < v0, v1, v3, v2>=< v0, v3, v1, v2>= − < v0, v3, v2, v1>=< v0, v2, v3, v1>= − < v0, v2, v1, v3>= < v2, v0, v1, v3>= − < v2, v0, v3, v1>=< v2, v3, v0, v1>= − < v2, v3, v1, v0>=< v2, v1, v3, v0>= − < v2, v1, v0, v3>= < v1, v2, v0, v3>= − < v1, v2, v3, v0>=< v1, v3, v2, v0>= − < v1, v3, v0, v2>=< v1, v0, v3, v2>= − < v1, v0, v2, v3>= < v3, v0, v2, v1>= − < v3, v0, v1, v2>=< v3, v1, v0, v2>= − < v3, v1, v2, v0>=< v3, v2, v1, v0>= − < v3, v2, v0, v1>
The second line can be read as ‘an edge directed from point v0 to point v1has
opposite orientation to the edge directed from point v1to point v0. This charac-
teristic can be used to change simplex orientation by performing a transposition (a single permutation), thus eliminating the need of using signed simplex de- scriptions. Section 4.2 will describe simplex orientation in more detail.
2. A face of Sn is a simplex of lower dimension whose nodes form a non-empty
subset of {v0, . . . , vn}. In other words, a simplex consists of simplexes of lower
dimension and these simplexes are defined by some of the points that define the original simplex. For instance, a tetrahedron S3 =< v0, v1, v2, v3 >consists of
four triangles < v1, v2, v3 >, < v0, v2, v3 >, < v0, v1, v3 > and < v0, v1, v2 >.
The formula to derive these less dimensional boundaries will be given in defini- tion 2 below.
3. If the subset is proper (i.e. not the whole of {v0, . . . , vn}), than the face is called
a proper face (Giblin 1977).
4. A n-simplex has in total 2(n+1)− 2 proper faces. For instance a triangle has six
proper faces (three edges and three nodes), while a tetrahedron has 14 proper faces (four triangles, six edges and four nodes).
5. For the number of faces of a specific dimension the following holds: a n-simplex has
n+ 1 p+ 1
faces of dimension p with (0 ≤ p < n). For instance: a tetrahe- dron consists of four triangles, six edges and four points.
6. The 0- and 1-dimensional faces (i.e. nodes and edges) of a n-simplex form a complete graph on n + 1 nodes.
7. The boundary of a n-simplex is defined by the following sum of n−1 dimensional simplexes (Poincar´e 1899) (the hat indicates omitting the specific node):
4.2. Orientation of simplexes 61