Impact of data representation rules on the robustness of topological relation evaluation

Abstract

A spatial object is characterized not only by its geometric extents, but also by the spatial relations existing with its surrounding objects. An important kind of spatial relations is represented by topological relations. Many models have been defined in literature for formalizing the semantics of topological relations between spatial objects in the Euclidean 2D and 3D space [3, 4, 7]. Nevertheless, when these relations are evaluated in available systems many robustness problems can arise, which are essentially related to the discrete representations adopted by such systems. In a Spatial Data Infrastructure (SDI) the perturbations introduced by the exchange of data between different systems can increase the robustness problems. This paper deals with a set of rules for the representation of spatial datasets which allow to evaluate topological relations in a robust way using existing systems. These rules are well-known and described in literature and are based on a few basic assumptions on the system behavior which are fulfilled by today’s systems. The main contribution of this paper is to determine in detail which rules are sufficient in order to make each topological relation robust; it turns out that the rules depend not only on the topological relation being considered, but also on the geometric types of the involved geometries and on the dimension of the space in which they are embedded, thus giving rise to a very large number of possible combinations. The paper analyses the topological relations and a significant subset of the geometric types defined in the most recent version of the Simple Feature Access (SFA) model published by OGC, considering both a 2D and a 3D space. The extension of the work to the types which have been left out can be done using the same concepts and methodology.

Appendices

Appendix A: Derived vector predicates and operations

This section presents the semantics of derived vector predicates and derived operations introduced in Section 2.1.

Table 11 Expressions for derived operations and predicates regarding vertices. In the table symbols v, $v_{1}, \dots , v_{n}$ denote a vertex, while V, V ₁, V ₂ are sets of vertices.

Table 12 Expressions for derived operations and predicates regarding segments. In the table symbols s, $s_{0}, s_{1}, \dots , s_{n}$ denote a segment, while S is a set of segments.

Notice that the operation s.diff(S) produces a result only if the set S contains some segments that overlap s, otherwise the result is always the empty geometry, since the second condition is not satisfied.

Fig. 5

Examples of possible cases that make true the predicate p.cnt(s). Notice that segments s _A and s _B do not satisfy the predicate p.cnt _int (), while all the other segments do.

Table 13 Expressions for derived operations and predicates regarding patches. In the table symbols p, p _i denote patches, s, s _i denote a segment, S is a set of segments, while v, v _i denote a vertex.

Table 14 Expressions for derived operations and predicates regarding linestrings (line), polygons (poly) and polyhedral surfaces (psur). In the table the following notation is used: $\textit {DR}(\textit {line})=\textit {ln}=\{s_{1}, \dots , s_{k}\}$ (k > 0), $\textit {DR}(\textit {poly})=\textit {pg}=\{\textit {ring}_{1}, \dots , \textit {ring}_{l}\}= \{\textit {pat}_{1}, \dots , \textit {pat}_{l}\}$ (l > 1) and $DR(\textit {psur})=\textit {ps}=\{p_{1}, \dots , p_{m}\}$ (m > 0), where s _i is a segment, pat _i is a patch, and p _i is a polygon.

Appendix B: Proof tables

This section reports the proof tables for Proposition 1. The following notation is used inside such tables: pn, ln, pg, and ps represent the discrete representation in the vector model of point, linestring, polygon and polyhedral surface, respectively. Similarly, v, s, and p denotes a vertex, segment, and patch, respectively.

Table 15 Proof Interior-Interior Intersection (pt/∗)

Table 16 Proof Interior-Interior intersection (ln/pg)

Table 17 Proof Interior-Interior intersection (ps/ps) dim=T – (only in 3D space).

Table 18 Proof Interior-Interior Intersection (ln/ln) Notice that testing conditions are presented on several rows: each row describes a disjunct which is a conjuction of a necessary condition and an additional one (usually the last one is used to take into account the dimension requirement)

Table 19 Proof Interior-Boundary intersection (pt/ln) and (ln/ln) –

Table 20 Proof Interior-Interior Intersection (ln/ps) – (only in 3D space)

Table 21 Proof Interior-Interior Intersection (pg ₁/pg ₂ ) – (only in 2D space)

Table 22 Proof Interior-Interior intersection (ps ₁/ps ₂) dim=0/1/2 – (only in 3D space).

Table 23 Proof Interior-Exterior intersection (pt/ln), (pt/pg) and (pt/ps)

Table 24 Proof Interior-Exterior intersection (ln/ln), (ln/pg) and (ln/ps)

Table 25 Proof Interior-Exterior intersection (pg/pg) and (ps/ps)– Case C.3.11 and C.3.16.

Appendix C: Robustness levels of the relation Touches

This section analyses the robustness behaviour of the relation Touches. Tables 26 and 27 shows the minimum and maximum level of robustness required for all intersection tests. The notation f(c _xy ) means that the cell is a derived cell and its test can be avoided since it is implied by the test of c _xy . The term RobLev stands for robustness level.

Table 26 Analysis of the robustness behavior of a.TC(b) in 2D

Table 27 Analysis of the robustness behavior of a.TC(b) in 3D