Topological Relationships and Their Use
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Topological Space Negative Condition Building Information Model Spatial Object Topological ModelDefinition
Topological relations between spatial objects have been widely recognized, implemented, and used in GIS. They provide a notion of the general structure and the interactions of spatial objects. Topology avoids dealing with geometry by introducing topological primitives, namely, boundary, interior, and exterior. The topological primitives allow to define approximate topological relationships between 0D (point), 1D (linestring ), 2D (surface), and 3D (body) spatial objects in 0D, 1D, 2D, and 3D space. The nineintersection model is the wellknown framework for detecting binary topological relationships (Egenhofer and Herring, 1992) and is adopted by the Opengeospatial Consortium as a basic framework for implementation. Suppose two simple spatial objects A and B are defined in the same topological space X and their boundary, interior, and exterior are denoted by ∂ A, A ^{∘}, A ^{−}, ∂ B, B ^{∘}, and B ^{−}. The binary relationship R(A, B) between the two objects is then identified by composing all the possible set intersections of the six topological primitives, i.e., A ^{∘}∩ B ^{∘}, ∂ A ∩ B ^{∘}, A ^{−}∩ B ^{∘}, A ^{∘}∩ ∂ B, ∂ A ∩ ∂ B, A ^{−}∩ ∂ B, \(A^{\circ }\cap B^{}\,\partial A \cap B^{}\), and \(A^{}\cap B^{}\), and detecting empty (0) or nonempty (1) intersections. For example, if two objects have a common boundary, the intersection between the boundaries is nonempty, i.e., ∂ A ∩ ∂ B = 1; if they have intersecting interiors, then the intersection A ^{∘}∩ B ^{∘} is not empty, i.e., A ^{∘}∩ B ^{∘} = 1. Since each pair of intersections can have either the empty or nonempty value, different “patterns” define different relationships. The theoretical number of all the relationships that can be derived from the 9 intersections is 2^{9}, i.e., 512 relationships, but only a small number of them (approximately 70) can be realized in reality. Following the nineintersection model, operators to detect topological relationships are widely implemented in GIS and DBMS commercial software. The operations can be used for various purposes: checking inclusions and consistency between objects, finding neighboring objects, establishing connectivity (path), composing constraints, etc.
Historical Background
The topological relationships have gradually gained attention in the last four decades and became a major component of current spatial information applications (Breunig and Zlatanova, 2011; Gröger and George, 2012). The topic of research has shifted from issues related to the definition of a particular formalism to represent topological relationships, to implementation issues (Ellul and Haklay, 2007; Borrman and Rank, 2009; Xu and Zlatanova, 2013). The research in the clarification of the relationships between spatial objects in 2D space has extended to investigation and implementation of relationships in 3D space.
The establishment of suitable framework for representation of topological relations takes relatively long time. Early work on topological relationships (e.g., Freeman 1975) does not have sufficient formalism and generality. The need for sound foundation is recognized by many researchers (e.g., Pullar and Egenhofer 1988; Herring 1991). First sound models are developed on basis of pointset theory and the notion of boundary, interior, and closure op point sets. The first, socalled fourintersection model (4IM) is presented by Pullar and Egenhofer (1988). It considers only point and line objects in 1D space. The topological relationships between regions are introduced by Egenhofer and Franzosa (1991). The nineintersection model (9IM) is an extension of the framework by including exterior of objects (?). A critical comparison between 4IM and 9IM have motivated the further adoption of 9IM (Egenhofer et al., 1993). The 4IM and 9IM models have triggered many studies, discussion, comparisons, and criticism since then. The framework is recognized as a very powerful and sound, but also rising many questions and limitations. Topics of investigation are the number of possible relationships, the equivalent relationships, the type of objects (simple, with holes, or with broad boundaries), and the naming of topological relationships.
Negative conditions for eliminating impossible relationships in IR, IR^{2}, and IR^{3}
Group  R(A,B)  ∂A ∩ ∂B  A^{∘}∩B^{∘}  ∂A∩B^{∘}  A^{∘}∩ ∂B  A ̄∩B ̄  A ̄ ∩ ∂B  A ̄∩B^{∘}  ∂A∩B ̄  A^{∘}∩B ̄  

C1  −  −  −  −  0  −  −  −  −  
C2^{a}  −  −  −  −  −  −  −  1  0  
1  C2^{b}  −  −  −  −  −  1  0  −  −  
C3^{a}  0  −  0  −  −  −  −  0  −  
C3^{b}  0  −  −  0  −  0  −  −  −  
C4^{a}  −  0  −  −  −  −  −  −  0  
2  C4^{b}  −  0  −  −  −  −  0  −  −  
C5^{a}  −  −  −  1  −  −  −  −  0  
C5^{b}  −  −  1  −  −  −  0  −  −  
C6^{a}  −  −  −  −  −  −  −  −  0  
3  C6^{b}  −  −  −  −  −  −  −  0  −  
C6^{c}  −  −  −  −  −  0  −  −  −  
C6^{d}  −  −  −  −  −  −  0  −  −  
C7^{a}  −  0  −  0  −  −  −  −  0  
4  C7^{b}  −  0  0  −  −  −  0  −  −  
C8^{a}  −  0  1  −  −  −  −  −  −  
5  C8^{b}  −  0  −  1  −  −  −  −  −  
C9^{a}  −  1  −  0  −  −  −  −  1  
C9^{b}  −  1  0  −  −  −  1  −  −  
C10^{a}  1  −  −  1  −  1  −  −  −  
6  C10^{b}  1  −  1  −  −  −  −  1  −  
C11^{a}  0  −  1  −  −  −  −  1  −  
7  C11^{b}  0  −  −  1  −  1  −  −  −  
C12  0  −  −  −  −  0  −  0  −  
8  C13  −  0  −  −  −  0  −  0  −  
C14^{a}  −  −  −  −  −  −  −  0  1  
C14^{b}  −  −  −  −  −  0  1  −  −  
C15^{a}  −  −  −  −  −  0  1  −  0  
9  C15^{b}  −  −  −  −  −  −  0  0  1  
C16^{a}  −  −  0  1  −  −  −  0  −  
C16^{b}  −  −  1  0  −  0  −  −  −  
C17^{a}  −  −  0  0  −  1  −  0  −  
10  C17^{b}  −  −  0  0  −  0  −  1  −  
C18^{a}  −  1  1  1  −  0  −  0  −  
C18^{b}  1  −  1  1  −  0  −  0  −  
C19^{a}  1  1  1  1  −  −  −  0  −  
C19^{b}  1  1  1  1  −  0  −  −  −  
C20^{a}  −  0  −  1  −  −  0  −  −  
C20^{b}  −  0  1  −  −  −  −  −  0  
11  C21^{a}  1  0  0  −  −  −  0  −  −  
C21^{b}  1  0  −  0  −  −  −  −  0  
C22^{a}  1  −  0  −  −  1  −  0  
12  C22^{b}  1  −  −  0  −  0  −  1  −  
C23^{a}  1  −  −  −  −  −  −  −  −  
C23^{b}  −  −  1  −  −  −  −  −  −  
13  C23^{c}  −  −  −  −  −  −  −  1  −  
C23^{d}  −  −  −  1  −  −  −  −  −  
C23^{e}  −  −  −  −  −  1  −  −  −  
C24^{a}  −  1  −  1  −  −  −  −  −  
C24^{b}  −  1  −  −  −  −  −  −  1  
C24^{c}  −  −  −  1  −  −  −  −  1  
C24^{d}  −  1  1  −  −  −  −  −  −  
C24^{e}  −  1  −  −  −  −  1  −  −  
C24^{f}  −  −  1  −  −  −  1  −  −  
C25^{a}  −  0  −  −  −  −  0  −  −  
C25^{b}  −  0  −  −  −  −  −  −  0 
While quite straight forward for simple objects, the 9IM reveals to have high complexity for objects with holes. Until recently only regions with holes in 2D have been investigated (Egenhofer et al., 1994; Schneider and Behr, 2006). Studies in 3D space have not been presented so far. Clementini and di Felice (1997) address thickness of the boundary or realworld objects as many objects have a boundary which cannot be approximated with a line. They present a modification of the framework for objects with broad boundaries.
Scientific Fundamentals

any union of elements of t belongs to t

any finite intersection of elements of t belongs to t

the empty set and Xbelong to t.

closure is the set \(\overline{E} = \cap \{K \subset X\vert K\) is closed and E ⊂ K},

interior is the set E ^{∘} = ∪{G ⊂ X  G is open and E ⊂ G},

boundary is the set \(\partial E = \overline{E} \cap \overline{(X  E)}\) which is a closed set.

E ^{∘} ⊂ E

\(E \subset \overline{E}\)

E is closed in X if and only if \(\overline{E} = E\)

E is open in X if and only if E ^{∘} = E

\(\overline{E} = E \cup \partial E\)

\(E^{\circ } = E  \partial E\)

\(X = E^{\circ }\cup \partial E \cup (X  E)^{\circ }\)

X ^{∘} = X
The topological definitions given above are not convenient for particular use (e.g., representation of geometric properties) due to the regularity of open sets in the Euclidean space, i.e., each open set of one point has the same properties as any other set. Therefore it is wiser to investigate the space around a point (or several points) and conclude that around other points it is the same. Thus the notion about a neighborhood is introduced:
If Xa topological space and x ∈ X, a neighborhood of x is a set U, which contains an open set containing x, i.e., x ∈ U ^{∘} where U ^{∘}is the interior of U.
The practical importance of topology for the modeling of the spatial objects is twofold. It provides a formalism to define spatial relationships, and it allows a formal geometric description of objects. The notions of neighborhood, interior, boundary, and closure provide formalism for investigating relationships between objects without considering their shape and size. The interrelations between the topological primitives of two or more objects in their neighborhoods define the topological relationships among objects. For example, if the neighborhood of a point is completely inside the interior of an object, then the point is inside the object; however, if the neighborhood is partially in the object’s interior, then the point is on the boundary of the object. After applying to all of the points of two objects, this mechanism supplied information on the interrelation of two objects.

each (n − 1)dimensional cell is shared by exactly two ndimensional cells

given two ndimensional cellss _{ i } and s _{ j }, there exists a sequence of cells \(s_{i} = s_{h} =\ldots = s_{k} = s_{j}\) such that s _{ h } C _{ s, k } is an (n − 1)dimensional face

given any \((n  2)\) dimensional cell, the n and (\(n  1)\) cells adjacent to it can be arranged in a single connected alternating cycle.
The first property leads to a closed manifold. The closed manifolds can be orientable or nonorientable. An orientable manifold is the one in which each two ndimensional cells that intersect in a given (n − 1)dimensional cell induce opposite orientations on that cell. The notations on manifolds played a critical role in constructing models for computational geometry. The traditional GIS and CAD topological models (e.g., wingededge, halfwing edge) are built on subdivisions of 2manifolds, i.e., vertex (0cell), edge (1cell), and face (2cell). 4IM and 9IM are developed under the assumption that the definitions of spatial objects are conform to the properties of 2manifold.
Suppose two simple spatial objects A and Bdefined as nonempty sets in the same topological space X, then the boundary, interior, closure, and exterior will be denoted as \(\partial A,A^{\circ },\overline{A},A^{},\partial B,B^{\circ },\overline{B},B^{}\). The binary relations R(A, B) between the two object s is then defined as investigating the intersections between the boundary, interior, and the exterior: \(\partial A \cap \partial B,A^{\circ }\cap B^{\circ },\partial A \cap B^{\circ },A^{\circ }\cap \partial B,A^{}\cap B^{},A^{}\cap \partial B,A^{}\cap B^{\circ },\partial A \cap B^{}\), and A ^{∘}∩ B ^{−}. The intersection (0, empty and 1, nonempty) between the six topological primitives can be represented as a matrix:R(A, B) = \(\left (\begin{array}{*{20}c} A^{\circ }\cap B^{\circ }&A^{\circ }\cap \partial B &A^{\circ }\cap B^{} \\ \partial A \cap B^{\circ }&\partial A \cap \partial B&\partial A \cap B^{} \\ A^{}\cap B^{\circ }&A^{}\cap \partial B&A^{}\cap B^{}\\ \end{array} \right ) = \left (\begin{array}{*{20}c} 0&0&1\\ 0 &0 &1 \\ 1&1&1\\ \end{array} \right )\)
For simplicity, instead of the matrix representation, a row representation of all the intersections is used in the following text. The intersection of the matrix above is listed in the following order: \(\partial A\cap \partial B,A^{\circ }\cap B^{\circ },\partial A\cap B^{\circ },A^{\circ }\cap \partial B,A^{}\cap B^{},A^{}\cap \partial B,A^{}\cap B^{\circ },\partial A\cap B^{}\text{, and }A^{\circ }\cap \ B^{}\)
For example, the relationship between objects with nonintersecting boundaries and interiors can be represented as 000011111, which can be seen as a binary number that corresponds to the decimal number 31. This number is denoted as a decimal code R031. This relationship corresponds to the “disjoint” relationship. It is apparent that different ordering of the intersections will result in a different decimal code. This specific order is chosen, because the first four intersections represent the 4IM. The intersection \(A^{}\cap B^{}\) is always empty (i.e., always 0) and separates the relationships that can be detected with 4IM from 9IM.
The value of the intersections (empty, nonempty) between interior, boundary, and exterior depends on the following three parameters: the dimension of the objects, the dimension of the space (considered also codimension of the object), and the type of boundary (connected or disconnected). As mentioned previously, the theoretical number of relationships, i.e., 16 for 4IM and 512 for 9IM, is not possible for realworld spatial objects. To eliminate the impossible relationships, negative conditions are defined. Negative conditions are composed of intersection values that are not possible (therefore, the name “negative”). Examples of negative condition (C1, C2, and C3) are given below (Egenhofer and Herring 1992):

C1: The exteriors of two objects always intersect.

C2: If A’s boundary intersects with B’s exterior, then A’s interior intersects with B’s exterior too and vice versa.

C3: A’s boundary intersects with at least one topological primitive of B and vice versa.
The first condition restricts the intersection of the exteriors to be always nonempty, i.e., \(A^{}\cap B^{} = \neg \emptyset\). This means that an empty value (or 0) can never appear in a relationship. Therefore condition C1 is represented as C1=(,,,,0,,,,) in Table 1. After these three negative conditions, the number of possible binary relationships is reduced to 104 for spatial objects with equal dimensions (e.g., surface and surface) and to 160 for spatial objects with different dimensions (e.g., linestring and surface). The conditions are often symmetric. For example, condition C3 implies that it is not possible to have a relationship in which ∂ A ∩ ∂ Band A ^{∘}∩ ∂ B and A ^{−}∩ ∂ B can be at the same time nonempty (or 0). However, the same is true for ∂ A ∩ ∂ B and ∂ A ∩ B ^{∘} and ∂ A ∩ B ^{−}, which is included in the condition as vice versa. The conditions depend on the type of objects and the dimension of the space they are embedded. The codimension is often used to define the difference between the dimension of an object and the dimension of the space. For example, 2cell embedded in 3D space has codimension 1. For simplicity, we refer 0cell, 1cell, 2cell, and 3cell to as points, linestings, surfaces, and bodies with corresponding notations P, L, S, and B. The 1, 2, and 3 dimension of the space is denoted by ℝ, ℝ^{2}, and ℝ^{3}. Thus the notation R(L,S) means that the binary relationship concerns linestring and surface as the linestring is the first object. The relationship R(S,L) is the converse relationship, which is referred to by the vice versa part of the condition.
 1.
Any objects: R(L,L) in ℝ; R(L,L), R(S,S), R(L,S), and R(S,L) in ℝ^{2}; R(L,L), R(S,S), R(B,B), R(L,S), R(L,B), R(S,B), R(S,L), R(B,L), and R(B,S) in ℝ^{3.}
 2.
Objects with equal dimensions: R(L,L) in ℝ; R(S,S) and R(L,L) in ℝ^{2}; R(L,L), R(S,S), and R(B,B) in ℝ^{3}.
 3.
Objects with different dimensions: R(S,L) and R(L,S) in ℝ^{2}; R(B,L), R(L,B), R(B,S), and R(S,B) in ℝ^{3}.
 4.
Objects with different dimensions and one of the objects with zero codimension: R(L,S) and R(S,L) in ℝ^{2}; R(L,B), R(S,B), R(B,L), and R(B,S) in ℝ^{3}.
 5.
At least one of the objects has zero codimension: R(L,L) in ℝ ; R(S,S), R(L,S), and R(S,L) in ℝ^{2}; R(L,B) R(S,B), R(B,L), R(B,S), and R(B,B) in ℝ^{3}.
 6.
At least one of the objects has a disconnected boundary: R(L,L), R(S,L), R(S,L), R(B,L), R(L,S), R(L,S), and R(L,B).
 7.
Objects with connected boundaries and at least one of the objects has a zero codimension: R(S,S) in ℝ^{2}; R(S,B) and R(B,S), and R(B,B) in ℝ^{3}.
 8.
Objects with equal dimensions and zero codimensions: R(L,L) in IR; R(S,S) in ℝ^{2}; and R(B,B) in ℝ^{3}.
 9.
Conditions for binary relations between objects of the same dimension and nonzero codimensions: R(L,L) in ℝ^{2} and R(S,S) in ℝ^{3}.
 10.
Objects with equal dimensions, connected boundaries, and nonzero codimensions: R(S,S) in ℝ^{3}.
 11.
Objects with different dimensions, nonzero codimensions, and one of them with a disconnected boundary: R(S, L)
 12.
Objects with equal dimensions, nonzero codimension, and disconnected boundaries: R(L,L) in ℝ^{2} and ℝ^{3}.
 13.
Conditions for binary relations between objects with at least one empty interior: R(P,P), R(P,L), R(P,S), R(P.B), R(L,P), R(S,P), and R(B,P)
Possible relationships between spatial objects in IR, IR^{2} and IR^{3}
R(A,B)  ∂A ∩ ∂B  A^{∘}∩B^{∘}  ∂A∩B^{∘}  A^{∘}∩ ∂B  A ̄∩B ̄  A ̄ ∩ ∂B  A ̄∩B^{∘}  ∂A∩B ̄  A^{∘}∩B ̄  

1  Disjoint  R026  0  0  0  0  1  1  0  1  0  R026  
R027  0  0  0  0  1  1  0  1  1  R027  
R030  0  0  0  0  1  1  1  1  0  R030  
R031  0  0  0  0  1  1  1  1  1  R031  
2  R051  0  0  0  1  1  0  0  1  1  R051  
R055  0  0  0  1  1  0  1  1  1  R055  
R063  0  0  0  1  1  1  1  1  1  R063  
3  R092  0  0  1  0  1  1  1  0  0  R092  
R093  0  0  1  0  1  1  1  0  1  R093  
R095  0  0  1  0  1  1  1  1  1  R095  
4  R117  0  0  1  1  1  0  1  0  1  R117  
R119  0  0  1  1  1  0  1  1  1  R119  
R125  0  0  1  1  1  1  1  0  1  R125  
R127  0  0  1  1  1  1  1  1  1  R127  
5  Crosses  R159  0  1  0  0  1  1  1  1  1  R159  
6  Contains  R179  0  1  0  1  1  0  0  1  1  R179  
R183  0  1  0  1  1  0  1  1  1  R183  
R191  0  1  0  1  1  1  1  1  1  R191  
7  Inside  R220  0  1  1  0  1  1  1  0  0  R220  
R221  0  1  1  0  1  1  1  0  1  R221  
R223  0  1  1  0  1  1  1  1  1  R223  
8  Crosses?  R243  0  1  1  1  1  0  0  1  1  R243  
R245  0  1  1  1  1  0  1  0  1  R245  
R247  0  1  1  1  1  0  1  1  1  R247  
R252  0  1  1  1  1  1  1  0  0  R252  
R253  0  1  1  1  1  1  1  0  1  R253  
R255  0  1  1  1  1  1  1  1  1  R255  
9  Touch  R272  1  0  0  0  1  0  0  0  0  R272  
R275  1  0  0  0  1  0  0  1  1  R275  
R277  1  0  0  0  1  0  1  0  1  R277  
R279  1  0  0  0  1  0  1  1  1  R279  
R284  1  0  0  0  1  1  1  0  0  R284  
R285  1  0  0  0  1  1  1  0  1  R285  
R287  1  0  0  0  1  1  1  1  1  R287  
10  R311  1  0  0  1  1  0  1  1  1  R311  
R316  1  0  0  1  1  1  1  0  0  R316  
R317  1  0  0  1  1  1  1  0  1  R317  
R319  1  0  0  1  1  1  1  1  1  R319  
11  R339  1  0  1  0  1  0  0  1  1  R339  
R343  1  0  1  0  1  0  1  1  1  R343  
R349  1  0  1  0  1  1  1  0  1  R349  
R351  1  0  1  0  1  1  1  1  1  R351  
12  R373  1  0  1  1  1  0  1  0  1  R373  
R375  1  0  1  1  1  0  1  1  1  R375  
R381  1  0  1  1  1  1  1  0  1  R381  
R383  1  0  1  1  1  1  1  1  1  R383  
13  Equal  R400  1  1  0  0  1  0  0  0  0  R400  
R403  1  1  0  0  1  0  0  1  1  R403  
R405  1  1  0  0  1  0  1  0  1  R405  
R407  1  1  0  0  1  0  1  1  1  R407  
R412  1  1  0  0  1  1  1  0  0  R412  
R413  1  1  0  0  1  1  1  0  1  R413  
R415  1  1  0  0  1  1  1  1  1  R415  
14  Covers  R435  1  1  0  1  1  0  0  1  1  R435  
R439  1  1  0  1  1  0  1  1  1  R439  
R444  1  1  0  1  1  1  1  0  0  R444  
R445  1  1  0  1  1  1  1  0  1  R445  
R447  1  1  0  1  1  1  1  1  1  R447  
15  CoveredBy  R467  1  1  1  0  1  0  0  1  1  R467  
R471  1  1  1  0  1  0  1  1  1  R471  
R476  1  1  1  0  1  1  1  0  0  R476  
R477  1  1  1  0  1  1  1  0  1  R477  
R479  1  1  1  0  1  1  1  1  1  R479  
Overlaps  R499  1  1  1  1  1  0  0  1  1  R499  
R501  1  1  1  1  1  0  1  0  1  R501  
R503  1  1  1  1  1  0  1  1  1  R503  
R508  1  1  1  1  1  1  1  0  0  R508  
R509  1  1  1  1  1  1  1  0  1  R509  
R511  1  1  1  1  1  1  1  1  1  R511 

Clearly the 9IM is superior to 4IM. However the increased relationships are only between object of mixed dimensions and objects of the same dimension, which are embedded in higher dimension space such as R (L,L) and R(S,S) in ℝ^{3}. The relationships that appear between all objects are only seven (R031, R179, R220, R400, R287, R435, and R476), and they can be detected with 4IM (Figs. 3, 4, and 5). Note that “overlap” relationship R511 is not possible for R (L,L) in IR . The relationships that visually represents “overlap” is R255. Because the boundary of linestring is disconnected, the intersection of boundaries is nonempty, i.e., ∂ A ∩ ∂ B = 0 (Fig. 3).

The relationships are related to the dimension of objects, i.e., some of the relationships never occur between particular objects. For example, R509 (the interiors can intersect without touching the boundary) appears only in R(L,B)/R(B,L) and R(S,B)/R(B/S) or R277 (boundaries completely overlap but the interiors not) is possible only for R(L,L) and R(S,S) in higher dimension space. This implies that certain relationships (respectively topological operators) will be quite specific and will depend on the dimension of the objects and the space.

The relationships are related to the geometric representation of the objects. This is to say that some relationships may not be needed because some configurations of objects are too complex and normally they are simplified. For example, R455 performed for body and surface (Fig. 7) may never be needed for urban applications. Most probably the surface A will be represented as two surfaces: one on the top and one inside the body B.

As it can be realized, the names (given in Table 2) established for relationships according 4IM are not sufficient for 9IM. The relationships between surfaces in 3D space are one typical example. In the human perception of “intersection,” “cross” might be completely different: e.g., overlap stands for R511 (e.g., surface and surface in 2D) and R255 (e.g., surface and linestring in 3D). Most of the relationships are not associated with appropriate names and even it is difficult to specify the type of interaction. Further implementations topological operators would require establishment of new names.
As mentioned previously, the calculations of intersections will give equivalent results if using classical geometric representations (Fig. 1). In 2011, the OGC implementation specifications for geographical features have been adapted to incorporate the Dimensionally Extended NineIntersection Model (DE9IM) as presented by Strobl (2008). The matrix for representing the 9IM is augmented to include the dimension of the intersection between the two objects as follows (Herring, 2011):

p = T ⇒ dim(x) ∈ {0, 1, 2}, i.e., x ≠ ∅

p = F ⇒ dim(x) = −1, i.e., x = ∅

p = * ⇒ dim(x) ∈ {1, 0, 1, 2}, i.e., Don’t Care

p = 0 ⇒ dim(x) = 0 (point)

p = 1 ⇒ dim(x) = 1 (line sting)

p = 2 ⇒ dim(x) = 2 (polygon)
The OGC specifications provide further instructions how the functions have to be implemented for point, linestring, polygon, and area in 2D space.
Key Applications
The topological operators are commonly implemented for vector models (boundary representation) and more specifically on geometrical models. Following the OGC abstract specification, Topic 1 Feature Geometry, the geometry is represented as an ordered sequence of vertices that are connected by straight line segments or circular arcs, which are used to describe 0D, 1D, 2D, and 3D objects. The vertices can be given with different dimensions (1D, 2D, and 3D). For example, threedimensional points are elements composed of three ordinates, X, Y, and Z. Line strings are composed of one or more pairs of points that define line segments. Polygons are composed of connected line strings that form a closed ring and the area of the polygon is implied. Selfcrossing polygons are not supported, although selfcrossing line strings might be supported. If a line string crosses itself, it does not become a polygon and does not have any implied area.
The usage of the topological relations is wellrepresented in query languages. All database management systems with a spatial support have provided topological operations based on the 9IM. The operations are developed by referencing the name of the eight relationships (covers, coveredBy, contains, inside, overlaps, touch, equal, anyinteract). The operator anyinteract detects if two objects are disjoint. If the operator returns nonempty value, other operators can clarify what the relationship is. Oracle Spatial 11g provides operators to check the topological operators in two versions as one operator SDO_RELATE (with mask for each relationship) and as convenience operators named after the relationships. PostGIS provides operators per relationship.
Consistency and Validity
The topological operations are utilized for different purposes. The most common use is investigation if a geographical map is a planar partition, i.e., to detect gaps or overlaps between geographical objects. These operations are of major importance for organizations, which maintain topographic, land use, cadastral, hydrological, and other maps, such as national mapping organizations, cadastre, topographic offices, municipalities, etc.
It should be noticed that most of the topological operators are currently implemented for a 2D space. If an object is given with its 3D coordinates, it is first projected on the xyplane and then the operation is performed. The topological operators might even trigger an error if vertical objects are given as predicates. Oracle Spatial and Graph 11g provide few operators such as SDO_ANYINTERACT and SDO_INSIDE, which make use of the 3D coordinates.
The topological operations can be used to define constraints either for update of existing data sets, i.e., to avoid wrong overlaps and intersections between geographical object, or to restrict the use from performing certain actions, which will result in violation of semantic or topological rules. (Louwsma et al., 2006) present such a set of constraints, which are implemented in a landscape simulator SALIX2. Examples of a constraint on the topological relationships between two objects are “Yucca tree must never stand in water” and “the tree must never be planted on a paving.” These constraints are then checked each time when students plant trees in the landscape simulator. SALIX2 maintains trees, bushes, and ground surfaces (water, paving, soft paving, grass, and bridge). The constraints are implemented in Oracle Spatial 9i using triggers (before and after) and the operators SDO_ANYINTERACT and SDO_RELATE (mask = “inside”).
Visibility Analysis
Navigation
3D Topological Relationships
There are still many cases that true 3D operator are needed. 3D world contains more information than 2D and the validity and integrity of 3D models are getting of increasing importance (Fig. 13) (Ellul, 2013). Borrman and Rank (2009) present algorithms define to inside, contain, touch, overlap, disjoint, and equal in 3D space for the purpose of analyzing building information models. The operators are implemented in 3D raster domain (octreebased), i.e., prior performing the operations the boundary objects is voxelized.
Future Directions
Advances in data collection, processing, management, and visualization have led to and increased need for welldefined objects, consist data sets, and elaborated analysis. The size of current data sets and complexity of 3D models have increased the interest in database storage with centralized management of information with extended analytical capabilities. The topology and topological relationships are key component in this process. Future investigations and developments are envisaged under the following broad groups: algorithms, data structures, and applications.
Topological relationships are most welldeveloped for simple objects and implemented for 2D objects. Next step is the implementations of the DE9IM for 2D objects. Algorithms for 1D, 2D, and 3D objects embedded in 3D space are still investigated only in research environment. A first option of extending the operations in 3D is by reusing and combining existing operators. The complexity of realworld manmade objects clearly shows that simple objects with holes and tunnels have to be urgently included in research agendas. A small set of 3D operators on solids with tunnels could be the first easy step. Investigations in higherorder topology, which is beyond the immediate neighbors or considering semantic expressions, are very much of interest.
The implementation of topological relationships is highly depended on the underlying spatial model. Current implementations make use of geometrical model as defined in the OGC abstract specification. Research and prototype implementations have convincingly shown that topological models have advantages to the geometrical model in keeping consistency and especially facilitating adjacency relationships. The disadvantage of topological data structures is the lack of unified data structure in 3D. van Oosterom et al. (2002) propose maintenance of the existing topological data structures in a metadata table similar to the reference coordinate systems. A complete and detailed description of all parameters defining topological model, such as dimension, primitives used, existence of explicit relationships, number of tables, rules, etc., will make possible to transform data between topological models.
The topological operations are presently discussed only for vector models. However they can be implemented for 2D or 3D raster models as well. The topological primitives interior, boundary, and exterior remain the same but their importance changes. In vector models, the boundary is the most critical primitive, which is realized through the shape of the object. In raster/voxel domain, the interior will be the leading primitive. Considering the complexity of 3D objects, 3D raster domain with its unified primitives (voxels) and welldefined structures (e.g., octree) can be an attractive option for implementing 3D topological operations.
Topological operations have been traditionally developed to represent relationships between geographical objects. The interest in topology in more domains is increasing. One straightforward implementation is detecting connectivity between objects and deriving dual graph. This approach has been widely accepted for shortest path computations, spatial syntax analysis, and walk through. Recently a new standard IndoorGML has been adopted by OGC, which utilizes the primadual approach for indoor navigation (Lee et al., 2014). Topological relationships attract the attention of CAD and BIM domain as well. Indoor environments are truly 3D and contain much more objects with multifaceted shapes and complex relationships. Examples are pipes and cables in buildings, furniture in rooms, elevators, and escalators between floors. Such relationships will definitely need extended 3D topological relationships. The richer set of geometric primitives in CAD and BIM domains, the density of objects and complexity of relationships could be another reason for shift to the 3D raster domain as unified environment for 3D topological relationships.
CrossReferences
References
 Armstrong MA (1983) Basic topology. Springer, New YorkCrossRefzbMATHGoogle Scholar
 Billen R, Kurata Y (2008) Refining topological relations between regions considering their shapes. In: Raunbal M, Miller J, Frank AU et al (eds) Geographic information science. Lecture notes in computer science. Heidelberg, Berlin, pp 18–32Google Scholar
 Borrman A, Rank E (2009) Topological analysis of 3D building models using a spatial query language. Adv Eng Inform 23(4):370–385CrossRefGoogle Scholar
 Breunig M, Zlatanova S (2011) 3D geodatabase research: retrospective and future directions. Comput Geosci 37(7):791–803CrossRefGoogle Scholar
 Clementini E, di Felice P (1997) Approximate topological relations. Int J Approx Reason 16:173–204MathSciNetCrossRefzbMATHGoogle Scholar
 Clementini E, di Felice P, van Oosterom PJM (1993) A small set of formal topological relations suitable for enduser interaction. In: Proceedings of the 3th international symposium on large spatial databases. Springer, Berlin, pp 277–295Google Scholar
 de Hoop S, van de Meij L, Molenaar M (1993) Topological relations in 3D vector maps. In: Proceedings of 4th EGIS, Genoa, pp 448–455Google Scholar
 Deng M, Cheng T, Chen X et al (2007) Multilevel topological relations between spatial regions based upon topological invariants. Geoinformatica 11:239–267CrossRefGoogle Scholar
 Egenhofer MJ (1995) Topological relations in 3D. Technical report, University of MaineGoogle Scholar
 Egenhofer MJ, Franzosa RD (1991) Pointset topological spatial relations. Int J Geogr Inf Syst 5:161–174CrossRefGoogle Scholar
 Egenhofer MJ, Herring JR (1992) Categorising topological relations between regions, lines and points in geographic databases. In: Egenhofer MJ, Herring IR (eds) A framework for the definition of topological relationships and an approach to spatial reasoning within this framework, Santa Barbara, pp 1–28Google Scholar
 Egenhofer MJ, Sharma J, Mark D (1993) A critical comparison of the 4intersection and 9intersection models for spatial relations: formal analysis. In: Autocarto 11, Minneapolis, pp 1–11Google Scholar
 Egenhofer MJ, Clementini E, di Felice P (1994) Topological relations between regions with holes. Int J Geogr Inf Syst 8(2):129–144CrossRefGoogle Scholar
 Ellul C (2013) Can topological preculling of faces improve rendering performance of city models in Google Earth. In: Pouliot J, Daniel S, Hubert F, Zamyadi Z (eds) Progress and new trends in 3D geoinformation sciences. Springer, Heidelberg/New York, pp 133–154CrossRefGoogle Scholar
 Ellul C, Haklay M (2007) The research agenda for topological and spatial databases. Comput Environ Urban Syst 31:373–378CrossRefGoogle Scholar
 Freeman J (1975) The modelling of spatial relations. Comput Graph Image Process 4:156–171CrossRefGoogle Scholar
 Gröger G, George B (2012) Geometry and topology. In: Kresse W, Danilo DM (eds) Springer handbook of geographic information. Springer, Berlin/New York, pp 159–177Google Scholar
 Herring JR (1991) The mathematical modeling of spatial and nonspatial information in geographic information systems. In: Mark D, Frank A (eds) Cognitive and linguistic aspects of geographic space. Kluwer Academic, Dordrecht, pp 313–350CrossRefGoogle Scholar
 Herring JR (2011) OpenGIS implementation specification for geographic information – simple feature access – Part 1: Common architecture: Version: 1.2.1, OGC Doc. No OGC 06103r3Google Scholar
 Kufoniyi O (1995) Spatial coincidence modelling, automated database updating and data consistency in vector GIS. PhD thesis, ITC, The NetherlandsGoogle Scholar
 Lee J, Li KJ, Zlatanova S, Kolbe TH, Nagel C, Becker T (2014) IndoorGML, Version 1.0, OGC Doc. No OGC 14005r3Google Scholar
 Louwsma J, Zlatanova S, van Lammeren R, van Oosterom P (2006) Specifying and Implementing Constraints in GIS–with Examples from a GeoVirtual Reality System. In: GeoInformatica, vol 10, No 4, pp 531–550CrossRefGoogle Scholar
 Pullar DV, Egenhofer MJ (1988) Toward the definition and use of topological relations among spatial objects. In: Proceedings of the third international symposium on spatial data handling, Sydney, pp 225–242Google Scholar
 Schaap J, Zlatanova S, van Oosterom PJM (2012) Towards a 3D geodata model to support pedestrian routing in multimodal public transport travel advices, In: Zlatanova S, Ledoux H, Fendel EM, Rumor M (eds) Urban and regional data management. UDMS annual 2011. CRC press/Taylor and Francis Group, Boca Raton/London, pp 63–78Google Scholar
 Schneider M, Behr T (2006) Topological relationships between complex spatial objects. ACM Trans Database Syst 31(1):39–81CrossRefGoogle Scholar
 Strobl C (2008) Dimensionally extended nineintersection model (DE9IM). In: Shekhar S, Xiong H (eds) Encyclopaedia of GIS. Springer, Berlin, pp 240–245CrossRefGoogle Scholar
 van Oosterom PJM, Stoter J, Quak W, Zlatanova S (2002) The balance between geometry and topology. In: Richardson D, van Oosterom PJM (eds) Advances in spatial data handling. 10th international symposium on spatial data handling. Springer, Berlin, pp 209–224Google Scholar
 Willard S (1970) General topology. AddisonWesley Publishing Company, ReadingzbMATHGoogle Scholar
 Xu D, Zlatanova S (2013) Am approach to develop 3D GeoDBMS topological operators by reducing existing 2D operators, ISPRS annals – volume II2/W1, 2013, WG II/2, 8th 3D GeoInfo conference & ISPRS WG II/2 workshop, Nov 2013, pp 291–298Google Scholar
 Zlatanova S (2000a) 3D GIS for urban development. PhD thesis, Graz University of Technology, ITC, The NetherlandsGoogle Scholar
 Zlatanova S (2000b) On 3D topological relationships, In: Proceedings of the 11th international workshop on database and expert system applications (DEXA 2000), 6–8 Sept. Greenwich, London, pp 913–919Google Scholar
 Zlatanova S, Tijssen TPM, van Oosterom PJM, Quak WC (2003) Research on usability of Oracle spatial within RWS organisation, GISt No. 21, ISSN:1569–0245, ISBN:9077029079, AGIGAG200321, Delft, 75pGoogle Scholar
Recommended Reading
 Billen R, Zlatanova S (2003) 3D spatial relationship model: a useful concept for 3D cadastre? Comput Environ Urban Syst 27:411–425CrossRefGoogle Scholar
 Clementini E, Sharma J, Egenhofer MJ (1994) Modelling topological spatial relations: strategies for query processing. Comput Graph 18(6):815–822CrossRefGoogle Scholar
 de Almeida JP, Morley JG, Dowman IJ (2007) Graph theory in higher order topological analysis of urban scenes. Comput Environ Urban Syst 31:426–440CrossRefGoogle Scholar
 Egenhofer MJ, Herring JR (1990) A mathematical framework for the definition of topological relations. In: Proceedings of fourth international symposium on SDH, Zurich, pp 803–813Google Scholar
 EscobarMolano ML, Barret DA, Carson E et al (2007) A representation for databases of 3D objects. Comput Environ Urban Syst 31:409–425CrossRefGoogle Scholar
 Hazelton NW, Bennett L, Masel J (1992) Topological structures for 4dimensional geographic information systems. Comput Environ Urban Syst 16(3):227–237CrossRefGoogle Scholar
 Park J, Lee J (2008) Defining 3D spatial neighborhood for topological analyses using a 3D networkbased topological data modelCA building based evacuation simulation. The international archives of the photogrammetry, remote sensing and spatial information sciences, Beijing, vol XXXVII, Part B2, pp 305–310Google Scholar
 Whiting E, Battat J, Teller S (2007) Topology of urban environments. In: Dong A, Vande Moere, Gero JS (eds) Computeraided architectural design futures (CAADFutures). Springer, Berlin, pp 114–128Google Scholar