Egenhofer, M. (1993): A MODEL FOR DETAILED BINARY TOPOLOGICAL RELATIONSHIPS, Geomatica 47 (3&4), p. 261-273, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... connected regions in the plane This deceptively simple extension of propositional logic and its generalizations are often referred to in the literature as topological inference problems [CGP98a] CGP98b] CHK99] They have proved to be relevant in the area of geographic information systems [E93], EF91] and in graph drawing [DETT99] In spite of many e orts [K91a] K98] and false claims [SP92] ES93] no algorithm was found for their solution. It is known that these problems are at least NP hard [KM89] K91b] MP93] Since each element of a nite system of regions in the plane can ....

M. Egenhofer, A model for detailed binary topological relationships, Geomatica 47 (1993), 261-273.

....route traverses the sequence or rooms and must avoid obstacles and curves e.g. the two lines must cross and the guide surfaces [represented as lines] must be adapted . They can be defined based on connectivity properties. For the first category, the 4 intersection model, as described in (Egenhofer, 1993), is necessary and sufficient for the expression of the analysed design rules. Eight basic topological relations can be defined based on the intersection between the interior (q) and boundary (d) of each object. They are: disjoint, meet, equal, inside, contains, covered by, covers and overlap. For ....

Egenhofer, M. J.: 1993, A Model for Detailed Binary Topological Relationships, Geomatica, 47 (3&4), 261-273.

....( n 2 ) V k=1 (T time [h; j] k where h; j 2 f1; ng. 3. 2 Spatial Constraint System In this section we want to define a constraint system, based on a first order language, whose predicates are spatial relationships between spatial objects, namely binary topological relationships from [3]. Also in this case, the sentences are interpreted over a predefined domain of spatial objects O. 3.2.1 Spatial Domain and Predicates Let O be a set of well defined spatial objects. With well defined spatial objects we mean 2 dimensional convex spatial objects, that can be points, lines, ....

....two functions are defined bound and int, that return respectively the boundary and the interior of the object to which are applied. In the follow, we denote with o 1 ; o 2 ; on elements of O. The fig.3 shows the set of binary topological relationships between the objects in O defined by [4, 3]. disjoint) intersect) equal) touch) contains) inside) covered by) covers) Figure 3: The eight basic topological relationships between two spatial objects embedded in R 2 We extend the set of predicates adding four more unary predicates on I, to express points (point(i) not ....

M. Egenhofer. A model for detailed binary topological relationships. Geomatica, 47(3 and 4):261--273, 1993.

....disjointness of two sets, interior, exterior, boundary, and Boolean connectives. The 4 intersection invariant was further refined by Egenhofer and Franzosa, by taking into account additional information, such as the number and dimension of components of the boundary intersection of two regions [FE92, Ege93, EF95a]. In particular, the invariant exhibited in [EF95a] is claimed (without proof) to completely characterize two regions (discs) up to homeomorphism. Unlike the above topological invariants, the topological invariants we use are lossless, i.e. they completely characterize the topology of a set of ....

M. Egenhofer. A Model for Detailed Binary Topological Relationships. Geomatica, 47(3-4): 261-273, 1993.

....set: 1D (for ; 0D, 1D, or 2D, cf. Table 3 (Clementini et al. 1993) Table 3: The possible dimension of intersection sets. ffi B B B c ffi A f Gamma1D 2Dg f Gamma1D 1Dg f Gamma1D 2Dg A f Gamma1D 1Dg f Gamma1D 0D 1Dg f Gamma1D 1Dg A c f Gamma1D 2Dg f Gamma1D 1Dg f Gamma1D 2Dg Egenhofer (1993) also specifies topological relations in a more detailed way, by additional numerical topological invariants: the dimension of intersections, and the number of components of an intersection set. Other topological invariants are added in a multiple representation framework in Egenhofer et al. ....

Egenhofer, Max J. (1993): A Model for Detailed Binary Topological Relationships. Geomatica, 47(3 & 4):261--273, 1993.

....and no geometric analysis of the the data need be performed. Often the analysis of topological relations may reduce the burden of geometric computations. A formal analysis of relations between sets has been provided by Egenhofer in several publications (Egenhofer and Franzosa 1991, Egenhofer 1991, Egenhofer 1993, Egenhofer et al. 1994) The idea is to represent the mutual relations of two sets A and B by a 2 by 2 matrix, called the 4intersection F, which is given by F = A B A B ffi A ffi B A ffi B ffi (1) where A; B denote the boundaries, A ffi ; B ffi the interiors of ....

Egenhofer, Max J. (1993): A Model for Detailed Binary Topological Relationships. Geomatica, 47(3 & 4):261-- 273, 1993.

....disjointness of two sets, interior, exterior, boundary, and Boolean connectives. The 4 intersection invariant was further refined by Egenhofer and Franzosa, by taking into account additional information, such as the number and dimension of components of the boundary intersection of two regions [FraEg92, E93, EgFra95]. In particular, the invariant exhibited in [EgFra95] is claimed to completely characterize two regions (discs) up to homeomorphism (the result is stated without proof) The PLA model was proposed by the U.S. Census Bureau in [Cor79] see also [Par95] Its ability to capture topological ....

....equivalent iff names(I) names(I 0 ) and for every p; q 2 names(I) ext(p) and ext(q) stand in the same 4 intersection relationship in I as in I 0 . It is well known that there exist spatial instances I, I 0 which are 4 intersection equivalent but not topologically equivalent (e.g. see [FraEg92, E93, EgFra95]) Figure 1 contains two such examples: the instances in (a) and (b) are 4 intersection equivalent, but not H equivalent, and similarly for those in (c) and (d) In addition to H, we consider two other permutation groups: symmetries and piece wise linear functions. These are defined next. Call a ....

M. Egenhofer. A Model for Detailed Binary Topological Relationships. Geomatica, vol. 47, no. 3 & 4, pp. 261-273, 1993.

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M. Egenhofer, A Model for Detailed Binary Topological Relationships. Geomatica 47(3 & 4): 261-273, 1993b.

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Egenhofer, M. (1993): A MODEL FOR DETAILED BINARY TOPOLOGICAL RELATIONSHIPS, Geomatica 47 (3&4), p. 261-273, 1993.

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