M. Egenhofer, R. Franzosa, On the Equivalence of Topological Relations, International Journal of Geographical Information Systems, 9(2), pp.133-152 (1995)  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... have not missed a condition (an adhoc style of analysis might easily identify a sufficient condition but might not identify all sufficient conditions) This analysis is rather in the style of the 4 and 9 intersection model of Egenhofer [Egenhofer and Franzosa, 1991; Egenhofer and Herring, 1994; Egenhofer and Franzosa, 1995] where from a 2 x 2 and 3 x 3 matrix which determine whether various topological parts of two regions share points or not, then by imposing a variety of conditions (such as regularity or one pieceness) the or 2 possibilities are whittled down to just eight possibilities (corresponding to ....

M J Egenhofer and R D Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9(2):133--152, 1995.

....of connection through fiat boundaries [59] Yet another notion of connection is given by a pair of linked (interlocked) tori [27, 65] 34 b g a b c d Figure 14. Further connection relations with multi piece regions. It is also worth pointing to the work of Egenhofer and Franzosa [28, 29], who present a calculus that allows, at the cost of arbitrary complexity, the possibility of classifying any topological distinct situation. Compare [18] for a related proposal. The variety of mereotopological connection relations is very rich indeed. We hope to have gone some way in the ....

Egenhofer M. J., Franzosa R. D., 1995, `On the Equivalence of Topological Relations ', International Journal of Geographical Information Systems 9, 133--152.

....intersection of A with B, and in certain other regions determined by A and B. The investigation of this use of connected components is due to Galton [Gal98] Another way of extending the 4 intersection model, which provides for still finer distinctions than in Galton s technique, is described in [EF95] The RCC schemes have been extended to systems such as RCC15 and RCC23 [CB 97] which take account of the convex hulls of the regions involved. Another direction for extension is to vague regions. Two approaches based on the 9 intersection model, are the work of Clementini and Di Felice ....

M. J. Egenhofer and R. Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9:133--152, 1995.

.... object based [7] While the former deals with spatial distributions over a geographical region, the latter deals with discrete, identifiable entities on the geographical space [8] These two approaches have led to two separate areas of research: definition of spatial operations on discrete objects [9], and definition of operations on fields (the map algebra of Tomlin [10] Furthermore, most current GIS implementations provide different sub systems for map algebra, spatial queries and image processing. To improve upon this situation, there is a need for a comprehensive approach which unifies ....

M.J. Egenhofer and R.D. Franzosa, On the equivalence of topological relations. International Journal of Geographical Information Systems, 9(2), 133-152 (1995).

....determined by A and B. The investigation of this use of connected components is due to Galton [Gal98] Another way of describing the relationship of A to B, which provides for ner distinctions than in Galton s technique, is a re nement of the 4 intersection model also by Egenhofer and Franzosa [EF95] Some recent work in this area includes the investigation of relations between three dimensional regions [Zla99] Since it is possible to regard time as the third dimension, this is likely to provide one way of treating relationships between two dimensional spatial regions which vary over a one ....

M. J. Egenhofer and R. Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9:133-152, 1995.

.... have not missed a condition (an adhoc style of analysis might easily identify a sufficient condition but might not identify all sufficient conditions) This analysis is rather in the style of the 4 and 9 intersection model of Egenhofer [Egenhofer and Franzosa, 1991; Egenhofer and Herring, 1994; Egenhofer and Franzosa, 1995] where from a 2 x 2 and 3 x 3 matrix which determine whether various topological parts of two regions share points or not, then by imposing a variety of conditions (such as regularity or one pieceness) the 2 4 or 2 9 possibilities are whittled down to just eight possibilities (corresponding ....

M J Egenhofer and R D Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9(2):133--152, 1995.

.... of the forms such as: Given that a region a is in relation R 1 to region b, and region b is in relation R 2 to region c; what relations may or must hold between a and c Of course, it might be possible to adapt the conventional mathematical formalisms, and indeed this strategy has been adopted [60, 64, 184]. One existing approach to topology which has been espoused by the QSR community is the work to be found in the philosophical logic community [181, 51, 183, 23, 24, 13] This work has built axiomatic theories of space which are predominantly topological in nature, and which take regions rather ....

....of dimensions. Other theories which introduce the notion of boundaries of regions explicitly include [166, 175, 152, 167] 4.2.2. Topology via n intersections An alternative approach to representing and reasoning about topological relations has been promulgated via a series of papers [57, 60, 58, 26, 61, 64]. Three sets of points are associated with every region its interior, boundary and complement. The relationship between any two region can be characterized by a 3x3 matrix 7 called the 9 intersection. Although it would seem 5 Note, however, that this task becomes almost trivial once the ....

Egenhofer, M. J. and Frenzosa, R.: \On the equivalence of topological relations", International Journal of Geographical Information Systems, 9(2), 1995, pages 133-152

....induced by the FOL data model, a finite structure which is an abstraction that captures all the topological properties of the spatial database. The strategy of maintaining such a finite structure (from now on called topological invariant) is very useful and has been addressed in the literature [7, 8, 19, 23]. Different abstractions of topological properties have been considered, and each of them can be viewed as an augmentation of the model proposed by the U.S. Census Bureau, which contains topological properties on points, lines and areas [6, 20] Different classes of continuous topological ....

M. J. Egenhofer and R. D. Franzosa. On the equivalence of topological relations. International Journal of Geographic Information Systems, 9(2):133--152, 1995.

....intersection of A with B, and in certain other regions determined by A and B. The investigation of this use of connected components is due to Galton [Gal98] Another way of extending the 4 intersection model, which provides for still finer distinctions than in Galton s technique, is described in [EF95] The RCC schemes have been extended to systems such as RCC15 and RCC23 [CB 97] which take account of the convex hulls of the regions involved. Another direction for extension is to vague regions. Two approaches based on the 9 intersection model, are the work of Clementini and Di Felice ....

M. J. Egenhofer and R. Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9:133--152, 1995.

....where no spatial indeterminacy is admitted. For example, the relation between two regions may be that they are disjoint or that they meet only at their boundaries. The two principal approaches to these topological relations are those based on the RCC formalism [6] and on Egenhofer s 4 intersection [11, 12] and 9 intersection models [13] The relations that can be de ned on regions in the RCC approach are often grouped into sets of relations that are pairwise disjoint and jointly exhaustive (JEPD) meaning that for any two regions, one and only one of a particular JEPD set of relations will hold ....

....us a function: F : R R fDR; PO; PP; PPi; EQg; 1) where R is the set of regions, and F (a; b) takes a given one of the ve values if and only if the corresponding relation holds between the regions a and b. The approach to relations between crisp regions due to Egenhofer and his colleagues [11, 12, 13] is based on a treatment of regions constructed from sets of points, making it a topological rather than mereological approach. It takes into account the interior and boundary (and exterior in the case of the 9 intersection model) of pairs of crisp regions, and looks at the intersections of these. ....

M. J. Egenhofer and R. Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9:133{ 152, 1995.

.... detailed survey of mereotopology which also deals with the RCC and related systems can be found in the recent book by Casati Varzi (1999) Schemes for classifying relationships between spatial regions have arisen from the needs of geographic information systems (GIS) Egenhofer Franzosa 1991, Egenhofer Franzosa 1995, Galton 1998) The majority of all this work has been concerned with space which is in nitely divisible and continuous. For example, the RCC axioms stipulate (Cohn et al. 1997, p283) that every region has a non tangential proper part, thus ensuring that space is in nitely divisible. The ....

Egenhofer, M. J. & Franzosa, R. (1995), `On the equivalence of topological relations', International Journal of Geographical Information Systems 9, 133-152.

....search and retrieval therefore allows for raw imagery to be queried on line. Currently, we are extending our queries to include configurations of objects. We employ the well known concept of 9 intersection, describing the major topological relations between areal, linear, and point features [5]. According to this model, the topological relationships between two objects is described by a 3x3 matrix whose elements express whether the mutual relationships between the interior, exterior and outlines of two features are empty or non empty sets. Individual objects are queried separately, and ....

Egenhofer M. & R. Franzosa. On the Equivalence of Topological Relations. Int. Journal of Geographical Information Systems, Vol. 9, No. 2, pp. 133-152, 1995.

....introduce boundaries of regions explicitly (e.g. 119, 120, 125, 109] but which did not explicitly introduce dimensional reasoning. Topology via n intersections An alternative approach to representing and reasoning about topological relations has been promulgated via series of papers (e.g.[23, 39, 41, 41, 40, 46, 42]) In the most recent calculus three sets of points are associated with every region its interior, boundary and complement; the relationship between two regions can be characterized by a 3x3 matrix, 6 called the 9 intersection, each of whose elements denotes whether the intersection of the ....

M J Egenhofer and R D Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9(2):133--152, 1995.

....imprecise information by abstracting away from metrical details. However, specific formalisms have also been developed to facilitate representing and reasoning with indefinite information. For example, 10] and, independently, 6] have developed extensions of two related formalisms (i.e. 9] and [13] respectively) for representing and reasoning about mereological relations between spatial regions. Common to both these formalisms is the notion of an egg yolk 1 : intuitively, 1 In fact [6] do not use the term egg yolk ; however the concept is essentially the same. this is a pair of ....

M J Egenhofer and R D Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9(2):133--152, 1995.

....some part of itself or something it is holding, regions on a diagram that is to be drawn from a description, and so on. Of course, it might be possible to adapt the conventional mathematical formalisms for our purposes, and indeed this strategy is sometimes adopted (Egenhofer and Franzosa 1991, Egenhofer and Franzosa 1995, Worboys and Bofakos 1993) However, there are considerations which have led us to adopt an alternative approach. This section discusses some of the relevant issues. What we want from a topological formalism can be summarised as follows: ffl Logical Consistency ffl Expressiveness Flexibility ....

Egenhofer, M. J. and Franzosa, R. D.: 1995, On the equivalence of topological relations, International Journal of Geographical Information Systems 9(2), 133--152.

....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 ....

....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 regions. The invariants contain information similar ....

M. Egenhofer and R. Franzosa. On the Equivalence of Topological Relations. Int'l J. of Geographical Information Systems, 9(2):133-152, 1995.

....in Subset B Figure 3: Cost and Size Palette 3. 3 Semantics of Relative Spatial Queries in SEE Given two spatial objects A and B, our SVIQUEL interface distinguishes between eight possible primitive topological relationships (disjoint, meet, overlap, covers, coveredBy, contains, inside and equal) EF95] EM95] and eight possible directional relationships (north, south, east, west, north east, north west, south east and south west) between the two objects [JTS96] TP95] and two special relationships of same longitude and same latitude. Since it is possible for two regions to have both kinds of ....

M. Egenhofer and R. Franzosa. On the equivalence of topological relations. International Journal of Geographic Information Systems, 9:133--152, 1995.

....introduce dimensional reasoning. Topology via n intersections An alternative approach to representing and reasoning about topological relations has been promulgated via a series of papers (e.g. Clementini et al. 1994; Egenhofer, 1989; Egenhofer and Franzosa, 1991; Egenhofer, 1994; Egenhofer and Franzosa, 1995; Egenhofer and Herring, 1994 ] In the most recent calculus, three sets of points are associated with every region its interior, boundary and complement; the relationship between two regions can be characterized by a 3x3 matrix, 8 called the 9 intersection, each of whose elements denotes ....

M J Egenhofer and R D Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9(2):133--152, 1995.

....into our world consists chiefly of interwoven and superimposed schemata. He goes on to provide a list of image schemata , which includes part whole, centre periphery and contact. Three significant formal ontologies of space have been provided by Cohn et al. GGC96] by Egenhofer and Franzosa [FE91,FE95], and by Smith [Smi96] Cohn et al. have developed a theory of regions and their relationships, based upon a single primitive contact relation, called connection, between regions. Egenhofer and Franzosa provide their well known model of binary topological relations between point sets. Smith ....

....discussed, R R does correspond to a meaningful boundary. However, in the two stage sets the boundary is disjoint from the core: R R = 0, but in the network example this is rarely the case. 4 Modelling the Region Connection Calculus without using Points There have been several investigations [GGC96,FE91,FE95,Smi96] devoted to the question of how to characterize the essential properties of regions as used in spatial information systems. It has been observed that classical mathematical models of space, in particular point set topology, are unsatisfactory for many practical aspects of spatial reasoning. This ....

R. Franzosa and M. J. Egenhofer. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9:133--152, 1995.

....(0,0) translate the the objects of the relation 5 in the direction of the x axis; rotate the objects of the relation 90 degrees around (0,0) give the gravity center of each object. Max Egenhofer has studied the topological issues that are related with geomatic data types very extensively [Ege88,Ege89,Ege91,Ege93,Ege94a,Ege94b]. In [Ege91,Ege93] the 9 intersection model is given for topological relations in IR 2 . This model is based on the overlapping properties of the interior (A ,B ) the complement (A,B) and the boundary (dA,dB) of two two dimensional objects A and B. There are 3 x 3 = 9 combinations for the ....

Egenhofer M., R. Franzosa, On the equivalence of topological relations, Int. J. Geographical Information Systems, p. 523-542, 1994.

....those spatial relations which are preserved under groups of continuous transformations such as translation and rotation. In other words, if both the objects are translated or rotated in the same manner, then their topological relationship still remains the same. Egenhofer s 4 intersection model [EF95] provides us with a means of classifying the various topologies into eight distinct categories: Disjoint, Meet, Overlaps, Equal, Contains, Inside, Covers and Covered By. This model considers the intersection values of the objects boundaries and interiors. If the 4 intersection values of two ....

M. Egenhofer and R. Franzosa. On the equivalence of topological relations. International Journal of Geographic Information Systems, 9:133--152, 1995.

....ffl translate the the objects of the relation 5 in the direction of the x axis; ffl rotate the objects of the relation 90 degrees around (0; 0) ffl give the gravity center of each object. Max Egenhofer has studied the topological issues that are related with geomatic data types very extensively [14, 10, 11, 12, 15, 13]. In [11, 12] the 9 intersection model is given for topological relations in R 2 . This model is based on the overlapping properties of the interiors (A ffi ; B ffi ) the complements (A; B) and the boundaries ( A; B) of two two dimensional objects A and B. Although this model only ....

M. Egenhofer and R. Franzosa. On the equivalence of topological relations. Int. J. Geographical Information Systems, pages 523--542, 1994.

....Algebras 21 10 Conclusions and Further Work 21 10.1 Conclusions . 21 10.2 Further work . 22 1 Introduction There have been several recent investigations [GGC96, FE91, FE95, Smi96] devoted to the question of how to characterize the essential properties of regions as used in spatial information systems. It has been observed that classical mathematical models of space, in particular point set topology, are unsatisfactory for many practical aspects of spatial reasoning. This ....

R. Franzosa and M. J. Egenhofer. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9:133--152, 1995.

....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 ....

....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 information has not been formally studied. Models using ....

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M. Egenhofer and R. Franzosa. On the Equivalence of Topological Relations. Int'l J. of Geographical Information Systems, vol. 9, no. 2, pp. 133-152, 1995.

....formalism is concerned, the regions and the relation of connection have precisely the properties specified by the axioms, and nothing more. This axiomatic approach to developing a topological formalism can be contrasted with an alternative, constructive one, e.g. Egenhofer and Franzosa 1991, Egenhofer and Franzosa 1995, Worboys 1992, Worboys and Bofakos 1993) The constructive approach begins by singling out a class of elementary objects, with topological properties which are specified as those of the topologically simplest kind of region that is of interest. More complex objects can then be constructed ....

....was designed initially to apply only to pairs of discs embedded in Euclidean 2 space, while the RCC relations are intended to apply to any kind of regions whatever. Both approaches allow these relations to be subdivided further, but the strategies for doing this differ considerably. Egenhofer (Egenhofer and Franzosa 1995) introduces a collection of invariants (properties of relations which remain unchanged across topologically equivalent situations) which together are claimed fully to characterise each topologically distinct relation between two (topological) discs embedded in the plane. Relations between more ....

Egenhofer, M. J. and Franzosa, R. D.: 1995, On the equivalence of topological relations, International Journal of Geographical Information Systems 9(2), 133--152.

....designing efficient algorithms and data structures. These efforts give an evidence that there is an obvious need of a formal model for the representation of topological relationships. Recently there has been a growing interests in the computer science and GIS community in topological relationships [1, 5, 8, 10, 15] and topological queries [2, 4, 20 22] A large number of contributions in the literature have focused on the modeling of topological relationships between simple objects: simply connected homogeneous 2 dimensional areas, simple lines (i.e. piecewise algebraic curves with exactly two endpoints) ....

M. J. Egenhofer and R. Franzosa. On the equivalence of topological relations. Int. J. Geo. Info. Sys., 9(2):133--152, 1995.

....and feature libraries are updated to connect the metadata values of the new image to the features detected in it. To extend the queries in configurations of objects, we employ the well known concept of 9 intersection, describing the major topological relations between areal, linear, and point [4]. Thus, we break the query for a spatial scene into two tasks: identifying where individual objects similar to the input exist in the database, and then identifying the combinations of these locations that satisfy the given topological relationship. INPUT METADATA LIBRARY FEATURE LIBRARY SEMANTIC ....

Egenhofer M. & R. Franzosa "On the Equivalence of Topological Relations", International Journal of Geographical Information Systems, Vol. 9, No. 2, pp. 133-152, 1995.

.... r) DC(diff(r; y) s) SEPNUM(y;N 1) 8z[MAX P(z; y) EC(z; s) As a special case, we can define the number of boundaries a region (other than u) shares with its complement: CBNUM(r;N ) j def SBNUM(r; compl(r) N ) These subdivisions of RCC relations bear resemblances to the work of (Egenhofer and Franzosa 1995), who refine a closely related set of base relations in similar ways. However, that work claims to produce a complete classification of topological relations between pairs of a specific kind of region, homeomorphic 4 to a 2 dimensional disc (i.e. any topologically distinct relation between a ....

....on to establish how the RCC axioms and variations on them relate to the sets of axioms used in the point based systems of point set topology. Looking more widely afield, the RCC axiomatic approach to topology for spatial reasoning can be contrasted with the constructive approach developed by (Egenhofer and Franzosa 1995) among others. While RCC begins with a set of axioms which are intended to capture some very general features of spatiality (or perhaps spatio temporality , since temporal models are mentioned in (Randell et al. 1992) the constructive alternative starts from a set of elementary objects ....

Egenhofer, M. J. and Franzosa, R. D.: 1995, On the equivalence of topological relations, International Journal of Geographical Information Systems 9(2), 133--152.

....Lee and Chin [15] developed an iconic interface for accessing and querying spatial databases. This interface lets the user graphically specify spatial relationships and query constraints with icons selected from a toolbar. In conjunction with the topological data model defined by Egenhofer [7], a development of the Lee Chin approach could give us a convenient, general purpose and simple to use interface for bioinformatics research. ffl Storage efficiency. SIERRA in its current form pays little attention to issues of space consumption. To handle gigabyte sized data sets, better on disk ....

....queries, and over time evolve into a full fledged 3 D spatial information system for bioinformatics. For example, SIERRA already extends the relational data model to handle one topological operator, that of spatial intersection. A comprehensive and rigorous topological model of spatial data [7] may simplify the addition CHAPTER 4. CONCLUSIONS 30 other spatial operations to SIERRA while retaining the practicality and relative portability of the relational data model as a low level implementation substrate. Appendix A SQL pseudocode LC is column of location codes at resolution N ....

M. J. Egenhofer and R. D. Franzosa. On the equivalence of topological relations. International Journal of Geographic Information Science (IJGIS), 9(2):133--152, 1995. BIBLIOGRAPHY 33

.... domain: the topological properties of and relations between sharplydelimited patches of the Earth s surface (as defined in legal and diplomatic documents for example) The domain is highly relevant to geographical information systems (GIS) related work includes (Worboys and Bofakos 1993, Egenhofer and Franzosa 1995). Much of the analysis transfers to planar regions, with possible applications in image interpretation, diagrammatic reasoning, and robotics. Human spatial competence includes the ability to acquire novel systems of spatial representation and reasoning for particular purposes. This occurs both on ....

Egenhofer, M. J. and Franzosa, R. D.: 1995, On the equivalence of topological relations, International Journal of Geographical Information Systems 9(2), 133--152.

....by the same motivations as those concerning the large body of work done on spatial relationships in spatial databases. In particular the various topological relationships that can exist between two specified spatial objects have been extensively investigated by Egenhofer and his collaborators [2, 3, 4, 5]. One way to think of our results is that we generalize these ideas to global topological properties of the entire spatial database [14] The issues are also relevant from a user interface point of view: a topologically invariant, lossless representation of the database corresponds to an interface ....

M. Egenhofer and R. Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 523--542, 1994.

....to avoid higher order logics does not mesh well with quantifying over sets of points. 4 COHN et al. Of course, it might be possible to adapt the conventional mathematicalformalisms for our purposes, and indeed this strategy is sometimes adopted (see, for example (Egenhofer and Franzosa 1991, Egenhofer and Franzosa 1995, Worboys and Bofakos 1993) However, because we take the view that much if not all reasoning about the spaces occupied by physical objects would not, a priori, seem to require points to appear in one s ontology, we do not follow this route but rather prefer to take regions as primitive and ....

.... Cui, Bennett and Gooday 1997, Bennett 1996a) is geographical information systems (GIS) In fact, a parallel development of a system very similar to RCC8 has taken place within this field (Egenhofer and Franzosa 1991, Egenhofer 1991, Egenhofer, Clementini and Di Felice 1994, Egenhofer 1994, Egenhofer and Franzosa 1995, Clementini, Di Felice and Oosterom 1994, Haarslev and Moller 1997) but firmly based on a point set theoretic approach rather than our logic of regions approach. The topological reasoning algorithm based on encoding RCC relations in I 0 (described in section 6.2 has been implemented as part of ....

Egenhofer, M. J. and Franzosa, R. D.: 1995, On the equivalence of topological relations, International Journal of Geographical Information Systems 9(2), 133--152.

....determined by A and B. The investigation of this use of connected components is due to Galton [Gal98] Another way of describing the relationship of A to B, which provides for finer distinctions than in Galton s technique, is a refinement of the 4 intersection model by Egenhofer and Franzosa [EF95]. A recent development in the study of relationships between pairs of regions considers how regions which are in some sense vague, may be related to each other. Two particular approaches are the work of Clementini and Di Felice [CF96,CF97] on regions with broad boundaries , and Cohn and Gotts ....

M. J. Egenhofer and R. Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9:133--152, 1995.

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M. Egenhofer and R. Franzosa, On the Equivalence of Topological Relations. International Journal of Geographical Information Systems 9(2): 133-152, 1995.

.... In order to establish topological relationequivalencebetween two regions (i.e. to decide whether or not two pairs of objects have the same topological relations) it is sufficient to describe topological invariants for the components (or separations) of the boundary boundary intersection [14] and the approach generalizes to line line and line region relations. The necessary invariants to consider for region region relations are: Journal of Visual Languages and Computing, Vol. 8, No. 4, pp. 403 424, 1997. the sequence of components counted in a consistent orientation of the plane ....

.... Detailed topological relations between two regions are expressed by the componentinvariant table for non empty boundary boundary sequences, which lists the sequence of boundaryboundary components and each component s dimension, type, crossing direction, boundedness, and complement relationship [12, 14]. Journal of Visual Languages and Computing, Vol. 8, No. 4, pp. 403 424, 1997. 4.3 Metrical Refinements Occasionally, topology per se is insufficient to characterize the essence of spatial relations. For instance, in order to capture the semantics of the spatial relation between Interstate I 95 ....

M. Egenhofer and R. Franzosa, "On the Equivalence of Topological Relations," International Journal of Geographical Information Systems, vol. 9, no. 2, pp. 133-152, 1995.

.... between two regions (i.e. to decide whether or not two pairs of objects have the same topological relations) it is sufficient to describe such invariants for the components (or separations) of the boundary boundary intersection only, since the other intersections can be inferred from them [7]. The necessary invariants to consider are the sequence of components counted along the boundaries; the dimension of each component; the type of boundary boundary component intersection (touch or cross) where crossing may be further refined depending on whether the component crosses into or ....

M. Egenhofer and R. Franzosa, "On the Equivalence of Topological Relations," International Journal of Geographical Information Systems, vol. 9, no. 2, pp. 133-152, 1995.

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M. Egenhofer, R. Franzosa, On the Equivalence of Topological Relations, International Journal of Geographical Information Systems, 9(2), pp.133-152 (1995)

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M. Egenhofer and R. Franzosa. On the Equivalence of Topological Relations. International Journal of Geographical Information Science, 9(2):133--152, 1995.

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M. Egenhofer and R. Franzosa, "On the Equivalence of Topological Relations," International Journal of Geographical Information Systems, vol. 9, pp. 133-152, 1995.

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M. J. Egenhofer and R. D. Franzosa. On the equivalence of topological relations. International Journal of Geographic Information Systems, 9(2):133-152, 1995.

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M. Egenhofer, R. Franzosa, On the Equivalence of Topological Relations, International Journal of Geographical Information Systems, 9(2), pp.133-152 (1995)

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M. J. Egenhofer and R. Franzosa. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9:133-152, 1995.

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Egenhofer M J and Franzosa R D. On the equivalence of topological relations. Int. J. Geographical Information Systems, 9:133--152, 1995.

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M J Egenhofer and R D Franzosa, `On the equivalence of topological relations ', International Journal of Geographical Information Systems, 9(2), 133-- 152, (1995).

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M. Egenhofer and R. Franzosa. On the equivalence of topological relations. Int. J. Geographical Information Systems, 9(2):133--152, 1995.

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