Searching spatial sense in the ontological world: discovering spatial objects
Abstract
The search of semantic understanding in information systems has led us to take advantage from fields like Natural Language processing from the Artificial Intelligence. This is the case of ontologies which have been used to solve many problems such as interoperability and integration. This paper presents a point of view over specific-domain ontologies to be used in geospatial activities which are named spatial ontologies. An overview of ontologies which could cover this need is presented, focusing on their strengths and weaknesses. In the last part we present the ontology to be used for the semantic integration of spatial schemas in the SIT-SD (Semantic Integration Tool for Spatial Data) prototype. This Variable-Level Spatial Ontology is being developed as part of the SIT-SD. In this ontology we have defined the main spatial characteristics to be used in the inference with spatial information.
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Sear q ching the Spatial Sense in the Ontological World: Discovering Spatial Objects V r illie Morocho s , t Llu´ıs u P´erez-V v idal w , t and F`elix v Saltor x y Departament de LSI-SI, Universitat Polit`ecnica z de Catalunya, Jordi { Girona 1-3. E-08034, Barcelona, Spain. | vmorocho,saltor } ~ @lsi.upc.es Departament de LSI-IG, Universitat Polit`ecnica de Catalunya, Av. Diagonal ,647. E-08028, Barcelona, Spain. [email protected] Abstract. The search of semantic understanding in information systems has led us to take advantage from fields like Natural Language processing from the Artificial Intelligence. This is the case of ontolo gies which have been used to solve many problems such as interoperability and integration. This paper presents a point of view over specific-domain ontologies to be used in geospatial activities which are named spatial ontologies. An overview of ontologies which could cover this need is presented, focusing on their strengths and weaknesses. In the last part we present the ontology to be used for the semantic integration of spatial schemas in the SIT-SD (Semantic Integration Tool for Spatial Data) prototype. This Variable-Level Spatial Ontology is being developed as part of the SIT-SD. In this ontology we have defined the main spatial characteristics to be used in the inference with spatial information. 1 Introduction The actual trend in the new generation of information systems is toward semantic understanding. In order to face this challenge, various aspects of knowledge and understanding between humans, users-machine and machines have been explored. Knowing that global knowledge is almost impossible to be achieved, it is much rational to divide it in domain-dependent sets of knowledge. Firstly, we present an overview of the de velopment of spatial ontologies. Secondly, we present the construction of one such ontology specially focused on the integration process of spatial database schemas. Kno wledge relative to semantic integrity constraints could be represented inside the ontology . For example: rules that do not allow a building to be intercepted by a street se gment; or rules that do not allow a constructed area of a building to be intercepted by another building. Many rules obvious for the human, as above, will be included as additional knowledge for the integrator tool. In Cyc ontology[10], among others, we can find some general knowledge. But in a specific domain, it will be necessary to define its specific knowledge.One way could be by meansof relationships. Reed et al.[12] present the insertion 1of geographicalentities, FIPS 10-4ontology,intoCyc. This is the 1We use the term insertion rather than mapping to avoid confusion with the terminology used in this paper
simplest form of ontolo gymapping in which the missingterms in the reference Cyc Ontology are easily identified and created. The Reed’s work is an insertion of “instances” of some geographic class. In opposite, we propose to increase the semantic knowledge at the type level. For instance, a province should belong only to “one” country. In the section 4 we make a review of some other approaches that have been developed in this direction. Spatial relationships give more semantic information than relationships from traditional databases. In order to take advantage of this kind of additional knowledge in the geospatial domain, it is necessary to represent it in the ontology in a suitable way. We use in theSIT-SD prototypethis kind of informationandrelationships.This information will be used in the assessment of semantic similarities between classes at the integration level of the federated schema. In this prototype, we have applied some algorithms. T wo cases have been covered in previous works [14],[9],[8]. The first case, we applied a word matching between object class and parts of object. The second case, we applied assessment of semantic-neighborhoodmatching. The third case, which is the main subject of this paper, appears when we apply an Variable-Depth Level Spatial Ontology (VDLSO). VDLSO is being developed as part of the Semantic Integration Tool for Spatial Data (SIT-SD) project [7], [15], where we ha ve defined the main spatial characteristics to be used in the work with spatial objects. In this case, two special subdivisions (from the point of view of spatial features) should be considered: material and immaterial objects. Material objects: it is impossible for tw o such objects to occupy the same space (a downtown area can not share its space with another “different” downtown area). Immaterial objects: such objects may occupy the same space (a limit line with a river, or the limit line of a province with a limit line of a country). The remainder of the paper is organized as follows. Section 2 presents the foundation of spatial ontologies, how a spatial ontology will be used in order to infer with spatial knowledge. Section 3 presents the main aspects to be considered in a developed of a spatial ontology. It show the main difference with another approaches. Section 4 briefly reviews some attempts of including spatial sense in some ontologies. Section 5 presents how we have designed and constructed a spatial ontology to be used in a SIT -SD project. Although the scope of this paper only is the spatial sense. Finally, conclusions and future work are presented in Section 6. 2 Foundation of Spatial Ontologies There are many definitions and concepts about of an ontology. We agree in that the main purpose of a ontology is to specify the intended meaning of a vocabulary, i.e. its underlying conceptualization[4]. Normally, the meaningof the ontologyterms, is taken from the consensus of a users community. In our case, we are talking about one particular spatial community. As spatial data, t we are considering data from applications like Geographic Information Systems, computer-aided design (CAD), robotics, image processing, all of which have at their core spatial objects that must be stored, queried, and displayed. Although in this paper, much of the work will be focused on geospatial knowledge. Moreover, the lines opened in this work could be continued toward appli-
cations in very different domains like for instance the microbiology where topological relationships are important too. The semantic problems appear when the integration of different sources is necessary .Semantic ¡ heterogeneity have been one of the classical problems in various integration approaches (Multidatabases, Distributed Databases, Federated Databases) and in our days continues to be a challenge to face. Figure 1 presents the main concept of inte ¢ gration with an ontology approach (extracted from Guarino[4]). A I (L) I (L) B M(L) £ Fig.1. Two systems A and B using the same language L can communicate only if the set of intended models ¤ ¥ ¦ § ¨ and © ª « ¬ associated ® to their conceptualizations overlap. ¯ ° ± ² , set of all models. From Guarino[4] From the spatial point of view, and specially from the domain of GIS, we should consider some additionalaspects in the integrationproblem.A GIS commonlyis an “inte gration” of many communities ³ in a complex system. For instance, a simple touristic map of the city could include data about of streets, transport(subway, subway station, b us route, bus stops), historical buildings, touristic buildings, interest points, gardens and so on. Often the correct design of the map is developed by layer ´ s2 µ . The layers will be: streetLyr, transportLyr, hbuindingLyr, ibuildingLyr, interest pointL yr, gardenL ¶ yr and so on. W · e could state that terms used in various layers keep an univocal ¸ sense, t therefore could use the same ontology. However, would it be possible to talk about terms with uni vocal sense for a whole GIS community or a spatial community? We think that an additional classification is necessary. Some general ontologies have adopted this approach, such as Cyc defines micr ¹ o theories. All this illustrates that there are many ways of considering the integration problem. The basic case appears when different communities need to communicate, see figure 1. In º this case, the communication is possible only if the intended models associated with the conceptualizationsdo overlap[4].On the otherhand, ina GIS communitya common conceptualization over all themes may not be possible (see section 3). Or perhaps, it e v xists only in a very high level of conceptualization. As result we obtain terms which are semantically close for a theme , t while semantically far for another theme . 2 » We take the concept of layer from GIS where the layer, commonly, contain similar characteristic objects
For instance, in a design of the same touristic map, for the people designing the streetLyr, t all plants could be classified as garden. Then, the conceptualization for all of them will be the same, and semantically will be close. Meanwhile, for the people that design the gardenLyr the classification will be completely different and with more detail. Obviously, both of them have the same main conceptualization, for both are plants. Then, when could we say that these plants are a garden, and when these plants are a forest, or when these plants are farming? Perhaps adding to the object not only the wor ¼ d sense, t but also the spatial sense (this will be explained in section 3). Let us leave this question in this point in order to complete the example. Normally, each local government cares for this kind of geographic information for each city (it also could be at state level, or country level, and so on). Now, suppose a tourist that need inte grated information from city A and the neighborhood city B (for example Barcelona and L’Hospitalet). In an utopian and wonderful world, the entire conceptualization of the city A GIS will correspond to city B GIS but this is impossible. This paperdoes not try to solve all problems derivedfrom this integrationprocess. It is the case of situations as, what happend when not correspond the layers from city A to city B, and so on. 3 ½ Main aspects to be represented in a spatial ontology In the representation of geographic information, it is necessary to take into account the main terms defined in this field and how they should be represented in the spatial ontology . (We have reviewed some works for designing our theory such as [13]). 3.1 ¾ Theme In º a GIS, the geospatial information corresponding to a particular topic is gathered in a theme . It is like defining the context or specific domain of the knowledge. The most common display of a theme is on a map. Maps of topography, railway network, road netw ¿ ork, city map and weather are examples of themes over a typical map. Certainly, most ontologies define certain themes in their structure. For example, in Cyc ontology there are micro theories. 3.2 ¾ Geospatial Objects In º order to represent entities of the real world, we should abstract up to the conceptual le À vel. A geospatial object corresponds to an entity. A theme is a collection of geospatial objects. Thus, if a geospatial object is formed by two components, the geospatial ontology must be a conjunction of word sense and spatial sense. –W · ORD SENSE: Usually a geospatial object is described by a set of descriptive attrib utes.It should be possible to make inferences with the name of a geospatial object in order to find the semantic of the word. For some years and until our days, most ontologies have been designed to cover enough this sense. Many researchers from the Natural Language processing field have left a valuable legacy that is presently improved to be used in almost all fields of information systems.
SemanticW eb, Semantic Integration, Semantic Understanding and so on. However, the most of these ontologies have few characteristics (or none of them) on spatial sense. –SP ATIAL SENSE: For anobject tobe consideredgeospatial,this shouldhavea spatial component. That is to say, if we are talking about a geospatial object, we must be capable of defining spatial attributes and spatial relationships. Certainly, in this case all entities in our real world could be represented by means of a geospatial object. This representation depends on the degree of transcendence of the object to be represented and, therefore, depends on the the importance of the object for this particular theme. In the beginning,informationsystems, and more specifically Data Base Á systems, only have represented the word sense of the entity. In geographical and spatial information,it should search the way for not losingthe additional intrinsic semantic information. Therefore, a suitable spatial ontology should be capable to express the intended meaning of the terms used by the geospatial community. The ontology can be limited to those structural relationships among terms that are considered relevant for the query. In order to avoid confusion with the terminology used in this paper, we should employ the term spatial as a more generic term for objects with spatial component(i.e. such object could be associated with a location relative to the Earth). W · e could conclude from this analysis that, a spatial ontology should cover the follo À wing main aspects: Â Be Á able to reason or to infer about spatial concepts considering both parts of spatial objects. Although not necessarily with sufficient effectiveness, even if this happens only at the first stage. Ã Takes advantage from engines specialized in spatial reasoning. Ä Links ¶ with various databases, thesaurus, and so on, that contain specific information. Å Supports the query capabilities for spatial information from an ontology. This does not mean queries over the data as such, but to be capable to enrich the query Æ semantically. The main semantic information to be represented in a spatial ontology should be the following: Ç T OPOLOGICAL RELATIONSHIPS Position Attribute: abo ve, adjacent, below, vertical, horizontal, left, right, near, on. Spatial ¡ Relationship: inFrontOf, inBackOf, connect, between, distance, cross, through. È DIMENSION É Measur Ê e: length, À area, angle, and more physical quantity Ë SHAPE Dealing with spatial primitives like line, point, polygon, circle. Ì REFERENCE Í SYSTEM Latitude and longitude, elevation, altitude. Î GEOPOLITICAL Ï SUBDIVISION For instance, country contains provinces, country contains states, and so on. At this point we believe that a spatial ontology that takes advantage of spatial characteristics could aid to solve semantic heterogeneity. For example, if we can infer that the limit of apark Ð normally Ñ is an str Ò eet;orasubway Ò station is always Ó near,o t rof- ten to ov Ô er the subway Ò b ut at a different altitude Õ ;atunnel Ö is a railway tunnel if a
railway crosses × through it, or it is a subway Ò tunnel if it o Ø verlaps a subway Ò ,o t ri s a con Ù ventional tunnel if ¢ o Ú verlaps a r Û oad. In all of them there are spatial Ò objects with topological Ü relationships besides afr Ý equency f actor which let us assess the similarity among spatial objects. 3.3 ¾ The problem of Multi representation and Multi resolution in a spatial ontology Ø The problem derived from the modeling of our world is the difference in the ways to abstract it. In the modelingprocess many problemsof subjectivity can be included.This fact could yield a wall impossible to cross in the interoperability and integration process. One way to face these problems is, to start the model from a common ontology. Which · means every designer should depart from the same knowledge base. Obviously this is an utopia. From the point of view of geospatial objects, there could be additional problems in the representation of spatial sense. Multi representation and multi resolution À in geographic objects have been challenges faced by many researches [16]. For instance, ¢ a city could be represented by a point. But in other abstraction, the same city could be represented by a polygon . Different shape , t from the spatial sense that we are talking in this paper, could be associated to the abstraction of the city. It will depend of the needand the possible use ofthe information.Therefore,a spatial ontologyshould be able to deal with multi-r ¹ epresentation and multi-r ¹ esolution of spatial object. We explain in the section 5 how to face this problem. 4 Reviewing ontologies to be used in spatial inference There are many attempts to developontologies as a tool for dealing with spatial objects. W · e make a brief analysis of Cyc[5], WordNet[6] and SUMO[11]looking for extensions to add “spatial knowledge” to the ontology. 4.1 OpenCyc Started in the mid eighties by D. Lenant (Microelectronics and Computer Consortium). It is, maybe, the most commonly used ontologyin our days, with enough commonsense kno Þ wledge to support natural language. Due mainly to consistency problems coming from a unique huge knowledge base, it was necessary to separate this in “microtheories”. Usually, a microtheory is a knowledge domain. However, when it must deal with se veral microtheories, it is not clear how Cyc solves the inconsistency. This is the case of GIS integration discussed in section 2. OpenCyc, the public version of Cyc technology ,now in release 1.0should include close to 6,000concepts and60,000 assertions. At the present time, it covers 90% of the expectations. It supplies other simple categories by means of links to synset structure of WordNet[6]. Cyc ß has a graph-like structure where the root is Thing from which Individual -> SpatialThing -> SpatialThing-Localized is derived, see figure 2 (of course we are taking into account only what is of interest to us). Then Cyc defines as SpatialThing “The collection of all things that have a spatial extent or
location relative to some other SpatialThing or in some embeddingspace ...”. But when we ask for any spatial thing like for example street à , t it only mentions “street is located inside ¢ a city” but without spatial attributes because it is only a sentence. It is also true that it uses common-language terms and locutions. We propose the representation of these spatial relationships and attributes inside the ontology suitable to be used in the inference ¢ process. SpatialThing (from Level −1) á SpatialThing−Localized (from Level −1) á FixedStructure (from Level Z0) GeographicalThing (from Level −1) á Thing â Individual Fig.2. Basic ã structure of Cyc from which start our extension 4.2 WordNet “WordNet is an on-line lexical reference system whose design is inspired by the psycholinguistic theories of human lexical memory”. It was developed by the Cognitive Science Laboratory (Princenton University) led by G. Miller [6]. It carries a lexical database with which it is capable to distinguish among the syntactic categories of noun, verb, adjective, and adverb. It could processes natural language although a bigger database will be necessary for a satisfactory result. In its 92 version, it included approximately ¢ 95,000 different word forms (51,500 simple words and 44,100 collocations) or ganized into almost 70,100 word meaning, or sets of synonyms. W · ordNet was designed for processing natural language, for which it is a well-suited tool, but this ontology presents lacks on the spatial sense . Thus, WordNet processing is enough v to cover the wor ¼ d sense needs of our work, but for the spatial sense we should look for other solutions.
It is interesting to refer to the EDR Electronic Dictionary too, which was designed to deal with natural japanese language by Yokoi[17]. Cyc and WordNet deal with the english v language. 4.3 SUMO an attempt toward Spatial Ontology Created ß by the IEEE Standard Upper Ontology (SUO) working group. The Suggested Upper ä MergedOntology [11] is an attempt to link categories and relations coming from dif ferent top level ontologies. The main focus of this ontology was over Semantic Web area. It appear with important goals and started with the possible spatial sense. There are attempts to deal with spatial information, but the work has stopped (and curiously we can not find news about the ontologies described above in their websites). SUMO presentstwoparticularontologies,one forgeographyandtheotherfortransportation. If we inspect its inner structure, we could find signs of these spatial attempts. But it will be impossible to satisfy the needs presented previously. Therefore, we take adv antage of a few characteristics developed here to launch our own vision of spatial ontolo gy. 5 Constructing the Spatial Ontology Re Í viewing the ontology classification made by Guarino in [4], [3], we collocate our w ork as a Variable-Depth Level Ontology It º results of a high-le å vel ontology , when we are working with general knowledge, and low-le ´ vel ontology , when we are working with detailed information. This approach is the way more natural of manage geospatial knowledge. Later we will explain our point of view. In order to formalize our concepts, we abstract the geographical world in some levels. First, the Spatial Object instance (e.g. Barcelona, Diagonal Avenue, Catalunya Square) in which the spatial object is initialized with values. Second, Spatial Object (e.g. city, way, garden) which was defined above (section 2). And third, Spatial Object Class ß (e.g. city class, street class, green area class) where the spatial objects are grouped by a particular characteristic and behavior. Furthermore, æ we define theme class as a set of spatial classes which depend of a domain-specific ç ontology (e.g. tourism theme class). An instance of tourism theme class could be tourism of Barcelona. Thus, the geospatial world will be defined as the set of theme classes which depend on a geospatial ontology, or geospatial knowledge base which is the result of a consensus among members of the geospatial community. In º figure 3 we show our approach. In º our approach, a spatial object could belong to more than one theme class. Although, the meaning of the term used to define this spatial object, will depend on the ontology behind it. F æ or instance, the spatial object city Ù could belong to some theme classes b ut the meaning and the representation will be different (remember the problem of multiresolution and multi-representation discussed in section 3.3). It is clear, that the upper part of the ontology for the city Ù term in the tourism theme is irrelevant. And only a few levels of g è eneralization are necessary (such as Localized-Thing and Spatial-Thing é , t
–Based on experience, we have deduced that a tcm should have at least one spatial Ò object Ø member of the Street class. Therefore, it is possible to find a spatial class member of Axis class or CenterLine class3 . –In every tcm, a spatial Ò object member of centerLine class al ways crosses the boundaries of the map at least one time. The process to discover the spatial objects will be divided in the following steps: –By the first axiom there should be at least one basic spatial class for each theme. And this class is the easiest to recognize. –It is possible to apply topological rules from this basic class in order to discover the most common classes. –The number of the most common classes in a spatial system of a certain theme is limited and is known. –It is possible to continue increasing the knowledge in the ontology, with new topological À rules and new spatial objects in order to recognize more specialized classes in ¢ a spatial system. From æ our example to recognize a touristic city map, the process would be as follo À wing: 1. Searching the base axis for streets: Any map with streets should have the base axis. Normally ð , the distance between two consecutive crossings in the same line should be around 100m. Often, a line is not crossed by any other more than one time. 2. Searching of parcels: Often, a parcel is enclosed by a set of streets. 3. Searching buildings: Normally, a bounding is enclosed by a parcel. Topological rules between feature classes Everything leads to define rules between spatial objects. Normally these rules should be implemented in GIS tools in order to a void data inconsistence at the time of input data. The same rules could be part of our ontology . Thus, a process of feedback could be defined to take advantage from the fact that these same rules will be defined in the process of input. These rules will be grouped by the main geometry class which is part of the rule. The tables 3, 4, 5 show the rules ( = point, =line, =Polygon geometry type)4. In order to avoid confusion with term used in some software products, we take the definition of feature class from OpenGIS and ESRI. F eature is ¢ the representation of a real-w orld object on a map. a group of spatial objects which together represent a realw orld entity (e.g. a complex feature is a road network). F eature class is ¢ a collection of geographic features with the same geometry type (such as point, line, of polygon), the same attributes, and the same spatial reference. That is, taking our definition of spatial objects, the geometryrepresentation of the spatial sense. Therefore,feature class allo w homogeneous features to be grouped into a single unit. From the point of view of our ontology, the feature class will be the spatial objects with the same parent.Inour ontology (figure 6), road and its children will belong to the same feature class. 3 These two later classes are the base lines when a map is digitalized. 4This tables were based in the topology rules from some GIS software products
All rules could be o verload, t that is say, if the spatial object have different shape representation, it could be the same restriction. Thus, if a building ( ) is representedby polygon the rule should be, related to parcel ( ), (table 5). On the other hand, if it is represented by a point the rule should be from table 3 Thus, the rules for the main spatial objects in a touristic map will be well defined as follo wing: ! " # # $ " % & ' ( ) * + , - . / 0 1 0 2 3 4 5 6 7 8 9 : ; < = = > < ? : @ A B C D E F G H I J K J L M N O P Q H R S T U V W X X Y W Z U [ \ ] ^ _ ` a b c d e f g h g i c j k l m n n o m p k q r s t u v w x y z { | } ~ } y This knowledge was introduced in the ontologyby means of rules in the proprietary language of Cyc. By means of inference could be discover the spatial objects in the our e xample. 6 Conclusions and future work Inthisworkwe havepresentedthefoundationsof spatialontologies,taking intoaccount the main differences with classical ontologies. In our case, we have faced the multiresolution and mluti-representing of spatial objects by a division in ontolo gy levels, thus as representing this ontology by means UML diagrams. It take advantage of a tool which convert Cyc structure in a XMI file capable to be used by UML tools. We ha ve developed a variable level ontology based in Cyc structure capable to use spatial characteristics for inferring. Thus, the ontology generated can work with the spatial sense and wor d sense. Such behavior is the main difference with another approaches. This ontologywas developedas part of Semantic IntegrationTool forSpatial Data (SITSD) prototype.A formalizationof topologicalrules also havebeen presented.By means of this rules the ontology could be enriched with much knowledge. It knowledge is used in order to discover spatial objects by means of their spatial characteristics, in contrast with the word sense of classical ontologies. As future work, it is necessary to adapt a process into geographical tools (such as ArcGIS) where the technical people is introducing topological rules. This rules will feed the ontology in order to obtain a real applicable ontology capable to incorporate enough knowledge for applications. All of which have at their core spatial knowledge. Acknowledgments Part of this work has been supported by: Politechnic University of Catalonia; FEDER and MCYT under the project TIC2001-2099-C03-01.Special acknowledgments to European Environment Agency and ESRI Spain for their valuable contributions for this research. References 1. K. A. Borges, C. A. Davis, and A. H. Laender. OMT-G: An object-oriented data model for geographic applications. GeoInformatica , 5(3):221–260, Sep 2001.
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Table 3. Topology rules, point related, between feature classes Let ¡ ¢ £ ¤ £ ¥ be three pairs of spatial objects belonging to point, line and polygon class respectively. Define ¦ as § a function to able to return the feature line or point if exist self orv ¨ erlap. The meaning of he symbol in the left part of the table when there are two equal symbols (e.g. © ª « ¬ ) is that the each feature belong to different feature class. Therefore, at the z0 level are different parents. Obviously, different geometry type will lead to different feature class. Thus, be ® ¯ a function that return the feature class of the spatial object. For example, from the ontology at figure 6 ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ . Then: Point Rules À Á Â Ã Ä point inside polygon Å Æ Ç È É Ê Ë Ê Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å e.g. Country Capital must be inside each country æ ç è é ê point be covered by boundary of polygon ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ - Utility service points might be required to be on the boundary of a parcel point be covered by endpoint of line ! " # $ % & ' ( ) * + , - . / 0 1 2 3 1 4 5 6 6 7 8 9 : ; < = > ? @ A A - Street intersection must be covered by the endpoints of street centerline B C D E F point must be covered by line G H I J K L M L N O P Q R S T U V W X Y Z [ \ ] - Bus Stop/Station must fall along Bus Route
Table 4. Topology rules, line related, between feature classes Line Rules ^ _ ` lines must not overlap a b c d e d f g h i j k b l m n o p q r s t u v w x y z { | } } ~ - Sidewalk borders normally are lines cannot overlap. lines must not intersect ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â Ã Á Ä Å Æ Ç È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Õ Ø Ù Ú Û Ü Ý Þ ß à á â ã ã ã ã - Segments could never cross or occupy the same space with other lines. ä å æ lines must not have pseudo-nodes ç è é ê ë ì í î ï ð ê é ê ð î è ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ ! " # $ % & ' ( ) * + + - Segments of a river system might to only have nodes at endpoints of junctions. , - . lines must not self overlap / 0 1 2 3 4 5 6 7 2 8 5 9 6 : ; 0 < = > ? @ A B C D E F G H I J K L M - For transportation analysis, street and highway segments of the same object should no overlap themselves. N O P lines must not self intersect Q R S T U V W X Y Z [ T V V R \ ] ^ _ ` a b c d e f g h i j - Contour lines cannot intersect themselves. k l m lines must not intersect or touch interior n o p q r s t q u u v w x y z { | } ~ ¡ ¢ £ ¤ ¥ ¦ § ¨ © © ª « ¬ ® ¯ ° ± ² ° ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¾ ¾ ¾ - When the lines should touch at their ends and not intersect of overlap. ¿ À Á Â Ã line must not overlap with another line Ä Å Æ Ç È Ç É Ê Ë Ì Í Î Å Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ , ß where à á â ã ä å æ ç è é ê ë ì - Highway can cross and come close to rivers, but road segments cannot overlap their segments í î ï ð ñ line must be covered by another line ò ó ô õ ö ÷ ø ÷ ù ó ú û ü ý þ ÿ , ß where - Lines that make up bus routes must be on top of lines in a road network. line end point must be covered by point ! " # $ % & ' & ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 7 : ; < = > ? @ A B C C D E F G H I J J - End points of secondary electric lines must be capped by either a transformer or meter. K L M N O line must be covered by boundary of polygon P Q R S T U V U W X Y Z [ \ ] ^ _ ` a b c d e f g h - Polylines used for displaying block and lot boundaries must be covered by parcel boundaries.
Table 5. T i opology rules, line related, between feature classes Polygon Rules j k l polygons must not overlap m n o p q p r s t u v w n x y z { | } ~ , ß where e.g. A voting district map cannot have any overlaps in its coverage polygons must not have gaps Let, ¡ ¢ £ ¤ ¥ ¦ § ¨ ¨ ¨ © ª where « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ ¿ ¿ À Á Â Ã Ä Å Æ , ß and Ç È É Ê universe set Then: Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ì í î ï ð ñ - Soil polygons that cannot include gaps nor forms void. ò ó ô õ ö polygon contains point ÷ ø ù ú û ü ý þ û ü ÿ - Parcels must contains at least one address point polygon boundary must be covered by line ! " # $ % & ' ( ) * + , - - Major road lines form part of outlines for census blocks . / 0 1 2 polygon must cover to another polygon 3 4 5 6 7 8 9 4 : ; < = > ? @ A B C D E F G - Autonomous Communities cover to Provinces. H I J K L Must be covered by M N O P Q R S R T N U V W X Y Z [ \ ] ^ _ ` a b - Provinces must be covered by Autonomous Communities c d e f g polygon must not overlap with another polygon h i j k l k m n o p q r i s t u v w x y z { | } ~ , where - Lakes and land parcels from two different feature classes must not overlap. Must cover each other ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° , ß where ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ - Vegetation and soils must cover each other. Where: Objects are indicated by upper-case letters (e.g. ¾ ¿ À ), Á their boundaries are denoted as  à , and their interior area as Ä Å (therefore Æ Ç È É Ê Ë Ì Í ). The boundary’s point object is considered to be always empty (hence the point is equivalent to its interior), and the boundary’s line is comprised of its two endpoints. A function called Î Ï Ð , is used to return the dimension of an object, and returns 0 if the object is a point, 1 if it is a line, or 2 if it is a polygon.
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