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• W28918241 abstract "Martin Erwig & Markus SchneiderFernUniversitat¨ Hagen, Praktische Informatik IV58084 Hagen, Germany erwig, markus.schneider@fernuni-hagen.de1 INTRODUCTIONNumerous applications in spatially-oriented disciplines like geography, cartography, and related areas, aswell as in computer-assisted systems like geographical information systems and spatial database systemswitness the importance of maps or spatial partitions. A map is a fundamental and well-known metaphorand a widely recognized geometric and topological structure that is capable of carrying a large amountof information.In a former paper (Erwig et al., 1997) we have layed the foundation for a formal treatment of spatialpartitions. This formal framework rests on three basic and powerful partition operations to which allapplication-specific operations known in the literature can be reduced. In this paper, we identify advancedoperations on maps and extend the formal framework accordingly.1.1 Maps RevisitedA spatial partition is a subdivision of the plane into pairwise disjoint regions where regions are sepa-rated from each other by boundaries and where each region is associated with an attribute having simpleor even complex structure. That is, a region with an attribute incorporates all points of a spatial par-tition having this attribute. A spatial partition implicitly models topological relationships between theparticipating regions which can be regarded as integrity constraints. First, it expresses neighborhood re-lationships where different regions may have common boundaries. This property is immediately visibleon a map. A second related aspect is that different regions of a partition are always disjoint (if we neglectcommon boundaries) so that a visual representation of a partition has a very simple structure and is easyto grasp.The basic idea for modeling a spatial partition is to map the Euclidean space to some label or attributetype, that is, regions of a partition are assigned single labels. Adjacent regions have different labels intheir interior, and a boundary is assigned the pair of labels of both adjacent regions.A number of application-specific operations has been defined on maps. The most important operation isoverlay which allows to arrange two partitions with different attribute categories on top of each other andto combine them through geometric intersection into a new partition of disjoint and adjacent regions. An-other operation is reclassify which retains the geometric structure of the spatial partition and transformsall or some partition attributes to new or modified attributes. The operation fusion is a kind of groupingoperation with subsequent geometric union. It merges neighbored regions of a partition with respect topartially identical attributes. The geometric union of all regions of a partition is formed by the opera-tion cover; it yields a result partition consisting of a single region. With the operation clipping we cancompute the intersection of a partition and a given rectangular window. The difference operation takestwo spatial partitions defined over the same attribute domain and computes the geometric difference ofSpatial Analysis 8a.3" @default.
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• W28918241 date "2000-01-01" @default.
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• W28918241 title "Formalization of Advanced Map Operations" @default.
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