Topological information are the most important kind of qualitative spatial information. Current formalisms for the topological aspect of space focus on relations between regions or properties of regions. This work provides a qualitative model for representing the topological internal structure of complex regions, which could be of multiple pieces and or have holes and islands to any finite level. We propose a layered graph model for representing the internal structure of complex plane regions, where each node represents the closure of a connected component of the interior or the exterior of a complex region. The model provides a complete representation in the sense that the (global) nine-intersections between the interiors, the boundaries, and the exteriors of two complex regions can be determined by the (local) RCC8 topological relations between the associated simple regions. Moreover, this graph model has an inherent hierarchy which could be exploited for map generalization.
The notion of composite region is first introduced in @cite_6 , where two components are allowed to meet at more than one points. This implies that the region in Figure (c) is considered as a composite region. This is the only difference between ours and that of @cite_6 .
@cite_6
2951895579 5053679 12110447 1689232663
We consider the non-crossing connectors problem, which is stated as follows: Given n simply connected regions R_1,...,R_n in the plane and finite point sets P_i subset of R_i for i=1,...,n, are there non-crossing connectors y_i for (R_i,P_i), i.e., arc-connected sets y_i with P_i subset of y_i subset of R_i for every i=1,...,n, such that y_i and y_j are disjoint for all i different from j? We prove that non-crossing connectors do always exist if the regions form a collection of pseudo-disks, i.e., the boundaries of every pair of regions intersect at most twice. We provide a simple polynomial-time algorithm if the regions are axis-aligned rectangles. Finally we prove that the general problem is NP-complete, even if the regions are convex, the boundaries of every pair of regions intersect at most four times and P_i consists of only two points on the boundary of R_i for i=1,...,n. The 4-intersection, a model for the representation of topological relations between 2-dimensional objects with connected boundaries and connected interiors, is extended to cover topological relations between 2-dimensional objects with arbitrary holes, called regions with holes. Each region with holes is represented by its generalized region—the union of the object and its holes — and the closure of each hole. The topological relation between two regions with holes, A and B, is described by the set of all individual topological relations between (1) A ’s generalized region and B’s generalized region, (2) A ’s generalized region and each of B’s holes, (3) B’s generalized region with each of A ’s holes, and (4) each of A ’s holes with each of B’s holes. As a side product, the same formalism applies to the description of topological relations between 1-spheres. An algorithm is developed that minimizes the number of individual topological relations necessary to describe a configuration completely. This model of representing complex topological relations is suitable for a multi-level treatment of topological relations, at the least detailed level of which the relation between the generalized regions prevails. It is shown how this model applies to the assessment of consistency in multiple representations when, at a coarser level of less detail, regions are generalized by dropping holes. There are many situations in GIScience where it would be useful to be able to assign a region to characterize the space occupied by a set of points. Such a region should represent the location or configuration of the points as an aggregate, abstracting away from the individual points themselves. In this paper, we call such a region a ‘footprint' for the points. We investigate and compare a number of methods for producing such footprints, with respect to nine general criteria. The discussion identifies a number of potential choices and avenues for further research. Finally, we contrast the related research already conducted in this area, highlighting differences between these existing constructs and our ‘footprints'. We develop a theory for describing composite objects in physics. These can be static objects, such as tables, or things that happen in spacetime (such as a region of spacetime with fields on it regarded as being composed of smaller such regions joined together). We propose certain fundamental axioms which, it seems, should be satisfied in any theory of composition. A key axiom is the order independence axiom which says we can describe the composition of a composite object in any order. Then we provide a notation for describing composite objects that naturally leads to these axioms being satisfied. In any given physical context we are interested in the value of certain properties for the objects (such as whether the object is possible, what probability it has, how wide it is, and so on). We associate a generalized state with an object. This can be used to calculate the value of those properties we are interested in for for this object. We then propose a certain principle, the composition principle, which says that we can determine the generalized state of a composite object from the generalized states for the components by means of a calculation having the same structure as the description of the generalized state. The composition principle provides a link between description and prediction.