Rectangular Ribbons and Generalized Topological Relations

In Geographical information system (GIS), the management of geographic objects is very important, especially not only for urban and environmental planning, but also for any application in a rural environment. In other hand, in GIS, there are different relationships between geographic objects, such as topological, projective, metric, etc. for topological relations, if the scale of the map is changed and if some geographic objects are generalized, not only the shapes of those objects will change, but also their topological relations can vary according to scale. In addition, a mathematical framework which models the variety of this category of relationships does not exist. In the first part of this paper, a new topology is presented based on rectangular ribbons which are defined through a longish rectangle; so, a narrow rectangular ribbon will become a line and then will disappear. In this paper, the different components of this mathematical framework are discussed, as the evolution of geographic objects according the scale. For each case, some assertions are defined which formulate the transformation of the topological relationships into other ones, when downscaling. To conclude this paper, after having proposed an algorithm of displacement to maintain some topological relationships, an experimental example is discussed.