# Rectangular Ribbons and Generalized Topological Relations

Brahim Lejdel (Univeristy Echahid Hamma Lekhder of El-Oued, El-Oued, Algeria) and Okba Kazar (Smart Laboratory, University of Biskra, Biskra, Algeria)
DOI: 10.4018/IJAEIS.2016040104

## Abstract

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.
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## 1. Introduction

Now, for any application in environmental planning, not only new software products must be created, but overall a new conceptual framework must be set by integrating new notions as rectangular ribbons, evolution of geographic objects and mutation of topological relationships according the scale. Taking all those characteristics into account, the important scientific question of this work is to develop a mathematical framework in order to deal with geographic object management in a robust and consistent form. Thus, one of the problems is the link between topological relationships and scale, the topological relations between geographic objects can vary according to scale. When somebody is saying « this road runs along the sea », what are exactly the spatial or geographic relations which are concerned? Sometimes, either the road touches the sea or a small beach is located between the road and the sea, etc. From a mathematical point of view, mostly there is a disjoint relation between the road and the sea whereas for people the relation is different. In addition, when one is reading a map, according to scale, the topological relation can be different, disjoint or meet.

Suppose a decision-maker wants to create a new motorway running along a lake with the help of a computer. Taking this consideration into account, any reasoning system will generate difficulties because the spatial relations hold differently.

Another problem comes from mathematical modeling of streets and rivers. Often, they are considered as linear objects even if they have some areas. By considering a road as a line or as an area, topological relationships can be different. In order to solve this problem, the concept of rectangular ribbon will be developed. Depending on the scale, or more exactly on visual acuity and granularity of interest, a rectangular ribbon will be a longish rectangle (area), a line or will disappear. In other words, rectangular ribbons can be seen as an extension of polylines. Moreover, in order not to be stuck to cartography, the concept of granularity of interest will be introduced.

The Polyline used in actual GIS cannot represent well the reality because the rivers, streets or the road have a width that must be taken into account. To represent more exactly the reality, another representation of these objects must be given in order to get a more robust model. In Figure 1, we show the difference of representation of longish object such as rivers, by two different models; Rectangular ribbons and Polyline. In this figure, the representation that uses the rectangular ribbons shows all the detail of the object as the area and the borders of the river (see Figure 1a), whereas the second representation that uses the Polyligne illustrates only the general shape of the object (see Figure 1b). Thus, we choose the rectangular ribbon representation because it represents more exactly the reality.

Figure 1.

Two representations of the same object (river)

This paper will be organized as follows. First, definitions and a state of the art review for cartographic generalization will be given, and a second review will examine topological relations. Then, rectangular ribbons and ribbon topology will be defined. And a model will be detailed in which according to visual acuity, geographic objects will be generalized. Finally, we present a conclusion and future work.

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