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# Topology and spatial relations

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Topology – or the study of places – (from the greek tόpos “place” and lόgos “study”) is one of the branches of modern mathematics; studies the properties of geometric shapes and forms that do not change when deformed without tearing, overlapping or gluing. It is based on the concepts of topological space, continuous function and homeomorphism.

Topology studies spatial relationships between geometric objects, thanks to it, in the “technology” GIS, we can arrange in a very sophisticated information. Indeed, we can divide the information on two levels: a geometrically dedicated solely to the geometries and topological level where we explain tables through some spatial relationships between these topological primitives.

The topology is based on three main concepts:

1. bow-knot: the arches can be connected only at the extremes;
2. left-right: the arches have a verse and then a right side and a left;
3. polygon-arc: the areas are defined as portions of the surface enclosed in polygons formed by arches.

To satisfy both the need for storage and representation of graphics data that the structuring of the connection with the alphanumeric data, using the models of storage of graphics primitives organized according to structures topologically defined. The models of storage are:

• the point, that depending on the scale can be used both to represent a position (such as the intersection of roads) that objects with dimensions not null (such as a tree); it is commonly packaged with a pair or a set of coordinates;
• arcs, represented as a broken or polygonal open, are used for the representations of linear elements and are stored as sequences of coordinates of the end nodes and their intermediate vertices;
• polygons, identified to represent closed areas, are represented with broken with the first and last vertices coincident, are stored as a sequence of arcs and lines that define them.

The connection between two arcs is represented by the fact of possessing a common node, the adjacency (or proximity) between two polygons is represented by the fact of having an arc in common.

Thanks to the topological structure, to features, are associated attributes allowing spatial analysis of the area we are studying; with the topology leave unchanged the spatial relationships between elements, namely:

• adjacency (or contiguous): Consider two polygons A and B, neighbors; whatever the manipulation that I make on A, the polygon B will continue to be adjacent to A. For example if deform the political map of Europe, Spain will continue to border with France and Portugal;

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• intersection: a polygon can be intersected by a linear element, for example. To give a practical example, consider a highway running through the territory of a certain town;

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• containment: Consider two polygons A and B; B contains A. For example the polygon Italy contains the polygon San Marino;

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• connectivity: if two linear elements are connected to each other, connectivity allows us to switch between them;
• relative position, once established a direction, a reference system, the rectangular polygon in the first instance will always be to the left of the hexagonal polygon B.

Il blog di Massimiliano Moraca ( Massimiliano Moraca ) / CC BY-SA 4.0