# Vector data

Vector data in corresponding vector models describe spatial objects using points. According to the Latin translation, a vector is a “carrier” or “driver” of geometric information. Vector data are represented by geometric entities such as

• Point and knot
• Lines and edges

as well as via coordinates (position / height, 2D / 3D), connections ( topology ), spatial properties (attributes) and display rules (color, line thickness, line type, symbols, area fill patterns, text heights, etc.). Neighborhood relationships, such as B. The start and end point of a line or surfaces adjacent to the points are taken into account. Vector data are used across the entire scale of geographic information systems, but are particularly common in the range from 1: 100 to 1: 10,000. A distinction is made between vector and raster data.

## Geometric entities

### Points and knots

The geometric information is represented in the vector model using points . Areas and lines, among others, are defined as higher structures and use the point as a basis. The coordinates of points are important in order to be able to derive geometric statements about higher structures, such as the length of connections, the surface area, the distance between geometric figures in the model, etc. In GIS and other surveying applications, on the one hand, the coordinates are make the points interesting and appropriate point attributes that are correlated with the point: e.g. B. Point height, point number, point type, point error, reliability and fixation. There is a requirement for vector models that points have a unique meaning in the model, i. H. a point may actually only appear once in the coordinate system. If so, there is topological integrity; H. the legal uniqueness of a point in terms of topology is called a node and thus fulfills the requirement that the point can only occupy a single position in space. If this is not the case, computationally correct and legally valid coordinates must be administered in parallel. Vector data can be displayed in topological form as a graph.

### Lines and edges

The points, which occupy different coordinates in the vector model, are connected to different structures by lines. These lines are edges that connect individual nodes and create a topological relationship between the points. The literature also speaks of an adjacency of nodes or of an incidence of edges. The structures mentioned can take multiple (geometric) shapes and create straight or circular connections. A common example of the representation of a topology of a transport network is z. B. the underground in London (Tube map).

The illustration shows the traffic point Baker Street, where three lines meet and play the role of nodes and the section between these nodes are edges. On the basis of this topological structure, a traffic network is designed and is used by GIS for displaying maps. The concept of edges and nodes is known in the literature as the edge-node structure and has its origin in graph theory. This is why it is increasingly used in geographic information systems, since the essential aspects of a model can be highlighted with a graph. For the use of geographic information systems, it is particularly helpful that, like points, the edges must be assigned a unique position.

Edges may not partially have the same course (uniqueness of edges). If you consider two edges that cross each other, but no knots are formed, the question of the dimension can also be important. Especially with z. B. Bridges and subway tunnels, the third dimension plays an important role and has not yet found its way into GIS. It is pointed out that something meaningful, three-dimensional is viewed with two-dimensional topological tools. The advantage of GIS is that it can also be used to model abstract worlds that cannot be perceived by the human eye.

### Mesh and mosaic

If the described elementary geographical structures point, line and surface are put together, complex structures emerge, which are called networks. Networks have several classification characteristics. A distinction is made between linear and flat networks, the latter also being called mosaic. There are also networks that combine the two, such as B. a tariff zone plan. The connections shown are a linear network, while the tariff zones above are considered to be an extensive network. In the case of mosaics, the aspect of the surface is rated differently. The subject of the network determines whether openings are useful or not. In addition, mosaics can be laid out across the board. Then every point belongs to exactly one area, so that there is neither overlapping of areas nor uncovered remaining areas. One speaks here of area division or partition. Linear networks can be connected as well as non-connected. Geodetic point networks z. B., for which an adjustment calculation is to be carried out, must be connected.

### The height in the vector model

Geographic information systems are models of the earth's surface that can only be projected locally into one plane. Even if a height is included for points, GIS are almost always only two-dimensional. The height is only a descriptive feature that is less important than the position coordinates. Measures to improve the quality of the third dimension consequently inevitably lead to the conventional GIS being abandoned.

## Basic geometric tasks of vector data

In the following, common geometric basic tasks for the practical use of vector data are presented. In general, this includes the geometrical position comparison, the geometrical intersection and the spatial transformation.

### Geometric comparison of positions

The determination of topological relationships, i.e. the positional relationships of nodes, edges and meshes or points, lines and polygons in topological systems that do not yet exist, is one of the most important tools that a GIS can provide using vector data. This basic function is particularly relevant for data acquisition, data tracking and analysis. For example, a point-in-polygon test is performed to find out whether a point is inside, outside, or on the edge of a polygon.

### Geometric intersection

With the geometric intersection of polygons (areas) with points, polygons with lines and polygons with each other, the individual location information of the respective structures is superimposed in order to answer various analytical questions: When points (e.g. groundwater quality measuring points in a federal state) with areas ( e.g. agricultural communities in this federal state) the correlation between agricultural use and the quality of the groundwater can be measured in order to adjust the use of fertilizers, for example. If lines intersect with an area, the question of how many power lines (lines) are on private land (area) can be answered. This can be interesting for dealing with possible damage cases and relevant for the question of which party has to pay for the costs. If areas and areas are intersected with one another, a GIS can use the vector data to determine, for example, how many agricultural areas are in a federal state (areas 1) sloping properties (areas 2). This can be essential for possible entitlements to agricultural compensation payments.

### Spatial transformation

The term spatial transformation encompasses the terms rotation (rotation), displacement (translation), scaling (enlargement or reduction) of vectors as well as the perspective to represent a three-dimensional object in a plane. Terrain models can thus be displayed in perspective in a GIS. The following briefly describes how these transformations can be represented mathematically in the Cartesian coordinate system . The rotation of a vector is represented by multiplying a rotation matrix, which consists of sums and products of the sine and cosine of the respective rotation angles. The displacement vector is added during the translation.

The scaling in the plane as well as in space is expressed by multiplying a matrix with a diagonal shape, with its individual elements being determined by the respective expansion factors of the coordinates. In summary, the spatial transformation, caused by rotation, translation and transformation, can be represented mathematically by multiplying and adding the respective displacement vector. Accordingly, nine parameters are relevant for this representation: three parameters for representing the rotation, three for displaying the scale and three for clarifying the displacement. In simplified terms, spatial transformations can be represented in a homogeneous coordinate system, since in this case they can be represented in only one matrix.

## literature

• Ralf Bill: Basics of geographic information systems . Wichmann, Bad Langensalza 2010, ISBN 978-3-87907-489-1 .
• Ralf Bill: Basics of geographic information systems . Analysis, applications and new developments. Wichmann, Heidelberg 1996, Volume 2, ISBN 3-87907-228-0 .
• Bibliographisches Institut (Ed.): Meyer's large pocket dictionary . New edition. Zechnersche Buchdruckerei, Speyer 1983, ISBN 3-411-02123-3 , p. 97, volume 23.
• Norbert Bartelme: GEO models-structures-functions . Springer, Berlin 1995, ISBN 3-540-58580-X .
• Stefan Lang and Thomas Blaschke: Landscape analysis with GIS . Eugen Ulmer, Stuttgart 2007, ISBN 978-3-8252-8347-6 .

Wiktionary: GIS  - explanations of meanings, word origins, synonyms, translations
Commons : Geographic information systems  - collection of images, videos and audio files

## Individual evidence

1. Bill (2010: 31)
2. Bartelme (1995: 61)
3. Bartelme (1995: 62)
4. Bartelme, (1995: 81)
5. Bartelme (1995: 82); Lang & Blaschke (2007: 60)
6. Bartelme; (1995: 82)
7. Bill; (1996: 28)
8. Bill; (1996: 101)
9. Bill; (1996: 102)
10. Bartelme; (1995: 89)
11. Bill; (1996: 102)