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2 Projections

Before creating your own project region (Location) it is important to consider which projection will be used. The following chapters will give a brief introduction to the most common projections, their properties, and parameters.

Figure 10 shows the steps involved in deriving a map projection from real spatial data.

Figure 10: Projection of the earth's surface on a map according to (7)

1 Geoid

A more precise definition of the earth's shape, especially of the heights can be made via the identification of the geoid. This considerably more complex physical calculation results from the quantity of the earth masses, which occur differently strong from region to region and therefore have different gravitation forces, which affect on the earth's shape. Thus, the geoid represents the gravitation field of the earth. In case of the view in Figure 10 the shape of the earth looks really 'distorted'. Due to the mathematical complexity the earth's shape is usually displayed by ellipsoids in Geographic Information Systems.

2 Ellipsoid

The simplified acceptance of the earth in spherical shape (sphere) is not exactly enough for creating maps in higher scales than 1:2 Mio. Rotation ellipsoids or else spheroids try to readjust the complex form of the earth mathematically as exactly as possible. Thereby, the distance between the poles and the geocenter is smaller than between the equator and the geocenter (see Fig. 10).
There is a range of ellipsoid models existing that should give optimized results for the different regions of the earth. In general, a sufficient exact basic for the localization of the situation will be possible.

Table 2: Dimensions of some internationally used ellipsoids (data rounded) and examples of their usage location according to (7)

Earth dimension according to	Semi-major axis (m)	Semi-minor axis (m)	Usage location
Bessel 1841	6377397	6356079	Germany, Chile, Netherlands, Sweden, ...
Clarke 1880	6378249	6356515	Afrika, France
Hayford 1909	6378388	6356912	Belgium, Finland, Italy, Spain, ...
WGS 1984	6378137	6356752	North America, worldwide

3 Datum

There are numerous surveying points that are indicated as Datum and that can be calibrated by their heights data. The following table contains a few examples for global and also regional datums. The versions GRASS 5.4 6.0support datum transformation (see Fig. 6).

Table 3: Some datums with their general application

Datum	Region	Point of origin	Ellipsoid
WGS 84	Global	Mass center of the earth	WGS 84
NAD 1983	North America, the Caribbean	Mass center of the earth	GRS 80
European 1950	Europe, North Afrika	Potsdam	International

4 Map projection types

In order to transfer the 3 dimensional form of the earth into a 2 dimensional plane (to project), a projection is needed.

Figure 11: Various projection models (cylindric, conic and azimuthal)

Depending on the regional situation different models are available (see Fig. 11) in order to hold the unavoidable distortions as low as possible.

Cylindric Figure: This is the simplest of these figure variants. In this figure the map plane at the equator is lain around the globe in order to create a cylinder. Meridians and parallels are projected on the plane in such a way that a rectangular grid is created during the 'phase out'. This figure is especially used for displaying regions near to the equator. A transverse variant is also common in other regions.

Conic Figure: If a cone is lain over the earth and is unrolled in the plane, a conic figure is created. With the simplest and most frequent application the apex lies on one line with both geographical poles and illustrates the pole nearby. Thence, the meridians go off in the same angle and the parallels form concentric circles around the intersection. The cone contacts the globe on one or two parallels, the standard parallels. This form is frequently applied to figure regions in the middle latitudes.

Azimuthal Figure: Here, the map plane is put on the earth's shape as tangent. Imagine a source of light in infinite distance at the opposite side shining through the globe and projecting the shadows of the parallels and meridians on the map plane.

The basic projections represented here can additionally be varied by the situation of the illustration plane to the earth. Normal (0^ to the axis of the earth), oblique (45^ to the axis of the earth) or diagonal axial (90^ to the axis of the earth).

5 Choosing the projection type

The decision for a projection depends on the illustration properties needed in the project. Despite the fact that a map projection is generally never displayable without distortions of one or several characteristics (area, form, distance, scale, direction or relation) it must be specified before, which properties have priority at the planned usage of the geodata.

Orthomorphic Projection: The scale remains the same from one point in all directions. The meridians and parallels intersect by 90^. The scale is locally the same and thus the form of the areas remain. Additionally, the angles between the lines remain unchanged. These maps are mainly used in the navigation and the surveying technology.

Equal-area Projection: Here, the area dimensions remain scaled as well as the proportional relations of the areas also remain. Scale, form and angle are distorted. The meridians do not intersect the parallels in the right angle. But this is not very important concerning smaller areas. These projections are often used for land use mapping and population mapping as well as for other researches that are related to one specific area.

Equidistant Projection: The distance between sites on the map remains undistorted on equidistant illustrations. This is especially important for traffic maps.