Chapter 5.  Maps and projections

Table of Contents

5.1 Map projections and their parameters
5.2 Transformation between projected coordinates
5.3 Substituting projections
5.4 Sheet labeling system of maps, the geo-reference provided by the labels

For the geographical information systems, maps are important data sources. In many cases, the map is represented by a scanned image, and the important data occurs as image information. Sometime we need to digitize a part of this information in vector format. To apply this information content it is needed to put this image into a pre-defined coordinate system, using the metadata and auxiliary information of the map. In this chapter we discuss the necessary meta- and auxiliary data and the methods to handle them.

Maps are planar, projected versions of the data on the base surface. Every map has a base ellipsoid, a datum (a representation of this ellipsoid), whose surface is projected to the plane of the map using some projection. The GIS packages usually know the equation information of the important map projection types. So, in this chapter we try to show the projections without to mention the projection equations.

5.1 Map projections and their parameters

For mapped representation, the surface of the Earth, the geoid, or rather its simplification, the ellipsoid should be projected to a plane. This procedure cannot be accomplished without distortion neither from the sphere, nor from the ellipsoid or from the geoid. Because of practical considerations (cf. Chapter 2), the geoid is not an input shape; the Earth is represented by sphere or ellipsoid in these computations. The procedure is called ‘projection’. The points of the surface of the sphere or the ellipsoid can be projected to a plane, to a cone or to a cylinder. The cone and the cylinder can be smoothed to the plane of the map (Fig. 21).

Earth with different projection types

Fig. 21. To project the sphere to a plane: cylindric, conic and planar projections.

Projections are realized by projection equations. These equations create the connection between the map plane coordinates (projected coordinates) and the spherical or ellipsoidal coordinates. The most general form of the projection equations is:

E=f1(Φ,Λ,p1,…,pn);

(5.1.1)

N=f2(Φ,Λ p1,…,pn).

(5.1.2)

where E and N are the projected coordinates of a point, p1...pn are the parameters of the projection.. Using this nomenclature (the Easting and Northing) we assume that the coordinates increase to east and to north, so the projected system has north-eastern orientation. This is true in most cases, however we discuss below the most important exceptions. The exact definition of the scale of the map is the number (usually much more less than one), which we have to multiply the resulted E and N coordinates with, to draw the map in the small piece of paper. As the Equations (5.1.1) and (5.1.2) are the direct projection equations, their inverse counterparts are

Φ=g1(E,N,p1,…,pn);

(5.1.3)

Λ=g2(E,N,p1,…,pn).

(5.1.4)

The mathematical form of the functions f1, f2, and g1, g2 are based on the type of the projection. Sometimes their form is quite complicated, in some cases they are implicit functions. However, in the GIS practice it is usually not necessary to work with these equations or even to know them – in most GIS packages or GPS receivers, they are pre-programmed. All we have to know is to handle them, giving them correct parameters. The projection equations can be assumed to be exact; the successive application of the direct and inverse projection equations provides the input coordinates with an error less than a millimeter.

Parameters p1,…,pn are based on the realized projection and their number n is a function of the projection type. In most cases n=5; however in some early projections e.g. the Cassini-Soldner, n=4; while in case of complicated ones, as the oblique Mercator or the oblique conic projections, n=6. These parameters should be known by the software – or, which is much more assuring, by ourselves. Let’s see, what parameters are needed for the projections.

Every projection has a so called projection origin or in other words, projection center. This point is the touching point of the plane/cylinder/cone and the ellipsoid. If the touching occurs along a line in cylindrical symmetric case (the central line of the projection), a point of this line should be assigned as projection center. The ellipsoidal latitude and longitude of this point are two necessary parameters.

The projected coordinates of the projection origin are the third and fourth mandatory parameters. As a default, they are both zeroes, however for practical considerations, they are often set to different values, e.g. to obtain positive or distinguishable coordinates throughout the mapped area. Because of this shift, these parameters are called FE (False Easting) and FN (False Northing), and they are expressed usually in meters.

A further, fifth parameter is the scale factor. In some cases, the plane/cylinder/cone is not placed in toughing but in secant position, in order to enlarge the low-distortion area around the projection center or around the central line of projection (Fig. 22). The scale factor is 1 in touching cases, and it is usually less than one, showing the reduction ratio. The only exception is Ireland, where the scale factor is more than one, because of historical reasons (to have a similar scale at the center as it was provided by the British system). In case of conic projections, instead of the scale factor, the standard parallels, where the cone cuts the base surface, can be also given.

Secant lines at different projections

Fig. 22. If the cone, the cylinder or the plane is placed to secant position, the length distortion is zero along the secant lines.

In case of oblique cylindrical projections, the projection origin is on the central line of the projection. It can be its farthest point from the equator (called Laborde projection) or its intersection with the equator (called Hotine projection). In general case, however, any point of the central line can be a projection center; that’s why a sixth parameter is needed for this projection type: the azimuth of the central line at the center (Fig. 23).

Position of the obliqe cylinder and the Earth at oblique Mercator projection

Fig 23. The tangent line is an oblique great circle in case of the oblique Mercator projection. The length distortion is minimum along this line.

It is important to mention that a projection means the type of the projection, but also its realized form, when the above mentioned parameters are fixed. On the maps, we see coordinates of realized projections.

The standards, describing the projecting procedures, often mention double projections. In these cases, the projection equations can be written in two steps: first projecting from the ellipsoid to an aposphere, then in the second projection, from the aposphere to the plane/cylinder/cone. Obviously, this was needed for the computations of the pre-computer age. In most cases, this raises no practical problems; the equations used in the GIS packages are good approximations of the double projections. If the centers of the two projections are not in the same place, the method of approximate projections (see below) can be applied.

There are a few tens of projection types that are used anywhere in the world. However, only a few of them are widespread used. Here we discuss the three most important ones; a cylindrical, a conic and a planar projection, the transversal Mercator, the Lambert conformal conic and the oblique stereographic ones, respectively. All of these discussed types are conformal projections.

At the transverse Mercator projection, the axis of the cylinder is in the plane of the equator. The origin of projection is at the equator. If the scale factor is the unity (e.g. in case of the former Warsaw Pact’s Gauss-Krüger projection or the WWII German military grid), the cylinder touches the base surface along a meridian, this is the central line of the projection. In this case, the low-distortion are, where the length distortions remain under 1/10000, expands to about 180 kilometers on both sides of the central line. It can be extended by applying a scale factor less than one: e.g. in case of the UTM (Universal Transverse Mercator), where k=0.9996. The False Northing is usually (but not exclusively!) set to zero, while the False Easting is defined to avoid negative coordinates, e.g. FE=500000 m.

In case of the Lambert conformal conic projection, the axis of the cone is in the semi-minor axis of the base ellipsoid. The central line of the projection is the parallel line where the cone touches the base surface (also known as normal parallel). The projection origin is a selected point of this parallel. This projection is usually used with reduction (scale factor is less than one). This projection can be defined by the projection origin and the scale factor or by the projection origin and the two parallels where the cone cuts the ellipsoid (standard parallels).

In case of the oblique stereographic projection (also known as Roussilhe-projection) we put a plane to a selected point of the base surface, perpendicular to its normal direction at that point, which is the projection origin. If the scale is unity, the low-distortion zone is a circle with a radius of about 127 kilometers around the projection origin.

In case of every projection, there is a zone with low distortion. As we have seen, in case of the transverse Mercator, it is a stripe along the central meridian, at the conformal conic projection this stripe is along the normal parallel, while it is a circle around the origin, using stereographic projection. If the mapped area extends beyond this range (in case of larger countries, or even the whole surface of the Earth), there usually are more projections defined with different origins, for different zones. The projection types are usually the same in these zones but the parameters are different, realizing different projections of course. In France, there are 4 zones of the Lambert conformal conic projection, each zone is elongated from west to east along the respective normal parallels. Using transverse Mercator projection, Austria defined 3, Germany 5 (prior to the territory losses, 7) zones, along central meridians. The zone system of Poland uses 4 stereographic and one transverse Mercator projections. These groups of projections, used as zones to map a larger area, are called projection systems.

In case of smaller countries, one zone is often enough to make low distortion maps. In the Netherlands, one stereographic projection is defined. The situation is similar in Romania, however this country is larger than the low-distortion area of the stereographic projection. The shape of the countries or regions suggests the projection type to be selected for the only zone. If the area is elongated from north to south (e.g. Chile, Portugal), the transverse Mercator projection is a good choice. For east-west elongated countries, e.g. Belgium or Estonia, the natural selection is the Lambert conformal conic projection. Switzerland and Hungary opted for oblique Mercator for the same reason, however for both countries the conic projection would have been also a good or even better possibility. The territory of Czechoslovakia between the two world wars could been mapped using one zone just by an oblique conic projection.

In the projected maps, the points of the same Easting or same Northing values are linear lines. The map grid or projection grid consists of these lines. The meridians and parallels are usually curves in the maps, just some distinct meridians and parallels can be linear. At every map points, there is an angle between the grid north and the geographic north; it is called the meridian convergence. Usually the meridian convergence is varying from place to place (an exception is the true Mercator projection where it is zero everywhere). We have to know, even seeing a low-scale map without projection grid indicated, that the invisible projection grid is there, behind the curves of parallels and meridians.