Chapter 2. Planar and spatial coordinate systems

Table of Contents

2.1 Units used in geodetic coordinate systems
2.2 Prime meridians
2.3 Coordinate systems and coordinate frames

2.1 Units used in geodetic coordinate systems

It is an old tradition that in our maps the angles can be read in the degree-minute-second system. The whole circle is 360 degrees, a degree can be divided into 60 minutes, a minute can be further divide into 60 seconds, so a degree consists of 3600 seconds.

Along the meridians, the physical distances connected to the angular units – supposing the Earth as a sphere – are practically equal. Along a meridian, and using the first definition of the meter, one degree distance is 40,000 km / 360 degrees = 111.111 kilometers. One second along the meridian is a 3600th part of this distance, 30.86 meters; this is the distance between two parallels, one second from each other. Along the parallels, the similar distance is also a function of the latitude and the above figures should be divided (in case of spherical Earth) by the cosine of the latitude. At the latitude of Budapest (latitude: 47.5 degrees), a longitudinal degree is 75,208 meters, a longitudinal second is 20.89 meters.

However, the degree-minute-second system is not the exclusive one. In the maps of France and the former French colonies, e.g. of Lebanon, the system of new degrees (gons or grads) is often used (Fig. 2). A full circle is 400 new degrees. One new degree consists of 100 new minutes or 10,000 new seconds.

In many cases, the GIS software packages ask some projection parameters or other coordinates in radians. Radian is also the default angular unit of the Microsoft Excel software. The full circle is, by definition, 2π radians, so one radian is approximately 57.3 degrees and one radian is 206264.806 arc seconds (this is the so called σ”).

Degrees and grads as angular units in frame of a Lebanese map

Fig. 2. In the map of Leban, a former French colony, the latitudes and longitudes are given in degrees (internal frame) and also in grads (indicated by ’G’ in the external frame).

The standard international length unit is the meter. In the history, it had three different definitions. After the first one, both newer descriptions made it more accurate, keeping the former measurements practically untouched. First, the meter was introduced as the one ten millionth part of the meridian length between the pole and the equator. As this definition was far too abstract for everyday use, later a metric etalon was produced and stored in France as the physical representation of the unit. The countries have replicas of it and maintain their own national systems to calibrate all local replicas to the national ones. Nowadays, the new definition of the unit is based on quantum-physical constants that are as far from the everyday use as the first definition is. However, as it is calibrated exactly in the GPS system, it is more and more a part of our everyday life.

Using the replica system was not without side effects. During 1870s, in the newly conquered Alsace and Lorraine, the Germans connected the geodetic networks of Prussia and France. The fitting of the two systems showed an error around ten meters. Later it occurred, that French and Prussian networks was constructed using different meter replicas as scale etalons at the baselines. The length of the German metric etalon (brought also in Paris) was longer by 13.55 microns than the original French one. This makes no problem in the most cases, but in long distances, it counts: in a distance of several hundred kilometers, the error of ten meters occurs easily. The length of the German replica was later the definition of the ’legal meter’, which is 1.00001355 ’international’ meters. There is an ellipsoid (see point 3.2), called ’Bessel-1841-Namibia, used for the German survey of southwestern Africa (Namibia); its semi-major axis is the one of the Bessel-1841 ellipsoid multiplied by this counting number between the meter and the legal meter. Thus, the legal meter is also known as ’Namibia-meter’.

In the Anglo-Saxon cartography, different length units are also used. In the former Austro-Hungarian Monarchy, the basic unit was the ’Viennese fathom’ (Wiener Klafter). Table 1 shows the length of these units in meters.

Lenght unit

In meters

Legal meter

1.0000135965

Viennese fathom

1.89648384

Viennese mile

7585.93536

Toise

1.94906

Imperial foot

0.3047972619

US Survey foot

0.30480060966

Sazhen (Russian fathom)

2.1336

Russian Verst

1066.78

Table 1. Historical and imperial/US units in meters.