# Gauss-Krüger coordinate system

The Gauß-Krüger coordinate system is a Cartesian coordinate system that enables sufficiently small areas of the earth to be located conformally ( angularly ) with metric coordinates (easting and northing) . It is a true-angle transverse cylinder image (transverse Mercator projection ).

## origin

The system was developed by Carl Friedrich Gauß , published by Johann Heinrich Louis Krüger and is mainly used in German-speaking countries. From 1935 to 2010 the Gauss-Krueger system was used throughout Germany as the basis for the representation of the triangulation results, later the Netherlands , Sweden and South Africa followed .

Much earlier, however, the military used the Gaussian triangular network . The Osnabrück region was from 1834 to 1847, the Emsland 1853-1860 and East Friesland in 1866 included , in the same year took over the Prussian General Staff trigonometric, topographic and cartographic works. Many official topographic maps , especially large and medium-sized scales , are based on the Gauß-Krüger coordinate system.

## construction

### Meridian stripes

A grid in the Gauß-Krüger coordinate system with a line spacing of 50 km is placed over a map of Germany.
The red lines mark the central meridians, the blue lines the edges of the meridian strips.

The grid of the geographical coordinates is divided into 3 ° wide meridian strips (a 6 ° division is also used). Each meridian strip runs parallel to its central meridian from the north to the south pole. The central meridians of neighboring meridian strips are accordingly 3 ° (or 6 °) apart.

Each meridian strip is given a code number (only with the Gauß-Krüger meridian strip system with 3 ° strips). This is derived from the number of degrees east of the central meridian (0 °, 3 °, 6 °, ...):

Code number = {0 °, 3 °, 6 °,…, 351 °, 354 °, 357 °} / 3 °.
Central meridian west longitude Eastern length
Longitude ... 9 ° 6 ° 3 ° 0 ° 3 ° 6 ° 9 ° 12 ° 15 ° ...
Code number ... 117 118 119 0 1 2 3 4th 5 ...

The meridian stripe is mapped onto a cylinder surface with the correct angle (conformal), the axis of which lies in the equatorial plane and the radius of which is equal to the meridian radius of curvature of the reference ellipsoid . The cylinder touches the figure of the earth along the central meridian.

#### overlap

According to a resolution of the Working Group of Surveying Administrations of the Federal Republic of Germany  (AdV) in 1966, each meridian strip has an extension of 1 ° 40 'on both sides and not half the width of 1 ° 30'. As a result, two adjacent strips overlap with a strip 20  minutes in length (on average about 23 km) wide. In this overlap zone, the coordinates in the respective meridian strip and in the adjacent meridian strip are specified for each point. So geodetic measurements and calculations are possible to a certain extent beyond the edge of the strip without changing the strip.

#### 6 ° wide stripes

Alternatively, a numbering system is used in Eastern Europe and Asia, which assigns zone 1 also the central meridian 3 ° east longitude, but uses zones with 6 ° latitude; Zone 0 is omitted in this convention; the highest zone code is 60 with the central meridian at 3 ° west longitude.

The central meridian of the Gauss-Krüger zone 21 in this system is not at 63 ° east longitude, but at 123 ° east longitude. The easternmost areas of Siberia are thus in zones 31 and 32 with the central meridians at 183 ° east longitude = 177 ° west longitude and 189 ° east longitude = 171 ° west longitude.

### Legal and high value

fictitious coordinate system with easting and northing .

One coordinate ( easting ) counts from the origin positive to the east, the other ( high value ) counts positively to the grid north . The letter Y is often used for easting and X is used for northing. When naming coordinates, they are usually given in the order easting and northing . Right and north values ​​are given in the SI unit of the meter . As in any Cartesian coordinate system, the right and north values ​​are read parallel to the axes and not to the arcuate lines of the longitude and latitude .

The origin of the coordinate system is:

The eastward value indicates the (distorted) distance from the central meridian to the point under consideration and the high value the distance from the equator or from the south pole on the true-to-length central meridian to the ordinate base.

The reference number of the central meridian or the meridian strip is written in front of the legal value; i.e. at the position of the seventh place in front of the decimal point.

#### False easting

In order to avoid negative easting values, a constant value is added to the easting value in the Gauß-Krüger meridian stripe system ( false easting ).

For Germany this is 500,000 m.

In the Austrian Federal Network  (BMN) - depending on the reference meridian - the following values ​​are added:

• Strip M28: 150,000 m
• Strip M31: 450,000 m
• Strip M34: 750,000 m.

#### example

The Paradeplatz in Mannheim has the following coordinates

• in degrees according to  WGS84 : 49 ° 29 ′ 13.6 ″ N / 8 ° 27 ′ 58.6 ″ E
• Gauß-Krüger: easting Y = 3461404 m, high value X = 5483498 m.

From the eastward value, the code number "3" shows that 9 ° East is used as the reference meridian. Since the “remainder” 461,404 is less than 500,000, it can be seen that the position is west of the reference meridian.

### Difference to UTM

The UTM coordinates - and the Gauss-meridian strip system use up to a scale factor , the same imaging equations (transverse compliant cylinder figure) to Verebnung the surface of the earth ellipsoid.

The main difference is that

• the Gauß-Krüger meridian strip system in Germany (also in Austria and other countries) is based on the Bessel or Krassowski ellipsoid and 3 ° wide meridian strips,
• while UTM coordinates refer to the WGS84 or GRS80 ellipsoid and use 6 ° wide meridian strips.

With this conformal type of mapping, the line distortions towards the outer edge of the stripes increase significantly with increasing stripe width :

${\ displaystyle {\ frac {1} {\ cos (y '/ R)}}}$

With

• the distance from the central meridian${\ displaystyle y '}$
• the radius of the earth .${\ displaystyle R}$

To compensate for the greater length and angle distortions at the zone edges caused by the wider meridian strips, a scale factor of 0.9996 is applied to the UTM system. The central meridian is shown shortened by the factor 0.9996 = 40 cm / km. With increasing distance from the central meridian to the east or to the west, this shortening decreases due to the growing image distortion within the zone; at a distance of about 180 km from the central meridian, the length distortion disappears.
In the case of the Gauß-Krüger coordinates, due to the only 3 ° wide stripe, such a scale factor is dispensed with, since the maximum distortions are still within the practically relevant accuracies or, for very precise requirements, with simple mathematical means (breaking off the necessary series expansion of the formulas after the second summand ) can be taken into account.

The UTM system also differs from the Gauß-Krüger system in terms of the naming of the strips and the coordinates. Since the UTM system was originally introduced as a reporting system for the American military , the designation here is plan-quadrant . The coordinate values ​​should be designated as East and North or East and North values to distinguish them from those of the Gauß-Krüger system (right and north).

Simplified summary:

Coordinate system Gauss-Kruger UTM
Width of the meridian strips 3 ° (or 6 °) 6 °
Reference ellipsoid Bessel and
Krassowski
WGS84 or
GRS80
Scale factor 1 0.9996
Coordinate values Legal and high value East and North values

## use

### In Germany

In German cartography and geodesy , the Bessel ellipsoid (in parts also the Krassowski ellipsoid) is used as the reference ellipsoid.

The spatial definition of the Bessel ellipsoid in relation to the earth's body - the positioning of the ellipsoid in the center of mass of the earth and its orientation to the earth's axis of rotation - was carried out for what was then Prussia with the help of the central point Rauenberg in Berlin. After its destruction, the central point of the network was mathematically transferred to the Helmert Tower in Potsdam , which is why the geodetic date of this system is often incorrectly referred to as the Potsdam date . This Rauenberg date is also the basis of the German Main Triangle Network (DHDN) .

In the GDR , the Krassowski ellipsoid was used as the basis. In the new federal states it was still used on a transitional basis, for example in Mecklenburg-Western Pomerania until around 2007 and in Saxony until 2014. In contrast, the state survey offices have switched from Gauß-Krüger to UTM coordinates since the 1990s . The aim of the change was to create a uniform geodetic reference system in the united Germany.

As an internationally standardized name are by the  OGC , the  EPSG ( European Petroleum Survey Group codes used). The following designations apply to the meridian strips used in Germany or in the area of ​​the former German Empire :

• 31466 for the meridian strip with the code number 2
• 31467 for the meridian strip with the code number 3
• 31468 for the meridian strip with the code number 4
• 31469 for the meridian strip with the code number 5

There are currently a number of spatial data services with incorrect (31492–31495) or old identifiers (31462–31465). The systems with the old and new ID differ in the order of the coordinate values:

• old: legal value, high value
• new: high value, legal value.

### In Russia

Traditionally, the Krassowski ellipsoid was used in the Soviet Union . This also largely applies to the successor states . How to use z. B. in Russia the Gauss-Krüger projection using the Krassowski ellipsoid.

### In Austria

In Austria , the date Austria is used for the Austrian Federal Registration Network , which is based on a shifted Bessel ellipsoid. The UTM coordinate system is increasingly being used by the authorities and other organizations, while the armed forces , based on NATO, also use the MGRS system.

In practice, the Gauß-Krüger coordinate system is mainly used. Accordingly, in civil engineering , hydraulic engineering and the like, Gauss-Krüger coordinates in CAD systems are used as a reference (e.g. the world coordinate system of the CAD software AutoCAD ). The coordinate system of the CAD software (e.g. world coordinates from AutoCAD) then corresponds to the Gauß-Krüger coordinate system.

### In Finnland

The topographic maps of Finland published between 1970 and 2005 (or nautical maps produced up to 2003 ) use the nationally unique YKJ coordinate system (yhtenäiskoordinaattijärjestelmä) .

The system indicates the location with an accuracy of one meter using two seven-digit numbers ( easting and northing ). It refers to the 27 eastern longitude with a Ostverschiebung (false easting) of 3,500,000 meters and a scale factor of 1. As a geodetic datum is reference ellipsoid from 1924 to Hayford used.

The YKJ coordinate system is currently being replaced by EUREF-FIN, the national implementation of ETRS89 .

## literature

• Walter Großmann: Geodetic calculations and images in the national survey . 3rd edition based on numerical calculations. Wittwer, Stuttgart 1976.
• Bernhard Heck: Calculation methods and evaluation models for national surveying. Classic and modern methods . 3., rework. u. exp. Edition. Wichmann, Karlsruhe 2003, ISBN 3-87907-347-3 .
• Bernhard Heckmann: Introduction of the ETRS89 / UTM position reference system when switching to ALKIS . In: Communications from the DVW Hessen-Thuringia . No. 1 , 2005, p. 17th ff .
• Louis Krüger: Conformal mapping of the earth ellipsoid into the plane . In: Publ. Kgl. Prussia. Geod. Inst . No. 51 , 1912 ( PDF ).
• Ralf Strehmel: Official reference system for the situation - ETRS89 . In: Brandenburg surveying . No. 1 , 1996, ISSN  1430-7650 ( PDF ).
• NIMA - National Imagery and Mapping Agency (Ed.): Department of Defense World Geodetic System 1984 . 3. Edition. January 2000 (Technical Report, TR 8350.2).
• Defense Mapping Agency (Ed.): The Universal Grids - Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) . September 1989 (DMA Technical Manual, DMATM 8358.2).
• Manfred Spata: The universal transversal Mercator image (UTM image) in the German national survey. In: Duisburger Forschungen, Volume 59, 2013, pp. 269–299.

## Individual evidence

1. ^ Book: CF Gauss and the land survey in Lower Saxony (1955) pp. 53–57
2. http://www.lung.mv-regierung.de/daten/lls_vortrag_10_06_08_holz_1.pdf
3. http://www.landesvermessung.sachsen.de/inhalt/etrs/etrs.html
4. Archived copy ( memento of the original from January 24, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.
5. Archived copy ( memento of the original from January 24, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.