# Gauss-Krüger

## Description

Gauss-Krüger is also known as an ellipsoidal version of the transverse Mercator projection because it is similar to the Mercator projection, except that in Gauss-Krüger, the cylinder touches the sphere or ellipsoid along a meridian instead of along the equator. The result is a conformal projection that does not maintain true directions. The central meridian is placed in the center of the region of interest. This centering minimizes distortion of all properties in that region. This projection is best suited for north-south areas.

The spherical version of the projection was presented by Johann H. Lambert in 1772. The first formulas with ellipsoidal correction were developed by Carl F. Gauss in 1822. The Gauss-Krüger name refers to the ellipsoidal form reevaluated by Louis Krüger in 1912. Gauss-Krüger coordinate systems and the Universal Transverse Mercator (UTM) coordinate system are based on this projection while the State Plane Coordinate System uses it for all north-south zones. Various countries use this projection for their topographic maps and large-scale coordinate systems. It is available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 8.0 and later.

## Projection properties

The subsections below describe the Gauss-Krüger projection properties.

### Graticule

Gauss-Krüger is a transverse cylindric projection. The equator and central meridian project as straight lines. Other meridians project as complex curves concave toward the central meridian. Other parallels are also complex curves, concave toward the nearest pole. Both poles are projected as points. The graticule is symmetric across the equator and the central meridian. The graticule is limited to 45 degrees away from the central meridian due to the math instability.

### Distortion

Gauss-Krüger is a conformal map projection. It generally does not maintain true directions, but angles and shapes are maintained at infinitesimal scale. Distances are accurate along the central meridian if the scale factor is 1.0. If it is less than 1.0, there are two approximately (when using an ellipsoid) straight lines with accurate scale equidistant from and on each side of the central meridian. Area, distance, and scale distortions rapidly grow with the distance from the central meridian or two standard lines as specified above. Distortion values are symmetric across the equator and the central meridian.

## Usage

This projection is appropriate for mapping large-scale or smaller areas with north-south predominant extents. It is very commonly used. Many countries use it for their topographic maps and large-scale coordinate systems. Gauss-Krüger coordinate systems, the Universal Transverse Mercator (UTM), and State Plane all use this map projection.

## Limitations

The Gauss-Krüger projection is limited to project data only within 45 degrees from the central meridian due to the math instability. In fact, the extent on a spheroid or ellipsoid should be limited to 10 to 12 degrees on both sides of the central meridian. Beyond that range, data projected may not project back to the same position. Data on a sphere does not have these limitations.

To map larger data extents, two implementation variants of transverse Mercator are available in ArcGIS: Transverse Mercator complex, available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 9.0 and later, and Transverse Mercator NGA 2014 available in ArcGIS Pro 1.2 and later and ArcGIS Desktop 10.4 and later.

## Parameters

Gauss-Krüger parameters are as follows:

• False Easting
• False Northing
• Central Meridian
• Scale Factor
• Latitude Of Origin

## Gauss-Krüger coordinate systems

The Gauss-Krüger coordinate system is a specialized application of the Gauss-Krüger projection and is used in Eurasia, including Russia and China. It divides the world into zones 6 degrees wide. Each zone has a scale factor of 1.0 and a false easting of 500,000 meters. The central meridian of zone 1 is at 3 degrees east. Some places also add the zone number times 1 million to the 500,000 false easting value. Gauss-Krüger zone 5 could have a false easting value of 500,000 or 5,500,000 meters. Three-degree Gauss-Krüger zones exist also.

## Sources

Snyder, J. P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper 1395. Wash

Snyder, J. P. and Voxland, P. M. (1989). An Album of Map Projections. U.S. Geological Survey Professional Paper 1453. Washington, DC: United States Government Printing Office.

The Universal Grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS). Defense Mapping Agency Technical Manual 8358.2 (1989). Accessed March 19, 2021. https://apps.dtic.mil/dtic/tr/fulltext/u2/a266497.pdf

The Universal Grids and the Transverse Mercator and Polar Stereographic Map Projections. NGA.SIG.0012_2.0.0_UTMUPS (2014). Accessed on April 19, 2021. https://earth-info.nga.mil/