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In cartography, the term **Gauss–Krüger**, named after Carl Friedrich Gauss and Johann Heinrich Louis Krüger,^{ [1] } is used in three slightly different ways.

**Cartography** is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.

**Johann Carl Friedrich Gauss** (; German: *Gauß*[ˈkaɐ̯l ˈfʁiːdʁɪç ˈɡaʊs]; Latin: *Carolus Fridericus Gauss*; was a German mathematician and physicist who made significant contributions to many fields in mathematics and sciences. Sometimes referred to as the *Princeps mathematicorum* and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and is ranked among history's most influential mathematicians.

**Johann Heinrich Louis Krüger** was a German mathematician and surveyor. He became director of the Prussian geodetic institute in 1917 and wrote several books on geodesy, operational and theoretical. In 1912 he presented his "Konforme Abbildung des E，rdellipsoids in der Ebene", one of the works that led to the 1923 Gauss–Krüger coordinate system and projection.

- Often, it is just a synonym for the transverse Mercator map projection. Another synonym is
*Gauss conformal projection*. - Sometimes, the term is used for a particular computational method for transverse Mercator: that is, how to convert between latitude/longitude and projected coordinates. There is no simple closed formula to do so when the earth is modelled as an ellipsoid. But the
*Gauss–Krüger*method gives the same results as other methods, at least if you are sufficiently near the central meridian: less than 100 degrees of longitude, say. Further away, some methods become inaccurate. - The term is also used for a particular set of transverse Mercator projections used in narrow zones in Europe and South America, at least in Germany, Turkey, Austria, Slovenia, Croatia, Bosnia-Herzegovina, Serbia, Montenegro, Macedonia, Finland and Argentina. This
*Gauss–Krüger*system is similar to the universal transverse Mercator system, but the central meridians of the Gauss–Krüger zones are only 3° apart, as opposed to 6° in UTM.

A **map projection** is a systematic transformation of the latitudes and longitudes of locations from the surface of a sphere or an ellipsoid into locations on a plane. Maps cannot be created without map projections. All map projections necessarily distort the surface in some fashion. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. There is no limit to the number of possible map projections.

In geography, **latitude** is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or *parallels*, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the *geodetic latitude* as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six *auxiliary latitudes* which are used in special applications.

**Longitude**, is a geographic coordinate that specifies the east–west position of a point on the Earth's surface, or the surface of a celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter lambda (λ). Meridians connect points with the same longitude. By convention, one of these, the Prime Meridian, which passes through the Royal Observatory, Greenwich, England, was allocated the position of 0° longitude. The longitude of other places is measured as the angle east or west from the Prime Meridian, ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. Specifically, it is the angle between a plane through the Prime Meridian and a plane through both poles and the location in question.

The **Mercator projection** is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for nautical navigation because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as straight segments that conserve the angles with the meridians. Although the linear scale is equal in all directions around any point, thus preserving the angles and the shapes of small objects, the Mercator projection distorts the size of objects as the latitude increases from the Equator to the poles, where the scale becomes infinite. So, for example, landmasses such as Greenland and Antarctica appear much larger than they actually are, relative to landmasses near the equator such as Central Africa.

A **geographic coordinate system** is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined three-dimensional Cartesian vector. A common choice of coordinates is latitude, longitude and elevation. To specify a location on a plane requires a map projection.

The **Ordnance Survey National Grid reference system** is a system of geographic grid references used in Great Britain, distinct from latitude and longitude. It is often called **British National Grid** (**BNG**).

The **transverse Mercator** map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the UTM. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

The **scale** of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways. The first way is the ratio of the size of the **generating globe** to the size of the Earth. The generating globe is a conceptual model to which the Earth is shrunk and from which the map is projected.

The **equirectangular projection** is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, and circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia and NASA World Wind, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth.

**ED50** is a geodetic datum which was defined after World War II for the international connection of geodetic networks.

The **Universal Transverse Mercator** (**UTM**) conformal projection uses a 2-dimensional Cartesian coordinate system to give locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, i.e. it is used to identify locations on the Earth independently of altitude. However, it differs from that method in several respects.

The **universal polar stereographic** (**UPS**) coordinate system is used in conjunction with the universal transverse Mercator (UTM) coordinate system to locate positions on the surface of the earth. Like the UTM coordinate system, the UPS coordinate system uses a metric-based cartesian grid laid out on a conformally projected surface. UPS covers the Earth's polar regions, specifically the areas north of 84°N and south of 80°S, which are not covered by the UTM grids, plus an additional 30 minutes of latitude extending into UTM grid to provide some overlap between the two systems.

The **State Plane Coordinate System** is a set of 124 geographic zones or coordinate systems designed for specific regions of the United States. Each state contains one or more state plane zones, the boundaries of which usually follow county lines. There are 110 zones in the contiguous US, with 10 more in Alaska, 5 in Hawaii, and one for Puerto Rico and US Virgin Islands. The system is widely used for geographic data by state and local governments. Its popularity is due to at least two factors. First, it uses a simple Cartesian coordinate system to specify locations rather than a more complex spherical coordinate system. By using the Cartesian coordinate system's simple XY coordinates, "plane surveying" methods can be used, speeding up and simplifying calculations. Second, the system is highly accurate within each zone. Outside a specific state plane zone accuracy rapidly declines, thus the system is not useful for regional or national mapping.

**Jordan Transverse Mercator (JTM)** is a grid system created by the Royal Jordan Geographic Center (RJGC). This system is based on 6° belts with a Central Meridian of 37° East and a Scale Factor at Origin (mo) = 0.9998. The JTM is based on the Hayford ellipsoid adopted by the IUGG in 1924. No transformation parameters are presently offered by the government. However, Prof. Stephen H. Savage of Arizona State University provides the following parameters for the projection:

**Israeli Transverse Mercator** is the new geographic coordinate system for Israel. The name is derived from the Transverse Mercator projection it uses and the fact that it is optimized for Israel. ITM has replaced the old coordinate system ICS. This coordinate system is sometimes also referred as the "New Israeli Grid". It has been use since January 1, 1994.

In cartography, a **conformal map projection** is one in which any angle on Earth is preserved in the image of the projection, i.e. the projection is a conformal map in the mathematical sense.

The article Transverse Mercator projection restricts itself to general features of the projection. This article describes in detail one of the (two) implementations developed by Louis Krüger in 1912; that expressed as a power series in the longitude difference from the central meridian. These series were recalculated by Lee in 1946, by Redfearn in 1948, and by Thomas in 1952. They are often referred to as the Redfearn series, or the Thomas series. This implementation is of great importance since it is widely used in the U.S. State Plane Coordinate System, in national and also international mapping systems, including the Universal Transverse Mercator coordinate system (UTM). They are also incorporated into the Geotrans coordinate converter made available by the United States National Geospatial-Intelligence Agency. When paired with a suitable geodetic datum, the series deliver high accuracy in zones less than a few degrees in east-west extent.

In 1989 Bernard Russel Bowring gave formulas for the Transverse Mercator that are simpler to program but retain millimeter accuracy. Bowring rewrote the fourth order Redfearn series in a more compact notation by replacing the spherical terms, i.e. those independent of ellipticity, by the exact expressions used in the spherical transverse Mercator projection. There was no gain in accuracy since the elliptic terms were still truncated at the 1mm level. Such modifications were of possible use when computing resources were minimal.

The **Central cylindrical projection** is a perspective cylindrical map projection. It corresponds to projecting the Earth's surface onto a cylinder tangent to the equator as if from a light source at Earth's center. The cylinder is then cut along one of the projected meridians and unrolled into a flat map.

The **Gauss–Boaga projection** is a map projection used in Italy that uses a Hayford ellipsoid.

- ↑ "Convert Gauss-Kruger (GK) coordinates to Latitude/Longitude" . Retrieved 3 October 2016.

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