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The **Universal Transverse Mercator** (**UTM**) is a system for assigning coordinates to locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, which means it ignores altitude and treats the earth as a perfect ellipsoid. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the *x*, *y* coordinate in that plane. The projection from spheroid to a UTM zone is some parameterization of the transverse Mercator projection. The parameters vary by nation or region or mapping system.

- History
- Definitions
- UTM zone
- Overlapping grids
- Latitude bands
- Latitude bands 2
- Notation
- Exceptions
- Locating a position using UTM coordinates
- Simplified formulae
- See also
- References
- Further reading

Most zones in UTM span 6 degrees of longitude, and each has a designated central meridian. The scale factor at the central meridian is specified to be 0.9996 of true scale for most UTM systems in use.^{ [1] }^{ [2] }

The National Oceanic and Atmospheric Administration (NOAA) website states the system to have been developed by the United States Army Corps of Engineers, starting in the early 1940s.^{ [3] } However, a series of aerial photos found in the Bundesarchiv-Militärarchiv (the military section of the German Federal Archives) apparently dating from 1943–1944 bear the inscription UTMREF followed by grid letters and digits, and projected according to the transverse Mercator,^{ [4] } a finding that would indicate that something called the UTM Reference system was developed in the 1942–43 time frame by the Wehrmacht. It was probably carried out by the Abteilung für Luftbildwesen (Department for Aerial Photography). From 1947 onward the US Army employed a very similar system, but with the now-standard 0.9996 scale factor at the central meridian as opposed to the German 1.0.^{ [4] } For areas within the contiguous United States the Clarke Ellipsoid of 1866^{ [5] } was used. For the remaining areas of Earth, including Hawaii, the International Ellipsoid ^{ [6] } was used. The World Geodetic System WGS84 ellipsoid is now generally used to model the Earth in the UTM coordinate system, which means current UTM northing at a given point can differ up to 200 meters from the old. For different geographic regions, other datum systems can be used.

Prior to the development of the Universal Transverse Mercator coordinate system, several European nations demonstrated the utility of grid-based conformal maps by mapping their territory during the interwar period. Calculating the distance between two points on these maps could be performed more easily in the field (using the Pythagorean theorem) than was possible using the trigonometric formulas required under the graticule-based system of latitude and longitude. In the post-war years, these concepts were extended into the Universal Transverse Mercator/Universal Polar Stereographic (UTM/UPS) coordinate system, which is a global (or universal) system of grid-based maps.

The transverse Mercator projection is a variant of the Mercator projection, which was originally developed by the Flemish geographer and cartographer Gerardus Mercator, in 1570. This projection is conformal, which means it preserves angles and therefore shapes across small regions. However, it distorts distance and area.

The UTM system divides the Earth into 60 zones, each 6° of longitude in width. Zone 1 covers longitude 180° to 174° W; zone numbering increases eastward to zone 60, which covers longitude 174°E to 180°. The polar regions south of 80°S and north of 84°N are excluded.

Each of the 60 zones uses a transverse Mercator projection that can map a region of large north-south extent with low distortion. By using narrow zones of 6° of longitude (up to 668 km) in width, and reducing the scale factor along the central meridian to 0.9996 (a reduction of 1:2500), the amount of distortion is held below 1 part in 1,000 inside each zone. Distortion of scale increases to 1.0010 at the zone boundaries along the equator.

In each zone the scale factor of the central meridian reduces the diameter of the transverse cylinder to produce a secant projection with two standard lines, or lines of true scale, about 180 km on each side of, and about parallel to, the central meridian (Arc cos 0.9996 = 1.62° at the Equator). The scale is less than 1 inside the standard lines and greater than 1 outside them, but the overall distortion is minimized.

Distortion of scale increases in each UTM zone as the boundaries between the UTM zones are approached. However, it is often convenient or necessary to measure a series of locations on a single grid when some are located in two adjacent zones. Around the boundaries of large scale maps (1:100,000 or larger) coordinates for both adjoining UTM zones are usually printed within a minimum distance of 40 km on either side of a zone boundary. Ideally, the coordinates of each position should be measured on the grid for the zone in which they are located, but because the scale factor is still relatively small near zone boundaries, it is possible to overlap measurements into an adjoining zone for some distance when necessary.

Latitude bands are not a part of UTM, but rather a part of the military grid reference system (MGRS).^{ [7] } They are however sometimes used.

Each zone is segmented into 20 latitude bands. Each latitude band is 8 degrees high, and is lettered starting from "C" at 80°S, increasing up the English alphabet until "X", omitting the letters "I" and "O" (because of their similarity to the numerals one and zero). The last latitude band, "X", is extended an extra 4 degrees, so it ends at 84°N latitude, thus covering the northernmost land on Earth.

Latitude bands "A" and "B" do exist, as do bands "Y" and "Z". They cover the western and eastern sides of the Antarctic and Arctic regions respectively. A convenient mnemonic to remember is that the letter "N" is the first letter in "northern hemisphere", so any letter coming before "N" in the alphabet is in the southern hemisphere, and any letter "N" or after is in the northern hemisphere.

The combination of a zone and a latitude band defines a grid zone. The zone is always written first, followed by the latitude band. For example, (see image, top right), a position in Toronto, Ontario, Canada, would find itself in zone 17 and latitude band "T", thus the full grid zone reference is "17T". The grid zones serve to delineate irregular UTM zone boundaries. They also are an integral part of the military grid reference system.

A note of caution: A method also is used that simply adds N or S following the zone number to indicate North or South hemisphere (the easting and northing coordinates along with the zone number supplying everything necessary to geolocate a position except which hemisphere). However, this method has caused some confusion since, for instance, "50S" can mean southern hemisphere but also *grid zone* "50S" in the northern hemisphere.^{ [8] } There are many possible ways to disambiguate between the two methods, two of which are demonstrated later in this article.

These grid zones are uniform over the globe, except in two areas. On the southwest coast of Norway, grid zone 32V (9° of longitude in width) is extended further west, and grid zone 31V (3° of longitude in width) is correspondingly shrunk to cover only open water. Also, in the region around Svalbard, the four grid zones 31X (9° of longitude in width), 33X (12° of longitude in width), 35X (12° of longitude in width), and 37X (9° of longitude in width) are extended to cover what would otherwise have been covered by the seven grid zones 31X to 37X. The three grid zones 32X, 34X and 36X are not used.

- Europe
- Africa
- South America
- Bering Sea with Alaska

A position on the Earth is given by the UTM zone number and the easting and northing planar coordinate pair in that zone. The point of origin of each UTM zone is the intersection of the equator and the zone's central meridian. To avoid dealing with negative numbers, the central meridian of each zone is defined to coincide with 500000 meters East. In any zone a point that has an easting of 400000 meters is about 100 km west of the central meridian. For most such points, the true distance would be slightly more than 100 km as measured on the surface of the Earth because of the distortion of the projection. UTM eastings range from about 167000 meters to 833000 meters at the equator.

In the northern hemisphere positions are measured northward from zero at the equator. The maximum "northing" value is about 9300000 meters at latitude 84 degrees North, the north end of the UTM zones. In the southern hemisphere northings decrease southward from the equator, set at 10000000 meters, to about 1100000 meters at 80 degrees South, the south end of the UTM zones. For the southern hemisphere, its northing at the equator is set at 10000000 meters so no point has a negative northing value.

The CN Tower is at 43°38′33.24″N79°23′13.7″W / 43.6425667°N 79.387139°W , which is in UTM zone 17, and the grid position is 630084 m east, 4833438 m north. Two points in Zone 17 have these coordinates, one in the northern hemisphere and one in the south; one of two conventions is used to say which:

- Append a hemisphere designator to the zone number, "N" or "S", thus "17N 630084 4833438". This supplies the minimum information to define the position uniquely.
- Supply the grid zone, i.e., the latitude band designator appended to the zone number, thus "17T 630084 4833438". The provision of the latitude band along with northing supplies redundant information (which may, as a consequence, be contradictory if misused).

Because latitude band "S" is in the northern hemisphere, a designation such as "38S" is unclear. The "S" might refer to the latitude band (32°N–40°N) or it might mean "South". It is therefore important to specify which convention is being used, e.g., by spelling out the hemisphere, "North" or "South", or using different symbols, such as − for south and + for north.

These formulae are truncated version of Transverse Mercator: flattening series, which were originally derived by Johann Heinrich Louis Krüger in 1912.^{ [9] } They are accurate to around a millimeter within 3,000 km of the central meridian.^{ [10] } Concise commentaries for their derivation have also been given.^{ [11] }^{ [12] }

The WGS 84 spatial reference system describes Earth as an oblate spheroid along north-south axis with an equatorial radius of km and an inverse flattening of . Let's take a point of latitude and of longitude and compute its UTM coordinates as well as point scale factor and meridian convergence using a reference meridian of longitude . By convention, in the northern hemisphere km and in the southern hemisphere km. By convention also and km.

In the following formulas, the distances are in kilometers. In advance let's compute some preliminary values:

First let's compute some intermediate values:

The final formulae are:

where is Easting, is Northing, is the Scale Factor, and is the Grid Convergence.

Note: Hemi=+1 for Northern, Hemi=-1 for Southern

First let's compute some intermediate values:

The final formulae are:

- Military grid reference system, a variant of UTM designed to simplify transfer of coordinates.
- Transverse Mercator projection, the map projection used by UTM.
- Universal Polar Stereographic coordinate system, used at the North and South poles.
- Open Location Code, a hierarchical zoned system
- MapCode, a hierarchical zoned system

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* that are used in special applications.

The **Mercator projection** is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred in marine navigation because ships can sail in a constant compass direction for long stretches, reducing the difficult, error-prone course corrections that otherwise would be needed frequently when sailing other kinds of courses. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator but accelerates with increasing latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

In mathematics, a **3-sphere**, or **glome**, is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere, the boundary of a ball in four dimensions is a 3-sphere. A 3-sphere is an example of a 3-manifold and an n-sphere.

In navigation, a **rhumb line**, **rhumb**, or **loxodrome** is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north.

In mathematics and physics, *n*-dimensional **de Sitter space** is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an *n*-sphere.

The **Gudermannian function**, named after Christoph Gudermann (1798–1852), relates the circular functions and hyperbolic functions without explicitly using complex numbers.

In geodesy, **conversion** among different **geographic coordinate** systems is made necessary by the different geographic coordinate systems in use across the world and over time. Coordinate conversion is composed of a number of different types of conversion: format change of geographic coordinates, conversion of coordinate systems, or transformation to different geodetic datums. Geographic coordinate conversion has applications in cartography, surveying, navigation and geographic information systems.

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 **history of Lorentz transformations** comprises the development of linear transformations forming the Lorentz group or Poincaré group preserving the Lorentz interval and the Minkowski inner product .

**Space-oblique Mercator projection** is a map projection devised in the 1970s for preparing maps from Earth-survey satellite data. It is a generalization of the oblique Mercator projection that incorporates the time evolution of a given satellite ground track to optimize its representation on the map. The oblique Mercator projection, on the other hand, optimizes for a given geodesic.

The **Cassini projection** is a map projection described by César-François Cassini de Thury in 1745. It is the transverse aspect of the equirectangular projection, in that the globe is first rotated so the central meridian becomes the "equator", and then the normal equirectangular projection is applied. Considering the earth as a sphere, the projection is composed of the operations:

In mathematics, the **spectral theory of ordinary differential equations** is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. It can also be related to the relativisic velocity addition formula.

In fluid dynamics, a **cnoidal wave** is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function *cn*, which is why they are coined *cn*oidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.

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.

In fluid dynamics, a **trochoidal wave** or **Gerstner wave** is an exact solution of the Euler equations for periodic surface gravity waves. It describes a progressive wave of permanent form on the surface of an incompressible fluid of infinite depth. The free surface of this wave solution is an inverted (upside-down) trochoid – with sharper crests and flat troughs. This wave solution was discovered by Gerstner in 1802, and rediscovered independently by Rankine in 1863.

In image analysis, the **generalized structure tensor (GST)** is an extension of the Cartesian structure tensor to curvilinear coordinates.. It is mainly used to detect and to represent the "direction" parameters of curves, just as the Cartesian structure tensor detects and represents the direction in Cartesian coordinates. Curve families generated by pairs of locally orthogonal functions have been the best studied.

**Moffatt eddies** are sequence of eddies that develop in corner bounded by plane walls due to arbitrary disturbance exerting on asymptotically large distances from the corner. Although the source of motion is the arbitrary disturbance at large distances, the eddies develop quite independently and thus solution of these eddies emerges from an eigenvalue problem, a Self-similar solution of the second kind.

- ↑ "Universal Transverse Mercator (UTM)".
*PROJ coordinate transformation software library*. - ↑ Snyder, John P. (1987).
*Map projections: A working manual*. U.S. Government Printing Office. - ↑ "NOAA History - Stories and Tales of the Coast & Geodetic Survey - Technology Tales/Geodetic Surveys in the US The Beginning and the next 100 years".
*www.history.noaa.gov*. Retrieved 4 May 2018. - 1 2 BUCHROITHNER, Manfred F.; PFAHLBUSCH, René. Geodetic grids in authoritative maps–new findings about the origin of the UTM Grid. Cartography and Geographic Information Science, 2016
- ↑ Equatorial radius 6,378,206.4 meters, polar radius 6,356,583.8 meters
- ↑ Equatorial radius 6,378,388 meters, reciprocal of the flattening 297 exactly
- ↑ "Military Map Reading 201" (PDF). National Geospatial-Intelligence Agency. 2002-05-29. Retrieved 2009-06-19.
- ↑ See "The Letter after the UTM Zone Number: Is that a Hemisphere or a Latitudinal Band?", page 7,
- ↑ Krüger, L. (1912).
*Konforme Abbildung des Erdellipsoids in der Ebene*. Royal Prussian Geodetic Institute, New Series 52. - ↑ Karney, Charles F. F. (2011). "Transverse Mercator with an accuracy of a few nanometers".
*J. Geodesy*.**85**(8): 475–485. arXiv: 1002.1417 . Bibcode:2011JGeod..85..475K. doi:10.1007/s00190-011-0445-3. - ↑ Kawase, K. (2012): Concise Derivation of Extensive Coordinate Conversion Formulae in the Gauss-Krüger Projection, Bulletin of the Geospatial Information Authority of Japan,
**60**, pp 1–6 - ↑ Kawase, K. (2011): A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection, Bulletin of the Geospatial Information Authority of Japan,
**59**, 1–13

- Snyder, John P. (1987).
*Map Projections – A Working Manual. U.S. Geological Survey Professional Paper 1395*. United States Government Printing Office, Washington, D.C.

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