Universal Transverse Mercator
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The Universal Transverse Mercator (UTM) is a
map projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...
system for assigning
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
to locations on the surface of the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
. Like the traditional method of
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
and
longitude Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...
, it is a
horizontal position representation A position representation is the parameters used to express a position relative to a reference. When representing positions relative to the Earth, it is often most convenient to represent ''vertical position'' (height or depth) separately, and to ...
, which means it ignores
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
and treats the earth as a perfect
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
. 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 A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
to a UTM zone is some
parameterization In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, de ...
of the
transverse Mercator The transverse Mercator map projection (TM, TMP) 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 Universal Transverse Mercat ...
projection. The parameters vary by nation or region or mapping system. Most zones in UTM span 6 degrees of
longitude Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...
, 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.


History

The
National Oceanic and Atmospheric Administration The National Oceanic and Atmospheric Administration (abbreviated as NOAA ) is an United States scientific and regulatory agency within the United States Department of Commerce that forecasts weather, monitors oceanic and atmospheric conditio ...
(NOAA) website states that the system was developed by the
United States Army Corps of Engineers , colors = , anniversaries = 16 June (Organization Day) , battles = , battles_label = Wars , website = , commander1 = ...
, starting in the early 1940s. However, a series of aerial photos found in the Bundesarchiv-Militärarchiv (the military section of the
German Federal Archives The German Federal Archives or Bundesarchiv (BArch) (german: Bundesarchiv) are the National Archives of Germany. They were established at the current location in Koblenz in 1952. They are subordinated to the Federal Commissioner for Culture and t ...
) apparently dating from 1943–1944 bear the inscription UTMREF followed by grid letters and digits, and projected according to the transverse Mercator, 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 a finding that would indicate that something called the UTM Reference system was developed in the 1942–43 time frame by the
Wehrmacht The ''Wehrmacht'' (, ) were the unified armed forces of Nazi Germany from 1935 to 1945. It consisted of the ''Heer'' (army), the ''Kriegsmarine'' (navy) and the ''Luftwaffe'' (air force). The designation "''Wehrmacht''" replaced the previous ...
. 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. For areas within the
contiguous United States The contiguous United States (officially the conterminous United States) consists of the 48 adjoining U.S. states and the Federal District of the United States of America. The term excludes the only two non-contiguous states, Alaska and Hawaii ...
the
Clarke Clarke is a surname which means "clerk". The surname is of English and Irish origin and comes from the Latin . Variants include Clerk and Clark. Clarke is also uncommonly chosen as a given name. Irish surname origin Clarke is a popular surname i ...
Ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
of 1866 was used. For the remaining areas of Earth, including
Hawaii Hawaii ( ; haw, Hawaii or ) is a state in the Western United States, located in the Pacific Ocean about from the U.S. mainland. It is the only U.S. state outside North America, the only state that is an archipelago, and the only stat ...
, the
International Ellipsoid The Hayford ellipsoid is a geodetic reference ellipsoid, named after the US geodesist John Fillmore Hayford (1868–1925), which was introduced in 1910. The Hayford ellipsoid was also referred to as the International ellipsoid 1924 after it had b ...
was used. The
World Geodetic System The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descri ...
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 In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted. ...
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 In the history of the 20th century, the interwar period lasted from 11 November 1918 to 1 September 1939 (20 years, 9 months, 21 days), the end of the World War I, First World War to the beginning of the World War II, Second World War. The in ...
. Calculating the distance between two points on these maps could be performed more easily in the field (using the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
) than was possible using the trigonometric formulas required under the graticule-based system of
latitude and longitude The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the various ...
. In the post-war years, these concepts were extended into the Universal Transverse Mercator/
Universal Polar Stereographic 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 sys ...
(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 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 it is unique in representing north as up and sou ...
, which was originally developed by the
Flemish Flemish (''Vlaams'') is a Low Franconian dialect cluster of the Dutch language. It is sometimes referred to as Flemish Dutch (), Belgian Dutch ( ), or Southern Dutch (). Flemish is native to Flanders, a historical region in northern Belgium; ...
geographer and cartographer
Gerardus Mercator Gerardus Mercator (; 5 March 1512 – 2 December 1594) was a 16th-century geographer, cosmographer and Cartography, cartographer from the County of Flanders. He is most renowned for creating the Mercator 1569 world map, 1569 world map based on ...
, in 1570. This projection is conformal, which means it preserves angles and therefore shapes across small regions. However, it distorts distance and area.


Definitions


UTM zone

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 The transverse Mercator map projection (TM, TMP) 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 Universal Transverse Mercat ...
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 Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to Science * Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon * ...
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 The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
. 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 line In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitu ...
s, 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.


Overlapping grids

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

Latitude bands are not a part of UTM, but rather a part of the
military grid reference system The Military Grid Reference System (MGRS)
Datums, Ellipsoids, Grids, and Grid Reference Sys ...
(MGRS). They are however sometimes used.


Latitude bands

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 English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...
alphabet An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syll ...
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, theoretically, as do bands "Y" and "Z". These cover the western and eastern sides of the Antarctic and Arctic regions respectively. A convenient
mnemonic A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding. Mnemonics make use of elaborative encoding, retrieval cues, and imag ...
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.


Notation

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 Toronto ( ; or ) is the capital city of the Canadian province of Ontario. With a recorded population of 2,794,356 in 2021, it is the most populous city in Canada and the fourth most populous city in North America. The city is the ancho ...
,
Canada Canada is a country in North America. Its ten provinces and three territories extend from the Atlantic Ocean to the Pacific Ocean and northward into the Arctic Ocean, covering over , making it the world's second-largest country by tot ...
, 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 The Military Grid Reference System (MGRS)
Datums, Ellipsoids, Grids, and Grid Reference Sys ...
. Occasionally only N or S following the zone number is added 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 notation is ambiguous since, for instance, "50S" can mean southern hemisphere but also ''grid zone'' "50S" in the northern hemisphere.


Exceptions

These grid zones are uniform over the globe, except in two areas. On the southwest coast of
Norway Norway, officially the Kingdom of Norway, is a Nordic country in Northern Europe, the mainland territory of which comprises the western and northernmost portion of the Scandinavian Peninsula. The remote Arctic island of Jan Mayen and t ...
, 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 Svalbard ( , ), also known as Spitsbergen, or Spitzbergen, is a Norwegian archipelago in the Arctic Ocean. North of mainland Europe, it is about midway between the northern coast of Norway and the North Pole. The islands of the group range ...
, 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. File:LA2-Europe-UTM-zones.png, Europe File:LA2-Africa-UTM-zones.png, Africa File:LA2-South-America-UTM-zones.png, South America File:LA2-Bering-Sea-UTM-zones.png, Bering Sea with Alaska


Locating a position using UTM coordinates

A position on the Earth is given by the UTM zone number and band letter and the
easting and northing A projected coordinate system, also known as a projected coordinate reference system, a planar coordinate system, or grid reference system, is a type of spatial reference system that represents locations on the Earth using cartesian coordina ...
planar coordinate pair in that zone and band. 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 meters East. In any zone a point that has an easting of 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 meters to meters at the equator. In the northern hemisphere positions are measured northward from zero at the equator. The maximum "northing" value is about meters at latitude 84 degrees North, the north end of the UTM zones. The southern hemisphere's northing at the equator is set at meters. Northings decrease southward from these meters to about meters at 80 degrees South, the south end of the UTM zones. Therefore, no point has a negative northing value. For example, the
CN Tower The CN Tower (french: Tour CN) is a concrete communications and observation tower in downtown Toronto, Ontario, Canada. Built on the former Railway Lands, it was completed in 1976. Its name "CN" referred to Canadian National, the railway c ...
is at , which is in UTM zone 17, and the grid position is east, north. Two points in Zone 17 have these coordinates, one in the northern hemisphere and one in the south; the non-ambiguous format is to specify the full zone and band, that is, "17T 630084 4833438". The provision of the latitude band along with northing supplies useful redundant information.


Simplified formulae

These formulae are truncated version of Transverse Mercator: flattening series, which were originally derived by
Johann Heinrich Louis Krüger Johann Heinrich Louis Krüger (21 September 1857 – 1 June 1923) was a German mathematician and surveyor/geodesist Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orient ...
in 1912.Krüger, L. (1912).
Konforme Abbildung des Erdellipsoids in der Ebene
'. Royal Prussian Geodetic Institute, New Series 52.
They are accurate to around a
millimeter 330px, Different lengths as in respect to the electromagnetic spectrum, measured by the metre and its derived scales. The microwave is between 1 meter to 1 millimeter. The millimetre (American and British English spelling differences#-re, -er, ...
within of the central meridian. Concise commentaries for their derivation have also been given.Kawase, K. (2012)
Concise Derivation of Extensive Coordinate Conversion Formulae in the Gauss-Krüger Projection
Bulletin of the
Geospatial Information Authority of Japan The , or GSI, is the national institution responsible for surveying and mapping the national land of Japan. The former name of the organization from 1949 until March 2010 was Geographical Survey Institute; despite the rename, it retains the same ...
, 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
The
WGS 84 The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descri ...
spatial reference system A spatial reference system (SRS) or coordinate reference system (CRS) is a framework used to precisely measure locations on the surface of the Earth as coordinates. It is thus the application of the abstract mathematics of coordinate systems and ...
describes Earth as an
oblate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circ ...
along north-south axis with an equatorial radius of a=6378.137 km and an inverse
flattening Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is ...
of 1/f=298.257\,223\,563. Let's take a point of latitude \,\varphi and of longitude \,\lambda and compute its UTM coordinates as well as point scale factor k\,\! and meridian convergence \gamma\,\! using a reference meridian of longitude \lambda_. By convention, in the
northern hemisphere The Northern Hemisphere is the half of Earth that is north of the Equator. For other planets in the Solar System, north is defined as being in the same celestial hemisphere relative to the invariable plane of the solar system as Earth's Nort ...
N_=0 km and in the southern hemisphere N_=10000 km. By convention also k_=0.9996 and E_=500 km. In the following formulas, the distances are in
kilometer The kilometre ( SI symbol: km; or ), spelt kilometer in American English, is a unit of length in the International System of Units (SI), equal to one thousand metres (kilo- being the SI prefix for ). It is now the measurement unit used for ex ...
s. In advance let's compute some preliminary values: :n=\frac,\quad A=\frac\left(1+ \frac +\frac +\cdots\right), :\alpha_1=\fracn-\fracn^2+\fracn^3,\,\,\,\alpha_2=\fracn^2-\fracn^3,\,\,\,\alpha_3=\fracn^3, :\beta_1=\fracn-\fracn^2+\fracn^3,\,\,\,\beta_2=\fracn^2+\fracn^3,\,\,\,\beta_3=\fracn^3, :\delta_1=2n-\fracn^2-2n^3,\,\,\,\delta_2=\fracn^2-\fracn^3,\,\,\,\delta_3=\fracn^3.


From latitude, longitude (''φ'', ''λ'') to UTM coordinates (E, N)

First let's compute some intermediate values: :t=\sinh\left(\tanh^\left(\sin\varphi\right)-\frac\tanh^\left(\frac \sin\varphi\right)\right), :\xi'=\tan^\left(\frac\right), \,\,\,\eta'=\tanh^\left(\frac\right), :\sigma=1+\sum_^3 2j\alpha_j\cos(2j\xi')\cosh(2j\eta'),\,\,\,\tau=\sum_^3 2j \alpha_j \sin(2j\xi') \sinh(2j\eta'). The final formulae are: :E=E_0+k_0 A\left(\eta'+\sum_^3 \alpha_j\cos(2j\xi')\sinh(2j\eta')\right), :N=N_0+k_0 A\left(\xi'+\sum_^3 \alpha_j\sin(2j\xi')\cosh(2j\eta')\right), :k=\frac\sqrt, :\gamma=\tan^\left(\frac\right). where E is Easting, N is Northing, k is the Scale Factor, and \gamma is the Grid Convergence.


From UTM coordinates (E, N, Zone, Hemi) to latitude, longitude (φ, λ)

Note: Hemi=+1 for Northern, Hemi=-1 for Southern First let's compute some intermediate values: :\xi=\frac,\,\,\,\eta=\frac, :\xi'=\xi-\sum_^\beta_j\sin\left(2j\xi\right)\cosh\left(2j\eta\right), \,\,\,\eta'=\eta-\sum_^\beta_j\cos\left(2j\xi\right)\sinh\left(2j\eta\right), :\sigma'=1-\sum_^2j\beta_j\cos\left(2j\xi\right)\cosh\left(2j\eta\right), \,\,\,\tau'=\sum_^2j\beta_j\sin\left(2j\xi\right)\sinh\left(2j\eta\right), :\chi=\sin^\left(\frac\right). The final formulae are: :\varphi=\chi+\sum_^\delta_j\sin\left(2j\chi\right), :\lambda_=\mathrm\mathrm\mathrm\mathrm\times 6^\circ - 183^\circ\, :\lambda=\lambda_+\tan^\left(\frac\right), :k=\frac\sqrt, :\gamma=\mathrm\mathrm\mathrm\mathrm\times\tan^\left(\frac\right).


See also

*
Military grid reference system The Military Grid Reference System (MGRS)
Datums, Ellipsoids, Grids, and Grid Reference Sys ...
, a variant of UTM designed to simplify transfer of coordinates. * Modified transverse Mercator, a variation of UTM used in Canada with zones spaced 3° of longitude apart as opposed to UTM's 6°. *
Transverse Mercator projection The transverse Mercator map projection (TM, TMP) 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 Universal Transverse Mercat ...
, the map projection used by UTM. *
Universal Polar Stereographic coordinate system 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 sys ...
, used at the North and South poles. *
Open Location Code The Open Location Code (OLC) is a geocode based in a system of regular grids for identifying an area anywhere on the Earth. It was developed at Google's Zürich engineering office, and released late October 2014. Location codes created by the OL ...
, a hierarchical zoned system * MapCode, a hierarchical zoned system


References


Further reading

* {{Authority control Geographic coordinate systems Cartography Geodesy