Universal Transverse Mercator coordinate system

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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. 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. Earth is the third planet from the Sun and the only astronomical object known to harbor life. According to radiometric dating and other sources of evidence, Earth formed over 4.5 billion years ago. Earth's gravity interacts with other objects in space, especially the Sun and the Moon, Earth's only natural satellite. Earth orbits around the Sun in 365.26 days, a period known as an Earth year. During this time, Earth rotates about its axis about 366.26 times. 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.

Contents

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. 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.

History

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.  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,  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.  For areas within the contiguous United States the Clarke Ellipsoid of 1866  was used. For the remaining areas of Earth, including Hawaii, the International Ellipsoid  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. The National Oceanic and Atmospheric Administration is an American scientific agency within the United States Department of Commerce that focuses on the conditions of the oceans, major waterways, and the atmosphere. The United States Army Corps of Engineers (USACE) is a U.S. federal agency under the Department of Defense (DoD) and a major Army command made up of some 37,000 civilian and military personnel, making it one of the world's largest public engineering, design, and construction management agencies. Although generally associated with dams, canals and flood protection in the United States, USACE is involved in a wide range of public works throughout the world. The Corps of Engineers provides outdoor recreation opportunities to the public, and provides 24% of U.S. hydropower capacity. The German Federal Archives or Bundesarchiv (BArch) are the National Archives of Germany. They were established at the current location in Koblenz in 1952.

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. In the context of the history of the 20th century, the interwar period was the period between the end of the First World War in November 1918 and the beginning of the Second World War in September 1939. This period is also colloquially referred to as Between the Wars. In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":

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 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 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 by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. 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 starts infinitesimally, but accelerates with 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.

The Flemish or Flemings are a Germanic ethnic group native to Flanders, in modern Belgium, who speak Flemish Dutch. They are one of two principal ethnic groups in Belgium, the other being the French-speaking Walloons. Flemish people make up the majority of the Belgian population. Historically, all inhabitants of the medieval County of Flanders were referred to as "Flemings", irrespective of the language spoken. The contemporary region of Flanders comprises a part of this historical county, as well as parts of the medieval duchy of Brabant and the medieval county of Loon. Gerardus Mercator was a 16th-century geographer, cosmographer and cartographer from Flanders. He is most renowned for creating the 1569 world map based on a new projection which represented sailing courses of constant bearing as straight lines—an innovation that is still employed in nautical charts.

Definitions

UTM zone Simplified view of contiguous US UTM zones, projected with Lambert conformal conic.

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. The 80th parallel south is a circle of latitude that is 80 degrees south of the Earth's equatorial plane. The 84th parallel north is a circle of latitude that is 84 degrees north of the Earth's equatorial plane, in the Arctic. It passes solely through the Arctic Ocean.

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.

Overlapping grids Universal Transverse Mercator (UTM) Grid Zones 31N thru 37N differ from the standard 6° wide by 84° zone for the northern hemisphere, in part to accommodate the southern half of the Kingdom of Norway. For more on its history, see Clifford J. Mugnier's article on Grids & Datums of The Kingdom of Norway that appeared in the October 1999 issue of PE&RS http://www.asprs.org/a/resources/grids/10-99-norway.pdf

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 (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 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.

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, 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.  There are many possible ways to disambiguate between the two methods, two of which are demonstrated later in this article.

Exceptions

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.

Locating a position using UTM coordinates

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 to about 1100000 meters at 80 degrees South, the south end of the UTM zones. The northing at the equator is set at 10000000 meters so no point has a negative northing value.

The CN Tower is at , 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:

1. 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.
2. 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°N40°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.

Simplified formulae

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

The WGS 84 spatial reference system describes Earth as an oblate spheroid along north-south axis with an equatorial radius of $a=6378.137$ km and an inverse flattening 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 $\lambda _{0}$ . By convention, in the northern hemisphere $N_{0}=0$ km and in the southern hemisphere $N_{0}=10000$ km. By convention also $k_{0}=0.9996$ and $E_{0}=500$ km.

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

$n={\frac {f}{2-f}},\quad A={\frac {a}{1+n}}\left(1+{\frac {n^{2}}{4}}+{\frac {n^{4}}{64}}+\cdots \right),$ $\alpha _{1}={\frac {1}{2}}n-{\frac {2}{3}}n^{2}+{\frac {5}{16}}n^{3},\,\,\,\alpha _{2}={\frac {13}{48}}n^{2}-{\frac {3}{5}}n^{3},\,\,\,\alpha _{3}={\frac {61}{240}}n^{3},$ $\beta _{1}={\frac {1}{2}}n-{\frac {2}{3}}n^{2}+{\frac {37}{96}}n^{3},\,\,\,\beta _{2}={\frac {1}{48}}n^{2}+{\frac {1}{15}}n^{3},\,\,\,\beta _{3}={\frac {17}{480}}n^{3},$ $\delta _{1}=2n-{\frac {2}{3}}n^{2}-2n^{3},\,\,\,\delta _{2}={\frac {7}{3}}n^{2}-{\frac {8}{5}}n^{3},\,\,\,\delta _{3}={\frac {56}{15}}n^{3}.$ From latitude, longitude (φ, λ) to UTM coordinates (E, N)

First let's compute some intermediate values:

$t=\sinh \left(\tanh ^{-1}\sin \varphi -{\frac {2{\sqrt {n}}}{1+n}}\tanh ^{-1}\left({\frac {2{\sqrt {n}}}{1+n}}\sin \varphi \right)\right),$ $\xi '=\tan ^{-1}\left({\frac {t}{\cos(\lambda -\lambda _{0})}}\right),\,\,\,\eta '=\tanh ^{-1}\left({\frac {\sin(\lambda -\lambda _{0})}{\sqrt {1+t^{2}}}}\right),$ $\sigma =1+\sum _{j=1}^{3}2j\alpha _{j}\cos \left(2j\xi '\right)\cosh \left(2j\eta '\right),\,\,\,\tau =\sum _{j=1}^{3}2j\alpha _{j}\sin \left(2j\xi '\right)\sinh \left(2j\eta '\right).$ The final formulae are:

$E=E_{0}+k_{0}A\left(\eta '+\sum _{j=1}^{3}\alpha _{j}\cos \left(2j\xi '\right)\sinh \left(2j\eta '\right)\right),$ $N=N_{0}+k_{0}A\left(\xi '+\sum _{j=1}^{3}\alpha _{j}\sin \left(2j\xi '\right)\cosh \left(2j\eta '\right)\right),$ $k={\frac {k_{0}A}{a}}{\sqrt {\left\{1+\left({\frac {1-n}{1+n}}\tan \varphi \right)^{2}\right\}{\frac {\sigma ^{2}+\tau ^{2}}{t^{2}+\cos ^{2}(\lambda -\lambda _{0})}}}},$ $\gamma =\tan ^{-1}\left({\frac {\tau {\sqrt {1+t^{2}}}+\sigma t\tan(\lambda -\lambda _{0})}{\sigma {\sqrt {1+t^{2}}}-\tau t\tan(\lambda -\lambda _{0})}}\right).$ 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:

$\xi ={\frac {N-N_{0}}{k_{0}A}},\,\,\,\eta ={\frac {E-E_{0}}{k_{0}A}},$ $\xi '=\xi -\sum _{j=1}^{3}\beta _{j}\sin \left(2j\xi \right)\cosh \left(2j\eta \right),\,\,\,\eta '=\eta -\sum _{j=1}^{3}\beta _{j}\cos \left(2j\xi \right)\sinh \left(2j\eta \right),$ $\sigma '=1-\sum _{j=1}^{3}2j\beta _{j}\cos \left(2j\xi \right)\cosh \left(2j\eta \right),\,\,\,\tau '=\sum _{j=1}^{3}2j\beta _{j}\sin \left(2j\xi \right)\sinh \left(2j\eta \right),$ $\chi =\sin ^{-1}\left({\frac {\sin \xi '}{\cosh \eta '}}\right).$ The final formulae are:

$\varphi =\chi +\sum _{j=1}^{3}\delta _{j}\sin \left(2j\chi \right),$ $\lambda _{0}=\mathrm {Z} \mathrm {o} \mathrm {n} \mathrm {e} \times 6^{\circ }-183^{\circ }\,$ $\lambda =\lambda _{0}+\tan ^{-1}\left({\frac {\sinh \eta '}{\cos \xi '}}\right),$ $k={\frac {k_{0}A}{a}}{\sqrt {\left\{1+\left({\frac {1-n}{1+n}}\tan \varphi \right)^{2}\right\}{\frac {\cos ^{2}\xi '+\sinh ^{2}\eta '}{\sigma '^{2}+\tau '^{2}}}}},$ $\gamma =\mathrm {H} \mathrm {e} \mathrm {m} \mathrm {i} \times \tan ^{-1}\left({\frac {\tau '+\sigma '\tan \xi '\tanh \eta '}{\sigma '-\tau '\tan \xi '\tanh \eta '}}\right).$ Related Research Articles 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.

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