Universal Transverse Mercator coordinate system

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The Universal Transverse Mercator (UTM) geographic coordinate system is a grid-based method of specifying locations on the surface of the Earth that is a practical application of a 2-dimensional Cartesian coordinate system. It is a horizontal position representation, i.e. it is used to identify locations on the earth independently of vertical position, but differs from the traditional method of latitude and longitude in several respects.

The UTM system is not a single map projection. The system instead employs a series of sixty zones, each of which is based on a specifically defined secant transverse Mercator projection.

File:Utm-zones.jpg

The UTM grid.

History

The Universal Transverse Mercator coordinate system was developed by the United States Army Corps of Engineers in the 1940s.[1] The system was based on an ellipsoidal model of Earth. For areas within the conterminous United States, the Clarke 1866 ellipsoid was used. For the remaining areas of Earth, including Hawaii, the International Ellipsoid was used. Currently, the WGS84 ellipsoid is used as the underlying model of Earth in the UTM coordinate system.

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 otherwise 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, so that it preserves angles and approximates shape but invariably distorts distance and area. UTM involves non-linear scaling in both Easting and Northing to ensure the projected map of the ellipsoid is conformal.

Definitions

UTM zone

File:Utm-zones.svg

Simplified view of US UTM zones.

The UTM system divides the surface of Earth between 80°S and 84°N latitude into 60 zones, each 6° of longitude in width, and centered over a meridian of longitude. Zone 1 is bounded by longitude 180° to 174° W and is centered on the 177th West meridian. Zone numbering increases in an eastward direction.

Each of the 60 longitude zones in the UTM system is based on a transverse Mercator projection, which is capable of mapping a region of large north-south extent with a low amount of distortion. By using narrow zones of 6° of longitude (up to 800 km) in width, and reducing the scale factor along the central meridian by only 0.0004 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 outer 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, located approximately 180 km on either side of, and approximately parallel to, the central meridian (ArcCos 0.9996 = 1.62° at the Equator). The scale factor is less than 1 inside these lines and greater than 1 outside of these lines, but the overall distortion of scale inside the entire zone 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 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 the 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, 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.

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.

• LA2-Europe-UTM-zones.png

  Europe

• 

  Africa

• LA2-South-America-UTM-zones.png

  South America

• LA2-Bering-Sea-UTM-zones.png

  Bering Sea with Alaska

Picture gallery: Grid zones in various parts of the world

Locating a position using UTM coordinates

A position on the Earth is referenced in the UTM system by the UTM zone, and the easting and northing coordinate pair. The easting is the projected distance of the position eastward from the central meridian, while the northing is the projected distance of the point north from the equator (in the northern hemisphere). Eastings and northings are measured in meters. The point of origin of each UTM zone is the intersection of the equator and the zone's central meridian. In order to avoid dealing with negative numbers, the central meridian of each zone is given a "false easting" value of 500,000 meters. Thus, anything west of the central meridian will have an easting less than 500,000 meters. For example, UTM eastings range from 167,000 meters to 833,000 meters at the equator (these ranges narrow towards the poles). In the northern hemisphere, positions are measured northward from the equator, which has an initial "northing" value of 0 meters and a maximum "northing" value of approximately 9,328,000 meters at the 84th parallel — the maximum northern extent of the UTM zones. In the southern hemisphere, northings decrease as you go southward from the equator, which is given a "false northing" of 10,000,000 meters so that no point within the zone has a negative northing value.

As an example, the CN Tower is located at the geographic position 43°38′33.24″N 79°23′13.7″W{{#coordinates:43|38|33.24|N|79|23|13.7|W| | |name=CN Tower }}. This is in zone 17, and the grid position is 630084m east, 4833438m north. There are two points on the earth with these coordinates, one in the northern hemisphere and one in the southern. In order to define the position uniquely, one of two conventions is employed:

1. Append a hemisphere designator to the zone number, "N" or "S", thus "17N 630084 4833438". This supplies the minimum additional 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).

Because latitude band "S" is in the northern hemisphere, a designation such as "38S" is ambiguous. 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.

Simplified formulas from latitude,longitude (φ,λ) to UTM coordinates (E, N)

Exact formulas are quite complex and not very usable. Here are some simplified formula with centimetric precision.

By convention, the WGS 84 geoid describe Earth as an ellipsoid along North-South axis with an equatorial radius of \(a=6378.137km\) and an orbital eccentricity of \(e=0.0818192\). Let's take a point of latitude φ and of longitude λ and compute its UTM coordinates using a reference meridian of longitude \(\lambda_{0}\). In the following formula, the angles are in radians, and the distances in kilometers.

First let's compute some intermediates values:

\( \nu(\varphi)=1/\sqrt{1-e^{2}\sin^{2}\varphi} \)

\(A=(\lambda-\lambda_{0})\,\cos\varphi \)

\( s(\varphi) = (1-\frac{e^{2}}{4}-\frac{3e^{4}}{64}-\frac{5e^{6}}{256})\varphi-(\frac{3e^{2}}{8}+\frac{3e^{4}}{32}+ \frac{45e^{6}}{1024})\sin2\varphi + (\frac{15e^{4}}{256}+\frac{45e^{6}}{1024})\sin4\varphi-\frac{35e^{6}}{3072}\sin6\varphi \)

\( T=\tan^{2}\varphi, \quad C=\frac{e^{2}}{1-e^{2}}\cos^{2}\varphi\)

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 \(E_{0}=500 km\). The final formulas are:

\( E = E_{0} +k_{0}a\nu(\varphi)\Big(A+(1-T+C)\frac{A^{3}}{6}+(5-18T+T^{2})\frac{A^{5}}{120}\Big) \)

\( N =N_{0}+k_{0}a\,\Big(s(\varphi)+ \nu(\varphi)\,\tan\varphi\Big(\frac{A^{2}}{2}+(5-T+9C+4C^{2})\frac{A^{4}}{24} +(61-58T+T^{2})\frac{A^{6}}{720}\Big)\Big) \)

See also

• The Military grid reference system is a variant of UTM designed to simplify transfer of coordinates.
• The Transverse Mercator projection is the map projection used by UTM.
• The Universal Polar Stereographic coordinate system are used at the North and South poles.

External links

• [2] TM8358.1: Datums, Ellipsoids, Grids and Grid Reference Systems
• [3] TM8358.2: Defense Mapping Agency Technical Manual 8358.2 The Universal Grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS)
• GeographicLib provides a utility GeoConvert (with source code) for conversions between geographic, UTM, UPS, and MGRS. Here is an online version of GeoConvert.
• jeotrans java port of NGA geotrans- provides conversions between geodetic (DMS), UTM, PS, and MGRS.ca:Projecció Universal Transversa de Mercator