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58-590: When the first standardized coordinate systems were created during the 20th century, such as the Universal Transverse Mercator , State Plane Coordinate System , and British National Grid , they were commonly called grid systems ; the term is still common in some domains such as the military that encode coordinates as alphanumeric grid references . However, the term projected coordinate system has recently become predominant to clearly differentiate it from other types of spatial reference system . The term

116-606: A = 6378.137 {\displaystyle a=6378.137} km and an inverse flattening of 1 / f = 298.257 223 563 {\displaystyle 1/f=298.257\,223\,563} . Let's take a point of latitude φ {\displaystyle \,\varphi } and of longitude λ {\displaystyle \,\lambda } and compute its UTM coordinates as well as point scale factor k {\displaystyle k\,\!} and meridian convergence γ {\displaystyle \gamma \,\!} using

174-419: A natural origin , e.g., at which the ellipsoid and flat map surfaces coincide, at which point the projection formulas generate a coordinate of (0,0). To ensure that the northing and easting coordinates on a map are not negative (thus making measurement, communication, and computation easier), map projections may set up a false origin , specified in terms of false northing and false easting values, that offset

232-458: A truncated grid reference may be used where the general location is already known to participants and may be assumed. Because the (leading) most significant digits specify the part of the world and the (trailing) least significant digits provide a precision that is not needed in most circumstances, they may be unnecessary for some uses. This permits users to shorten the example coordinates to 949-361 by concealing 05nnn34 56nnn74 , assuming

290-419: A weather map with atmospheric pressure , are also projected by conformal projections. Small scale maps have large scale variations in a conformal projection, so recent world maps use other projections. Historically, many world maps are drawn by conformal projections, such as Mercator maps or hemisphere maps by stereographic projection . Conformal maps containing large regions vary scales by locations, so it

348-439: A 6-figure grid reference identifies a square of 100-metre sides, an 8-figure reference would identify a 10-metre square, and a 10-digit reference a 1-metre square. In order to give a standard 6-figure grid reference from a 10-figure GPS readout, the 4th, 5th, 9th and 10th digits must be omitted, so it is important not to read just the first 6 digits. Universal Transverse Mercator The Universal Transverse Mercator ( UTM )

406-458: A digit from 0 to 9 (with 0 0 being the bottom left square and 9 9 being the top right square). For the church in Little Plumpton, this gives the digits 6 and 7 (6 on the left to right axis (Eastings) and 7 on the bottom to top axis (Northings). These are added to the four-figure grid reference after the two digits describing the same coordinate axis , and thus our six-figure grid reference for

464-461: A false Easting of −500 000 meters is added to the central meridian. Thus a point that has an easting of 400 000 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 166 000 meters to 834 000 meters at

522-415: A four-digit grid reference describing a one-kilometre square on the ground. The convention is the grid reference numbers call out the lower-left corner of the desired square. In the example map above, the town Little Plumpton lies in the square 6901, even though the writing which labels the town is in 6802 and 6902, most of the buildings (the orange boxed symbols) are in square 6901. The more digits added to

580-507: 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

638-455: A grid reference, the more precise the reference becomes. To locate a specific building in Little Plumpton, a further two digits are added to the four-digit reference to create a six-digit reference. The extra two digits describe a position within the 1-kilometre square. Imagine (or draw or superimpose a Romer ) a further 10x10 grid within the current grid square. Any of the 100 squares in the superimposed 10×10 grid can be accurately described using

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696-543: A map, but it is important to know how many digits the GPS displays to avoid reading off just the first six digits. A GPS unit commonly gives a ten-digit grid reference, based on two groups of five numbers for the Easting and Northing values. Each successive increase in precision (from 6 digit to 8 digit to 10 digit) pinpoints the location more precisely by a factor of 10. Since, in the UK at least,

754-550: A reference meridian of longitude λ 0 {\displaystyle \lambda _{0}} . By convention, in the northern hemisphere N 0 = 0 {\displaystyle N_{0}=0} km and in the southern hemisphere N 0 = 10000 {\displaystyle N_{0}=10000} km. By convention also k 0 = 0.9996 {\displaystyle k_{0}=0.9996} and E 0 = 500 {\displaystyle E_{0}=500} km. In

812-587: 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). They are however sometimes included in UTM notation. Including latitude bands in UTM notation can lead to ambiguous coordinates—as

870-452: Is a map projection 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 surface as a perfect ellipsoid . However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to

928-619: Is a navigational term referring to the direction northwards along the grid lines of a map projection . It is contrasted with true north (the direction of the North Pole ) and magnetic north (the direction in which a compass needle points). Many topographic maps , including those of the United States Geological Survey and Great Britain's Ordnance Survey , indicate the difference between grid north, true north, and magnetic north. The grid lines on Ordnance Survey maps divide

986-690: 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, and instead covered by

1044-552: Is deemed scale-optimal when it minimizes the ratio of maximum to minimum scale across the entire map. This occurs by assigning a unit scale to the boundary of the disc. Chebyshev applied this theorem to create a conformal map for the European part of the Russian Empire, which reduced scale errors to 1/50. Many large-scale maps use conformal projections because figures in large-scale maps can be regarded as small enough. The figures on

1102-583: Is given a two-digit code, based on the British national grid reference system with an origin point just off the southwest coast of the United Kingdom . The area is divided into 100 km squares, each of which is denoted by a two-letter code. Within each 100 km square, a numerical grid reference is used. Since the Eastings and Northings are one kilometre apart, a combination of a Northing and an Easting will give

1160-478: Is the Grid Convergence. Note: Hemi = +1 for Northern, Hemi = −1 for Southern First let's compute some intermediate values: The final formulae are: Conformal map projection In cartography , a conformal map projection is one in which every angle between two curves that cross each other on Earth (a sphere or an ellipsoid ) is preserved in the image of the projection; that is,

1218-695: Is used in international standards such as the EPSG and ISO 19111 (also published by the Open Geospatial Consortium as Abstract Specification 2), and in most geographic information system software. The map projection and the Geographic coordinate system (GCS, latitude and longitude) date to the Hellenistic period , proliferating during the Enlightenment Era of the 18th century. However, their use as

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1276-580: Is usually represented conventionally with easting first, northing second. For example, the peak of Mount Assiniboine (at 50°52′10″N 115°39′03″W  /  50.86944°N 115.65083°W  / 50.86944; -115.65083 on the British Columbia / Alberta border in Canada ) in UTM Zone 11 is at (0594934mE, 5636174mN) , meaning that is almost 600km east of the false origin for Zone 11 (95km east of

1334-610: The United Kingdom , the first version of the British National Grid was released in 1938, based on earlier experiments during World War I by the Army and the Ordnance Survey . During World War II , modern warfare practices required soldiers to quickly and accurately measure and report their location, leading to the printing of grids on maps by the U.S. Army Map Service (AMS) and other combatants. Initially, each theater of war

1392-627: The transverse mercator (used in Universal Transverse Mercator , the British National Grid , the State Plane Coordinate System for some states), Lambert Conformal Conic (some states in the SPCS ), and Mercator ( Swiss coordinate system ). Map projection formulas depend on the geometry of the projection as well as parameters dependent on the particular location at which the map is projected. The set of parameters can vary based on

1450-479: The universal polar stereographic (UPS) coordinate system. 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

1508-597: The UK into one-kilometre squares, east of an imaginary zero point in the Atlantic Ocean, west of Cornwall. The grid lines point to a Grid North, varying slightly from True North. This variation is zero on the central meridian (north-south line) of the map, which is at two degrees West of the Prime Meridian , and greatest at the map edges. The difference between grid north and true north is very small and can be ignored for most navigation purposes. The difference exists because

1566-698: The War, UTM gradually gained users, especially in the scientific community. Because UTM zones do not align with political boundaries, several countries followed the United Kingdom in creating their own national or regional grid systems based on custom projections. The use and invention of such systems especially proliferated during the 1980s with the emergence of geographic information systems . GIS requires locations to be specified as precise coordinates and performs numerous calculations on them, making cartesian geometry preferable to spherical trigonometry when computing horsepower

1624-464: The basis for specifying precise locations, rather than latitude and longitude, is a 20th century innovation. Among the earliest was the State Plane Coordinate System (SPCS), which was developed in the United States during the 1930s for surveying and engineering, because calculations such as distance are much simpler in a Cartesian coordinate system than the three-dimensional trigonometry of GCS. In

1682-646: The central meridian is specified to be 0.9996 of true scale for most UTM systems in use. The National Oceanic and Atmospheric Administration (NOAA) website states that the system was 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

1740-410: The church becomes 696017. This reference describes a 100-metre by 100-metre square, and not a single point, but this precision is usually sufficient for navigation purposes. The symbols on the map are not precise in any case, for example the church in the example above would be approximately 100x200 metres if the symbol was to scale, so in fact, the middle of the black square represents the map position of

1798-409: The conformal projection. In a conformal projection, parallels and meridians cross rectangularly on the map; but not all maps with this property are conformal. The counterexamples are equirectangular and equal-area cylindrical projections (of normal aspects). These projections expand meridian-wise and parallel-wise by different ratios respectively. Thus, parallels and meridians cross rectangularly on

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1856-552: The correspondence between a flat map and the round Earth is necessarily imperfect. At the South Pole , grid north conventionally points northwards along the Prime Meridian . Since the meridians converge at the poles, true east and west directions change rapidly in a condition similar to gimbal lock . Grid north solves this problem. Locations in a projected coordinate system, like any cartesian coordinate system, are measured and reported as easting/northing or ( x , y ) pairs. The pair

1914-417: The equator. In the northern hemisphere positions are measured northward from zero at the equator. The maximum "northing" value is about 9 300 000 meters at latitude 84 degrees North, the north end of the UTM zones. The southern hemisphere's northing at the equator is set at 10 000 000 meters. Northings decrease southward from these 10 000 000 meters to about 1 100 000 meters at 80 degrees South,

1972-570: 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

2030-473: The following formulas, the distances are in kilometers . First, here are some preliminary values: First we compute some intermediate values: The final formulae are: where E {\displaystyle E} is Easting, N {\displaystyle N} is Northing, k {\displaystyle k} is the Scale Factor, and γ {\displaystyle \gamma }

2088-437: The gaps. 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

2146-628: 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

2204-435: The letter "S" either refers to the southern hemisphere or a latitude band in the northern hemisphere—and should therefore be avoided. A position on the Earth is given by the UTM zone number and hemisphere designator 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,

2262-611: The map's center, then the map remains conformal. However, it is difficult to compare lengths or areas of two far-off figures using such a projection. The Universal Transverse Mercator coordinate system and the Lambert system in France are projections that support the trade-off between seamlessness and scale variability. Maps reflecting directions, such as a nautical chart or an aeronautical chart , are projected by conformal projections. Maps treating values whose gradients are important, such as

2320-408: The map, but these projections do not preserve other angles; i.e. these projections are not conformal. As proven by Leonhard Euler in 1775, a conformal map projection cannot be equal-area, nor can an equal-area map projection be conformal. This is also a consequence of Carl Gauss 's 1827 Theorema Egregium [Remarkable Theorem]. A conformal parameterization of a disc-like domain on the sphere

2378-414: The map. The projection preserves the ratio of two lengths in the small domain. All of the projection's Tissot's indicatrices are circles. Conformal projections preserve only small figures. Large figures are distorted by even conformal projections. In a conformal projection, any small figure is similar to the image, but the ratio of similarity ( scale ) varies by location, which explains the distortion of

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2436-399: The maps are nearly similar to their physical counterparts. A non-conformal projection can be used in a limited domain such that the projection is locally conformal. Glueing many maps together restores roundness. To make a new sheet from many maps or to change the center, the body must be re-projected. Seamless online maps can be very large Mercator projections , so that any place can become

2494-607: The northern hemisphere and one in the south; the non-ambiguous format is to specify the full zone and hemisphere designator, that is, "17N 630084 4833438". 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 3000 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

2552-601: 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

2610-412: The overall distortion is minimized. The UTM zones are uniform across the globe, except in two areas. On the southwest coast of Norway , zone 32 is extended 3° further west, and zone 31 is correspondingly shrunk to cover only open water. Also, in the region around Svalbard , the zones 32, 34 and 36 are not used, while zones 31 (9° wide), 33 (12° wide), 35 (12° wide), and 37 (9° wide) are extended to cover

2668-419: 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. Most zones in UTM span 6 degrees of longitude , and each has a designated central meridian. The scale factor at

2726-438: The position of a geographic location on a map , a map projection is used to convert geodetic coordinates to plane coordinates on a map; it projects the datum ellipsoidal coordinates and height onto a flat surface of a map. The datum, along with a map projection applied to a grid of reference locations, establishes a grid system for plotting locations. Conformal projections are generally preferred. Common map projections include

2784-417: The projection is a conformal map in the mathematical sense. For example, if two roads cross each other at a 39° angle, their images on a map with a conformal projection cross at a 39° angle. A conformal projection can be defined as one that is locally conformal at every point on the map, albeit possibly with singular points where conformality fails. Thus, every small figure is nearly similar to its image on

2842-437: The purpose of any coordinate system is to accurately and unambiguously measure, communicate, and perform calculations on locations, it must be defined precisely. The EPSG Geodetic Parameter Dataset is the most common mechanism for publishing such definitions in a machine-readable form, and forms the basis for many GIS and other location-aware software programs. A projected SRS specification consists of three parts: To establish

2900-447: The real church, independently of the actual size of the church. Grid references comprising larger numbers for greater precision could be determined using large-scale maps and an accurate Romer . This might be used in surveying but is not generally used for land navigating for walkers or cyclists, etc. The growing availability and decreasing cost of handheld GPS receivers enables determination of accurate grid references without needing

2958-823: The significant digits (3,4, and 5 in this case) are known to both parties. Alphanumeric encodings typically use codes to replace the most significant digits by partitioning the world up into large grid squares. For example, in the Military Grid Reference System , the above coordinate is in grid 11U (representing UTM Zone 11 5xxxxxx mN), and grid cell NS within that (representing the second digit 5xxxxxmE x6xxxxxm N), and as many remaining digits as are needed are reported, yielding an MGRS grid reference of 11U NS 949 361 (or 11U NS 9493 3617 or 11U NS 94934 36174). The Ordnance Survey National Grid (United Kingdom) and other national grid systems use similar approaches. In Ordnance Survey maps, each Easting and Northing grid line

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3016-518: The south end of the UTM zones. Therefore, no point has a negative northing value. For example, the CN Tower is at 43°38′33.24″N 79°23′13.7″W  /  43.6425667°N 79.387139°W  / 43.6425667; -79.387139  ( CN Tower ) , which is in UTM zone 17, and the grid position is 630 084  m east, 4 833 438  m north. Two points in Zone 17 have these coordinates, one in

3074-464: The true central meridian at 117°W) and 5.6 million meters north of the equator . While such precise numbers are easy to store and calculate in GIS and other computer databases, they can be difficult for humans to remember and communicate. Thus, since the mid 20th century, there have been alternative encodings that shorten the numbers or convert the numbers into some form of alphanumeric string. For example,

3132-432: The true origin. For example, in UTM, the origin of each northern zone is a point on the equator 500km west of the central meridian of the zone (the edge of the zone itself is just under 400km to the west). This has the desirable effect of making all coordinates within the zone positive values, being east and north of the origin. Because of this, they are often referred to as the easting and northing . Grid north ( GN )

3190-473: The type of project and the conventions chosen for the projection. For the transverse Mercator projection used in UTM, the parameters associated are the latitude and longitude of the natural origin, the false northing and false easting, and an overall scale factor. Given the parameters associated with particular location or grin, the projection formulas for the transverse Mercator are a complex mix of algebraic and trigonometric functions. Every map projection has

3248-485: 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

3306-545: Was at a premium. In recent years, the rise of global GIS datasets and satellite navigation , along with an abundance of processing speed in personal computers, have led to a resurgence in the use of GCS. That said, projected coordinate systems are still very common in the GIS data stored in the Spatial Data Infrastructures (SDI) of local areas, such as cities, counties, states and provinces, and small countries. Because

3364-523: Was mapped in a custom projection with its own grid and coding system, but this resulted in confusion. This led to the development of the Universal Transverse Mercator coordinate system , possibly adopted from a system originally developed by the German Wehrmacht . To facilitate unambiguous reporting, the alphanumeric Military Grid Reference System (MGRS) was then created as an encoding scheme for UTM coordinates to make them easier to communicate. After

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