Geodesy  

Fundamentals  
Concepts  
Standards (History)
 
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.^{ [n 1] } 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 threedimensional Cartesian vector. A common choice of coordinates is latitude, longitude and elevation.^{ [1] } To specify a location on a plane requires a map projection.^{ [2] }
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 significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the xcoordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.
Vertical position or vertical location is a position along a vertical direction above or below a given vertical datum. Vertical distance or vertical separation is the distance between two vertical positions. Many vertical coordinates exist for expressing vertical position: depth, height, altitude, elevation, etc. Each quantity may be expressed in various units: metres, feet, etc.
The nvector representation is a threeparameter nonsingular representation wellsuited for replacing latitude and longitude as horizontal position representation in mathematical calculations and computer algorithms.
The invention of a geographic coordinate system is generally credited to Eratosthenes of Cyrene, who composed his nowlost Geography at the Library of Alexandria in the 3rd century BC.^{ [3] } A century later, Hipparchus of Nicaea improved on this system by determining latitude from stellar measurements rather than solar altitude and determining longitude by timings of lunar eclipses, rather than dead reckoning. In the 1st or 2nd century, Marinus of Tyre compiled an extensive gazetteer and mathematicallyplotted world map using coordinates measured east from a prime meridian at the westernmost known land, designated the Fortunate Isles, off the coast of western Africa around the Canary or Cape Verde Islands, and measured north or south of the island of Rhodes off Asia Minor. Ptolemy credited him with the full adoption of longitude and latitude, rather than measuring latitude in terms of the length of the midsummer day.^{ [4] }
An invention is a unique or novel device, method, composition or process. The invention process is a process within an overall engineering and product development process. It may be an improvement upon a machine or product or a new process for creating an object or a result. An invention that achieves a completely unique function or result may be a radical breakthrough. Such works are novel and not obvious to others skilled in the same field. An inventor may be taking a big step in success or failure.
Eratosthenes of Cyrene was a Greek mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria. He invented the discipline of geography, including the terminology used today.
Cyrene was an ancient Greek and later Roman city near presentday Shahhat, Libya. It was the oldest and most important of the five Greek cities in the region. It gave eastern Libya the classical name Cyrenaica that it has retained to modern times. Located nearby is the ancient Necropolis of Cyrene.
Ptolemy's 2ndcentury Geography used the same prime meridian but measured latitude from the Equator instead. After their work was translated into Arabic in the 9th century, AlKhwārizmī's Book of the Description of the Earth corrected Marinus' and Ptolemy's errors regarding the length of the Mediterranean Sea,^{ [n 2] } causing medieval Arabic cartography to use a prime meridian around 10° east of Ptolemy's line. Mathematical cartography resumed in Europe following Maximus Planudes' recovery of Ptolemy's text a little before 1300; the text was translated into Latin at Florence by Jacobus Angelus around 1407.
An equator of a rotating spheroid is its zeroth circle of latitude (parallel). It is the imaginary line on the spheroid's surface, equidistant from its poles, dividing it into northern and southern hemispheres. In other words, it is the intersection of the spheroid's surface with the plane perpendicular to its axis of rotation and midway between its geographical poles.
The Mediterranean Sea is a sea connected to the Atlantic Ocean, surrounded by the Mediterranean Basin and almost completely enclosed by land: on the north by Southern Europe and Anatolia, on the south by North Africa and on the east by the Levant. Although the sea is sometimes considered a part of the Atlantic Ocean, it is usually identified as a separate body of water. Geological evidence indicates that around 5.9 million years ago, the Mediterranean was cut off from the Atlantic and was partly or completely desiccated over a period of some 600,000 years, the Messinian salinity crisis, before being refilled by the Zanclean flood about 5.3 million years ago.
Maximus Planudes was a Byzantine Greek monk, scholar, anthologist, translator, mathematician, grammarian and theologian at Constantinople. Through his translations from Latin into Greek and from Greek into Latin he brought the Greek East and the Latin West into closer contact with one another. He is now best known as a compiler of the Greek Anthology.
In 1884, the United States hosted the International Meridian Conference, attended by representatives from twentyfive nations. Twentytwo of them agreed to adopt the longitude of the Royal Observatory in Greenwich, England as the zeroreference line. The Dominican Republic voted against the motion, while France and Brazil abstained.^{ [5] } France adopted Greenwich Mean Time in place of local determinations by the Paris Observatory in 1911.
The United States of America (USA), commonly known as the United States or America, is a country composed of 50 states, a federal district, five major selfgoverning territories, and various possessions. At 3.8 million square miles, the United States is the world's third or fourth largest country by total area and is slightly smaller than the entire continent of Europe's 3.9 million square miles. With a population of over 327 million people, the U.S. is the third most populous country. The capital is Washington, D.C., and the largest city by population is New York City. Fortyeight states and the capital's federal district are contiguous in North America between Canada and Mexico. The State of Alaska is in the northwest corner of North America, bordered by Canada to the east and across the Bering Strait from Russia to the west. The State of Hawaii is an archipelago in the midPacific Ocean. The U.S. territories are scattered about the Pacific Ocean and the Caribbean Sea, stretching across nine official time zones. The extremely diverse geography, climate, and wildlife of the United States make it one of the world's 17 megadiverse countries.
The International Meridian Conference was a conference held in October 1884 in Washington, D.C., in the United States, to determine a prime meridian for international use. The conference was held at the request of U.S. President Chester A. Arthur. The subject to discuss was the choice of "a meridian to be employed as a common zero of longitude and standard of time reckoning throughout the world". It resulted in the recommendation of the Greenwich Meridian as the international standard for zero degrees longitude.
The Royal Observatory, Greenwich is an observatory situated on a hill in Greenwich Park, overlooking the River Thames. It played a major role in the history of astronomy and navigation, and is best known for the fact that the prime meridian passes through it, and thereby gave its name to Greenwich Mean Time. The ROG has the IAU observatory code of 000, the first in the list. ROG, the National Maritime Museum, the Queen's House and Cutty Sark are collectively designated Royal Museums Greenwich.
In order to be unambiguous about the direction of "vertical" and the "horizontal" surface above which they are measuring, mapmakers choose a reference ellipsoid with a given origin and orientation that best fits their need for the area they are mapping. They then choose the most appropriate mapping of the spherical coordinate system onto that ellipsoid, called a terrestrial reference system or geodetic datum.
In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network computations are performed and point coordinates such as latitude, longitude, and elevation are defined.
A geodetic datum or geodetic system is a coordinate system, and a set of reference points, used to locate places on the Earth. An approximate definition of sea level is the datum WGS 84, an ellipsoid, whereas a more accurate definition is Earth Gravitational Model 2008 (EGM2008), using at least 2,159 spherical harmonics. Other datums are defined for other areas or at other times; ED50 was defined in 1950 over Europe and differs from WGS 84 by a few hundred meters depending on where in Europe you look. Mars has no oceans and so no sea level, but at least two martian datums have been used to locate places there.
Datums may be global, meaning that they represent the whole Earth, or they may be local, meaning that they represent an ellipsoid bestfit to only a portion of the Earth. Points on the Earth's surface move relative to each other due to continental plate motion, subsidence, and diurnal Earth tidal movement caused by the Moon and the Sun. This daily movement can be as much as a metre. Continental movement can be up to 10 cm a year, or 10 m in a century. A weather system highpressure area can cause a sinking of 5 mm. Scandinavia is rising by 1 cm a year as a result of the melting of the ice sheets of the last ice age, but neighbouring Scotland is rising by only 0.2 cm. These changes are insignificant if a local datum is used, but are statistically significant if a global datum is used.^{ [1] }
Earth tide is the displacement of the solid earth's surface caused by the gravity of the Moon and Sun. Its main component has meterlevel amplitude at periods of about 12 hours and longer. The largest body tide constituents are semidiurnal, but there are also significant diurnal, semiannual, and fortnightly contributions. Though the gravitational forcing causing earth tides and ocean tides is the same, the responses are quite different.
The Moon is an astronomical body that orbits planet Earth and is Earth's only permanent natural satellite. It is the fifthlargest natural satellite in the Solar System, and the largest among planetary satellites relative to the size of the planet that it orbits. The Moon is after Jupiter's satellite Io the seconddensest satellite in the Solar System among those whose densities are known.
Scandinavia is a region in Northern Europe, with strong historical, cultural, and linguistic ties. The term Scandinavia in local usage covers the three kingdoms of Denmark, Norway, and Sweden. The majority national languages of these three, belong to the Scandinavian dialect continuum, and are mutually intelligible North Germanic languages. In English usage, Scandinavia also sometimes refers to the Scandinavian Peninsula, or to the broader region including Finland and Iceland, which is always known locally as the Nordic countries.
Examples of global datums include World Geodetic System (WGS 84), the default datum used for the Global Positioning System,^{ [n 3] } and the International Terrestrial Reference Frame (ITRF), used for estimating continental drift and crustal deformation.^{ [6] } The distance to Earth's center can be used both for very deep positions and for positions in space.^{ [1] }
Local datums chosen by a national cartographical organisation include the North American Datum, the European ED50, and the British OSGB36. Given a location, the datum provides the latitude and longitude . In the United Kingdom there are three common latitude, longitude, and height systems in use. WGS 84 differs at Greenwich from the one used on published maps OSGB36 by approximately 112 m. The military system ED50, used by NATO, differs from about 120 m to 180 m.^{ [1] }
The latitude and longitude on a map made against a local datum may not be the same as one obtained from a GPS receiver. Coordinates from the mapping system can sometimes be roughly changed into another datum using a simple translation. For example, to convert from ETRF89 (GPS) to the Irish Grid add 49 metres to the east, and subtract 23.4 metres from the north.^{ [7] } More generally one datum is changed into any other datum using a process called Helmert transformations. This involves converting the spherical coordinates into Cartesian coordinates and applying a seven parameter transformation (translation, threedimensional rotation), and converting back.^{ [1] }
In popular GIS software, data projected in latitude/longitude is often represented as a Geographic Coordinate System. For example, data in latitude/longitude if the datum is the North American Datum of 1983 is denoted by 'GCS North American 1983'.
The "latitude" (abbreviation: Lat., φ, or phi) of a point on Earth's surface is the angle between the equatorial plane and the straight line that passes through that point and through (or close to) the center of the Earth.^{ [n 4] } Lines joining points of the same latitude trace circles on the surface of Earth called parallels, as they are parallel to the Equator and to each other. The North Pole is 90° N; the South Pole is 90° S. The 0° parallel of latitude is designated the Equator, the fundamental plane of all geographic coordinate systems. The Equator divides the globe into Northern and Southern Hemispheres.
The "longitude" (abbreviation: Long., λ, or lambda) of a point on Earth's surface is the angle east or west of a reference meridian to another meridian that passes through that point. All meridians are halves of great ellipses (often called great circles), which converge at the North and South Poles. The meridian of the British Royal Observatory in Greenwich, in southeast London, England, is the international prime meridian, although some organizations—such as the French Institut Géographique National—continue to use other meridians for internal purposes. The prime meridian determines the proper Eastern and Western Hemispheres, although maps often divide these hemispheres further west in order to keep the Old World on a single side. The antipodal meridian of Greenwich is both 180°W and 180°E. This is not to be conflated with the International Date Line, which diverges from it in several places for political reasons, including between far eastern Russia and the far western Aleutian Islands.
The combination of these two components specifies the position of any location on the surface of Earth, without consideration of altitude or depth. The grid formed by lines of latitude and longitude is known as a "graticule".^{ [8] } The origin/zero point of this system is located in the Gulf of Guinea about 625 km (390 mi) south of Tema, Ghana.
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On the GRS80 or WGS84 spheroid at sea level at the Equator, one latitudinal second measures 30.715 metres , one latitudinal minute is 1843 metres and one latitudinal degree is 110.6 kilometres. The circles of longitude, meridians, meet at the geographical poles, with the westeast width of a second naturally decreasing as latitude increases. On the Equator at sea level, one longitudinal second measures 30.92 metres, a longitudinal minute is 1855 metres and a longitudinal degree is 111.3 kilometres. At 30° a longitudinal second is 26.76 metres, at Greenwich (51°28′38″N) 19.22 metres, and at 60° it is 15.42 metres.
On the WGS84 spheroid, the length in meters of a degree of latitude at latitude φ (that is, the distance along a north–south line from latitude (φ − 0.5) degrees to (φ + 0.5) degrees) is about
Similarly, the length in meters of a degree of longitude can be calculated as
(Those coefficients can be improved, but as they stand the distance they give is correct within a centimeter.)
An alternative method to estimate the length of a longitudinal degree at latitude is to assume a spherical Earth (to get the width per minute and second, divide by 60 and 3600, respectively):
where Earth's average meridional radius is 6,367,449 m. Since the Earth is not spherical that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude is
where Earth's equatorial radius equals 6,378,137 m and ; for the GRS80 and WGS84 spheroids, b/a calculates to be 0.99664719. ( is known as the reduced (or parametric) latitude). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 meter of each other if the two points are one degree of longitude apart.
Latitude  City  Degree  Minute  Second  ±0.0001° 

60°  Saint Petersburg  55.80 km  0.930 km  15.50 m  5.58 m 
51° 28′ 38″ N  Greenwich  69.47 km  1.158 km  19.30 m  6.95 m 
45°  Bordeaux  78.85 km  1.31 km  21.90 m  7.89 m 
30°  New Orleans  96.49 km  1.61 km  26.80 m  9.65 m 
0°  Quito  111.3 km  1.855 km  30.92 m  11.13 m 
To establish 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. Common map projections in current use include the Universal Transverse Mercator (UTM), the Military Grid Reference System (MGRS), the United States National Grid (USNG), the Global Area Reference System (GARS) and the World Geographic Reference System (GEOREF).^{ [10] } Coordinates on a map are usually in terms northing N and easting E offsets relative to a specified origin.
Map projection formulas depend in 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 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.^{ [11] } 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.^{ [11] }^{:4554}
The Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) coordinate systems both use a metricbased cartesian grid laid out on a conformally projected surface to locate positions on the surface of the Earth. The UTM system is not a single map projection but a series of sixty, each covering 6degree bands of longitude. The UPS system is used for the polar regions, which are not covered by the UTM system.
During medieval times, the stereographic coordinate system was used for navigation purposes.^{[ citation needed ]} The stereographic coordinate system was superseded by the latitudelongitude system. Although no longer used in navigation, the stereographic coordinate system is still used in modern times to describe crystallographic orientations in the fields of crystallography, mineralogy and materials science.^{[ citation needed ]}
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Vertical coordinates include height and depth.
Every point that is expressed in ellipsoidal coordinates can be expressed as an rectilinear x y z (Cartesian) coordinate. Cartesian coordinates simplify many mathematical calculations. The Cartesian systems of different datums are not equivalent.^{ [2] }
The earthcentered earthfixed (also known as the ECEF, ECF, or conventional terrestrial coordinate system) rotates with the Earth and has its origin at the center of the Earth.
The conventional righthanded coordinate system puts:
An example is the NGS data for a brass disk near Donner Summit, in California. Given the dimensions of the ellipsoid, the conversion from lat/lon/heightaboveellipsoid coordinates to XYZ is straightforward—calculate the XYZ for the given latlon on the surface of the ellipsoid and add the XYZ vector that is perpendicular to the ellipsoid there and has length equal to the point's height above the ellipsoid. The reverse conversion is harder: given XYZ we can immediately get longitude, but no closed formula for latitude and height exists. See "Geodetic system." Using Bowring's formula in 1976 Survey Review the first iteration gives latitude correct within 10^{11} degree as long as the point is within 10000 meters above or 5000 meters below the ellipsoid.
A local tangent plane can be defined based on the vertical and horizontal dimensions. The vertical coordinate can point either up or down. There are two kinds of conventions for the frames:
In many targeting and tracking applications the local ENU Cartesian coordinate system is far more intuitive and practical than ECEF or geodetic coordinates. The local ENU coordinates are formed from a plane tangent to the Earth's surface fixed to a specific location and hence it is sometimes known as a local tangent or local geodetic plane. By convention the east axis is labeled , the north and the up .
In an airplane, most objects of interest are below the aircraft, so it is sensible to define down as a positive number. The NED coordinates allow this as an alternative to the ENU. By convention, the north axis is labeled , the east and the down . To avoid confusion between and , etc. in this article we will restrict the local coordinate frame to ENU.
Similar coordinate systems are defined for other celestial bodies such as:
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.
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.
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 nautical navigation because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as straight segments that conserve the angles with the meridians. Although the linear scale is equal in all directions around any point, thus preserving the angles and the shapes of small objects, the Mercator projection distorts the size of objects as the latitude increases from the Equator to the poles, where the scale becomes infinite. So, for example, landmasses such as Greenland and Antarctica appear much larger than they actually are, relative to landmasses near the equator such as Central Africa.
In mathematics, a spherical coordinate system is a coordinate system for threedimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the threedimensional version of the polar coordinate system.
An azimuth is an angular measurement in a spherical coordinate system. The vector from an observer (origin) to a point of interest is projected perpendicularly onto a reference plane; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.
Earth radius is the distance from the center of Earth to a point on its surface. Its value ranges from 6,378 kilometres at the equator to 6,357 kilometres at a pole.
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 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 eastwest extent.
The use of orthographic projection in cartography dates back to antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection, in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges.
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 first way is the ratio of the size of the generating globe to the size of the Earth. The generating globe is a conceptual model to which the Earth is shrunk and from which the map is projected.
The equirectangular projection is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, and circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia and NASA World Wind, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth.
The Universal Transverse Mercator (UTM) conformal projection uses a 2dimensional Cartesian coordinate system to give locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, i.e. it is used to identify locations on the Earth independently of altitude. However, it differs from that method in several respects.
A Lambert conformal conic projection (LCC) is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and many national and regional mapping systems. It is one of seven projections introduced by Johann Heinrich Lambert in his 1772 publication Anmerkungen und Zusätze zur Entwerfung der Land und Himmelscharten.
ECEF, also known as ECR, is a geographic and Cartesian coordinate system and is sometimes known as a "conventional terrestrial" system. It represents positions as X, Y, and Z coordinates. The point is defined as the center of mass of Earth, hence the term geocentric coordinates. The distance from a given point of interest to the center of Earth is called the geocentric radius or geocentric distance.
Local tangent plane coordinates (LTP), sometimes named local vertical, local horizontal coordinates (LVLH), are a geographical coordinate system based on the local vertical direction and the Earth's axis of rotation. It consists of three coordinates: one represents the position along the northern axis, one along the local eastern axis, and one represents the vertical position. Two righthanded variants exist: east, north, up (ENU) coordinates and north, east, down (NED) coordinates. They serve for representing state vectors that is commonly used in aviation.
In cartography, the cylindrical equalarea projection is a family of cylindrical, equalarea map projections.
Web Mercator, Google Web Mercator, Spherical Mercator, WGS 84 Web Mercator or WGS 84/PseudoMercator is a variant of the Mercator projection and is the de facto standard for Web mapping applications. It rose to prominence when Google Maps adopted it in 2005. It is used by virtually all major online map providers, including Google Maps, Mapbox, Bing Maps, OpenStreetMap, Mapquest, Esri, and many others. Its official EPSG identifier is EPSG:3857, although others have been used historically.
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