Geographic coordinate system

Geographic coordinate system

Overview

A geographic coordinate system is a coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

 that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position
Altitude
Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context...

, and two or three
N-vector
n-vector is a three parameter non-singular horizontal position representation well-suited for replacing latitude and longitude in mathematical calculations and computer algorithms. Geometrically, it is a unit vector that is normal to the reference ellipsoid. The vector is decomposed in an Earth...

 of the numbers represent horizontal position
Horizontal position representation
A position representation is the parameters used to express a position relative to a reference. Representing position in three dimensions is often done by a Euclidean vector. However, when representing position relative to the Earth it is often more convenient to represent vertical position as...

. A common choice of coordinates is latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

 and elevation
Elevation
The elevation of a geographic location is its height above a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface ....

.



The geographic latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

 (abbreviation: Lat., φ, or phi) of a point on the Earth's surface is the angle between the equatorial plane and a line that passes through that point and is normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

 to the surface of a reference ellipsoid
Reference ellipsoid
In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body....

 which approximates the shape of the Earth
Figure of the Earth
The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earth's size and shape is to be defined. The actual topographic surface is most apparent with its variety of land forms and water areas. This is, in fact, the surface...

.The surface of the Earth is closer to an ellipsoid
Earth ellipsoid
An Earth ellipsoid is a mathematical figure approximating the shape of the Earth, used as a reference frame for computations in geodesy, astronomy and the geosciences...

 than to a sphere, as its equatorial diameter is larger than its north-south diameter.
This line passes a few kilometers away from the center of the Earth except at the poles and the equator where it passes through Earth's center.The greatest distance between an ellipsoid normal and the center of the Earth is 21.9 km at a latitude of 45°, using Earth radius#Radius at a given geodetic latitude and Latitude#Comparison of selected types: Lines joining points of the same latitude trace circles on the surface of the Earth called parallels
Circle of latitude
A circle of latitude, on the Earth, is an imaginary east-west circle connecting all locations that share a given latitude...

, as they are parallel to the equator and to each other.
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Encyclopedia

A geographic coordinate system is a coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

 that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position
Altitude
Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context...

, and two or three
N-vector
n-vector is a three parameter non-singular horizontal position representation well-suited for replacing latitude and longitude in mathematical calculations and computer algorithms. Geometrically, it is a unit vector that is normal to the reference ellipsoid. The vector is decomposed in an Earth...

 of the numbers represent horizontal position
Horizontal position representation
A position representation is the parameters used to express a position relative to a reference. Representing position in three dimensions is often done by a Euclidean vector. However, when representing position relative to the Earth it is often more convenient to represent vertical position as...

. A common choice of coordinates is latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

 and elevation
Elevation
The elevation of a geographic location is its height above a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface ....

.

Geographic latitude and longitude




The geographic latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

 (abbreviation: Lat., φ, or phi) of a point on the Earth's surface is the angle between the equatorial plane and a line that passes through that point and is normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

 to the surface of a reference ellipsoid
Reference ellipsoid
In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body....

 which approximates the shape of the Earth
Figure of the Earth
The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earth's size and shape is to be defined. The actual topographic surface is most apparent with its variety of land forms and water areas. This is, in fact, the surface...

.The surface of the Earth is closer to an ellipsoid
Earth ellipsoid
An Earth ellipsoid is a mathematical figure approximating the shape of the Earth, used as a reference frame for computations in geodesy, astronomy and the geosciences...

 than to a sphere, as its equatorial diameter is larger than its north-south diameter.
This line passes a few kilometers away from the center of the Earth except at the poles and the equator where it passes through Earth's center.The greatest distance between an ellipsoid normal and the center of the Earth is 21.9 km at a latitude of 45°, using Earth radius#Radius at a given geodetic latitude and Latitude#Comparison of selected types: Lines joining points of the same latitude trace circles on the surface of the Earth called parallels
Circle of latitude
A circle of latitude, on the Earth, is an imaginary east-west circle connecting all locations that share a given latitude...

, as they are parallel to the equator and to each other. The north pole
North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...

 is 90° N; the south pole
South Pole
The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where the Earth's axis of rotation intersects its surface. It is the southernmost point on the surface of the Earth and lies on the opposite side of the Earth from the North Pole...

 is 90° S. The 0° parallel of latitude is designated the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

, the fundamental plane
Fundamental plane (spherical coordinates)
The fundamental plane in a spherical coordinate system is a plane which divides the sphere into two hemispheres. The latitude of a point is then the angle between the fundamental plane and the line joining the point to the centre of the sphere....

 of all geographic coordinate systems. The equator divides the globe into Northern and Southern Hemispheres.

The Longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

 (abbreviation: Long., λ, or lambda) of a point on the Earth's surface is the angle east or west from a reference meridian
Meridian (geography)
A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

 to another meridian that passes through that point. All meridians are halves of great ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

s (often improperly called great circle
Great circle
A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

s), which converge at the north and south poles.

A line passing near the Royal Observatory, Greenwich
Royal Observatory, Greenwich
The Royal Observatory, Greenwich , in London, England played a major role in the history of astronomy and navigation, and is best known as the location of the prime meridian...

 (near London in the UK
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...

) has been chosen as the international zero-longitude reference line, the Prime Meridian
Prime Meridian
The Prime Meridian is the meridian at which the longitude is defined to be 0°.The Prime Meridian and its opposite the 180th meridian , which the International Date Line generally follows, form a great circle that divides the Earth into the Eastern and Western Hemispheres.An international...

. Places to the east are in the eastern hemisphere, and places to the west are in the western hemisphere. The antipodal
Antipodes
In geography, the antipodes of any place on Earth is the point on the Earth's surface which is diametrically opposite to it. Two points that are antipodal to one another are connected by a straight line running through the centre of the Earth....

 meridian of Greenwich is both 180°W and 180°E. The zero/zero point is located in the Gulf of Guinea about 625 km south of Tema, Ghana.

In 1884 the United States hosted the International Meridian Conference
International Meridian Conference
The International Meridian Conference was a conference held in October 1884 in Washington, D.C., in the United States to determine the Prime Meridian of the world. The conference was held at the request of U.S. President Chester A...

 and twenty-five nations attended. Twenty-two of them agreed to adopt the location of Greenwich as the zero-reference line. The Dominican Republic
Dominican Republic
The Dominican Republic is a nation on the island of La Hispaniola, part of the Greater Antilles archipelago in the Caribbean region. The western third of the island is occupied by the nation of Haiti, making Hispaniola one of two Caribbean islands that are shared by two countries...

 voted against the adoption of that motion, while France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

 and Brazil
Brazil
Brazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...

 abstained. To date, there exist organizations around the world which continue to use historical prime meridians which existed before the acceptance of Greenwich became common-place.The French Institut Géographique National (IGN) maps still use longitude from a meridian passing through Paris, along with longitude from Greenwich.

The combination of these two components specifies the position of any location on the planet, but does not consider altitude
Altitude
Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context...

 nor depth.

This latitude/longitude "webbing" is known as the conjugate graticule.

In defining an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

, the short (vertical) diameter is known as the conjugate diameter, and the long (horizontal) diameter—perpendicular, or "transverse", to the conjugate—is the transverse diameter. With a sphere or ellipsoid, the conjugate diameter is known as the polar axis
Semi-minor axis
In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis...

and the transverse as the equatorial axis
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

. The graticule perspective
Perspective (graphical)
Perspective in the graphic arts, such as drawing, is an approximate representation, on a flat surface , of an image as it is seen by the eye...

 is based on this designation: As the longitudinal rings — geographically defined, all great circles — converge at the poles, it is the poles that the conjugate graticule is defined. If the polar vertex is "pulled down" 90°, so that the vertex is on the equator, or transverse diameter, then it becomes the transverse graticule, upon which all spherical trigonometry
Spherical trigonometry
Spherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...

 is ultimately based (if the longitudinal vertex is between the poles and equator, then it is considered an oblique graticule).

Latitude and longitude in practice


Say you set up your Wild T4 next to the water tank north of the airport at Hilo, Hawaii, intending to determine its latitude and longitude by the stars. NGS predicts you will find the tank to be at 19.7323 deg North, 155.0412 deg West.

You cross the island and set the T4 next to the Keahole Point lighthouse; NGS estimates that by the stars the lighthouse will turn out to be 19.7244 N 156.0787 W. Calculating the distance from the water tank to the lighthouse using those lat-lons we get about 108.8 km, but if we measure the actual distance it turns out to be 105.5 km. What went wrong?

Hawaii is an extreme case of a problem that exists everywhere: when trying to measure latitude and longitude by the stars we can only orient our measuring device by gravity. We'd like the T4's axis to point to the center of the Earth, but the T4's level vials don't know where that is — all they know is the direction of gravity, which is much affected by that 4000-meter mountain 50 km away. So when we measure the lat-lons for two points the relationship between those two points can be distorted, which renders their lat-lons fairly useless for most people. When we measure the lat-lons of two points we want to be able to use those lat-lons to calculate the distance and direction from one to the other; we want to be able to draw a scale map and plot points on it by their lat-lons, and the distance between any pair of points on the map is supposed to closely match the actual distance we would measure on the ground.

So we need a different plan — a different definition of latitude and longitude. What they did in Hawaii circa 1930 was call the marker "Oahu West Base" 21 deg 18 min 13.889 sec North, 157 deg 50 min 55.796 West, and define the lat-lon of every other point by its distance and direction from there. NGS now says that in 1993 that point was 21-18-02.54891 N 157-50-45.90280 W in the present NAD83 system. Was the old lat-lon off by 300+ meters? Well, yes, but the relationships between points in the islands were much more accurate than that. C&GS triangulated from island to island, calculating each successive point's lat-lon by its distance and direction from the previous points in the chain. Eventually they deemed the Hilo water tank to be at 19-43-54.526 N 155-03-26.463 W, which would make it 339191.7 meters from Oahu West Base on the Clarke 1866 spheroid. NGS now figures those two points are 339192.8 meters apart.

Similarly in North America. If in 1980 you had asked NGS for the lat-lons for the Empire State Building and a certain water tank in Anchorage, the NAD27 lat-lons they would have given you would be different from the current ones, but the distance you would have calculated then is 8.2 meters different from now. A transcontinental triangulation cannot do better than that.

UTM and UPS systems



The Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) coordinate systems both use a metric-based 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 map projections, one for each of sixty 6-degree bands of longitude. The UPS system is used for the polar regions, which are not covered by the UTM system.

Stereographic coordinate system


During medieval times, the stereographic coordinate system was used for navigation purposes. The stereographic coordinate system was superseded by the latitude-longitude system.

Although no longer used in navigations, the stereographic coordinate system is still used in modern times to describe crystallographic orientations in the fields of crystallography
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...

, mineralogy
Mineralogy
Mineralogy is the study of chemistry, crystal structure, and physical properties of minerals. Specific studies within mineralogy include the processes of mineral origin and formation, classification of minerals, their geographical distribution, as well as their utilization.-History:Early writing...

 and materials science.

Geodetic height


To completely specify a location of a topographical feature on, in, or above the Earth, one has to also specify the vertical distance from the centre of the Earth, or from the surface of the Earth. Because of the ambiguity of "surface" and "vertical", it is more commonly expressed relative to a precisely defined vertical datum which holds fixed some known point. Each country has defined its own datum. For example, in the United Kingdom
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...

 the reference point is Newlyn
Newlyn
Newlyn is a town and fishing port in southwest Cornwall, England, United Kingdom.Newlyn forms a conurbation with the neighbouring town of Penzance and is part of Penzance civil parish...

, while in Canada, Mexico and the United States, the point is near Rimouski, Quebec
Quebec
Quebec or is a province in east-central Canada. It is the only Canadian province with a predominantly French-speaking population and the only one whose sole official language is French at the provincial level....

, Canada
Canada
Canada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...

. The distance to Earth's centre can be used both for very deep positions and for positions in space.

Cartesian coordinates


Every point that is expressed in ellipsoidal coordinates can be expressed as an (Cartesian) coordinate. Cartesian coordinates simplify many mathematical calculations. The origin is usually the center of mass of the earth, a point close to the Earth's center of figure
Figure of the Earth
The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earth's size and shape is to be defined. The actual topographic surface is most apparent with its variety of land forms and water areas. This is, in fact, the surface...

.

With the origin at the center of the ellipsoid, the conventional setup is the expected right-hand:

Z-axis along the axis of the ellipsoid, positive northward

X- and Y-axis in the plane of the equator, X-axis positive toward 0 degrees longitude and Y-axis positive toward 90 degrees east longitude

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/height-above-ellipsoid coordinates to X-Y-Z is straightforward—calculate the X-Y-Z for the given lat-lon on the surface of the ellipsoid and add the X-Y-Z 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 X-Y-Z we can immediately get longitude, but no closed formula for latitude and height exists. However, using Bowring's formula in 1976 Survey Review the first iteration gives latitude correct within degree as long as the point is within 10000 meters above or 5000 meters below the ellipsoid.

Shape of the Earth



The Earth is not a sphere, but an irregular shape approximating a biaxial ellipsoid
Earth ellipsoid
An Earth ellipsoid is a mathematical figure approximating the shape of the Earth, used as a reference frame for computations in geodesy, astronomy and the geosciences...

. It is nearly spherical, but has an equatorial bulge making the radius at the equator about 0.3% larger than the radius measured through the poles. The shorter axis approximately coincides with axis of rotation. Map-makers choose the true ellipsoid 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. In the United Kingdom there are three common latitude, longitude, height systems in use. The system used by GPS, WGS84
World Geodetic System
The World Geodetic System is a standard for use in cartography, geodesy, and navigation. It comprises a standard coordinate frame for the Earth, a standard spheroidal reference surface for raw altitude data, and a gravitational equipotential surface that defines the nominal sea level.The latest...

, differs at Greenwich from the one used on published maps OSGB36 by approximately 112m. The military system ED50
ED50
ED 50 is a geodetic datum which was defined after World War II for the international connection of geodetic networks....

, used by NATO, differs by about 120m to 180m.

Though early navigators thought of the sea as a flat surface that could be used as a vertical datum, this is far from reality. The Earth has a series of layers of equal potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

 within its gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

. Height is a measurement at right angles to this surface, roughly toward the centre of the Earth, but local variations make the equipotential layers irregular (though roughly ellipsoidal). The choice of which layer to use for defining height is arbitrary. The reference height we have chosen is the one closest to the average height of the world's oceans. This is called the geoid
Geoid
The geoid is that equipotential surface which would coincide exactly with the mean ocean surface of the Earth, if the oceans were in equilibrium, at rest , and extended through the continents . According to C.F...

.

The Earth is not static as points move relative to each other due to continental plate motion, subsidence, and diurnal movement caused by the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

 and the tide
Tide
Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the moon and the sun and the rotation of the Earth....

s. The daily movement can be as much as a metre. Continental movement can be up to a year, or in a century. A weather system high-pressure area can cause a sinking of . Scandinavia
Scandinavia
Scandinavia is a cultural, historical and ethno-linguistic region in northern Europe that includes the three kingdoms of Denmark, Norway and Sweden, characterized by their common ethno-cultural heritage and language. Modern Norway and Sweden proper are situated on the Scandinavian Peninsula,...

 is rising by a year as a result of the melting of the ice sheets of the last ice age
Ice age
An ice age or, more precisely, glacial age, is a generic geological period of long-term reduction in the temperature of the Earth's surface and atmosphere, resulting in the presence or expansion of continental ice sheets, polar ice sheets and alpine glaciers...

, but neighbouring Scotland
Scotland
Scotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...

 is rising by only . These changes are insignificant if a local datum is used, but are significant if the global GPS datum is used.

Expressing latitude and longitude as linear units


On the GRS80 or WGS84 spheroid at sea level
Sea level
Mean sea level is a measure of the average height of the ocean's surface ; used as a standard in reckoning land elevation...

 at the equator, one latitudinal second measures 30.715 metre
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

s
, 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 west-east width of a second naturally decreasing as latitude increases. On the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

 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

111132.954 - 559.822(cos 2φ) + 1.175(cos 4φ)

(Those coefficients can be improved, but as they stand the distance they give is correct within a centimeter.)

To estimate the length of a longitudinal degree at latitude we can 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 . Since the Earth isn't 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 parametric or reduced 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.
Longitudinal length equivalents at selected latitudes
Latitude Town Degree Minute Second ±0.0001°
60° Saint Petersburg
Saint Petersburg
Saint Petersburg is a city and a federal subject of Russia located on the Neva River at the head of the Gulf of Finland on the Baltic Sea...

55.65 km 0.927 km 15.42 m 5.56 m
51° 28' 38" N Greenwich
Greenwich
Greenwich is a district of south London, England, located in the London Borough of Greenwich.Greenwich is best known for its maritime history and for giving its name to the Greenwich Meridian and Greenwich Mean Time...

69.29 km 1.155 km 19.24 m 6.93 m
45° Bordeaux
Bordeaux
Bordeaux is a port city on the Garonne River in the Gironde department in southwestern France.The Bordeaux-Arcachon-Libourne metropolitan area, has a population of 1,010,000 and constitutes the sixth-largest urban area in France. It is the capital of the Aquitaine region, as well as the prefecture...

78.7 km 1.31 km 21.86 m 7.87 m
30° New Orleans 96.39 km 1.61 km 26.77 m 9.63 m
Quito
Quito
San Francisco de Quito, most often called Quito , is the capital city of Ecuador in northwestern South America. It is located in north-central Ecuador in the Guayllabamba river basin, on the eastern slopes of Pichincha, an active stratovolcano in the Andes mountains...

111.3 km 1.855 km 30.92 m 11.13 m


Datums often encountered



Latitude and longitude values can be based on different geodetic system
Geodetic system
Geodetic systems or geodetic data are used in geodesy, navigation, surveying by cartographers and satellite navigation systems to translate positions indicated on their products to their real position on earth....

s or datums, the most common being WGS 84
World Geodetic System
The World Geodetic System is a standard for use in cartography, geodesy, and navigation. It comprises a standard coordinate frame for the Earth, a standard spheroidal reference surface for raw altitude data, and a gravitational equipotential surface that defines the nominal sea level.The latest...

, a global datum used by all GPS equipment.WGS 84 is the default datum used in most GPS equipment, but other datums can be selected. Other datums are significant because they were chosen by a national cartographical organisation as the best method for representing their region, and these are the datums used on printed maps. The latitude and longitude on a map may not be the same as on a GPS receiver. Coordinates from the mapping system can sometimes be roughly changed into another datum using a simple translation
Translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...

. 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. More generally one datum is changed into any other datum using a process called Helmert transformation
Helmert transformation
The Helmert transformation is a transformation method within a three-dimensional space...

s. This involves converting the spherical coordinates into Cartesian coordinates and applying a seven parameter transformation (translation, three-dimensional rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

), and converting back.

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'.

Geostationary coordinates


Geostationary satellites (e.g., television satellites) are over the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

 at a specific point on Earth, so their position related to Earth is expressed in longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

 degrees only. Their latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

 is always zero, that is, over the equator.

See also


  • Automotive navigation system
    Automotive navigation system
    An automotive navigation system is a satellite navigation system designed for use in automobiles. It typically uses a GPS navigation device to acquire position data to locate the user on a road in the unit's map database. Using the road database, the unit can give directions to other locations...

  • Digital Earth Reference Model
    Digital Earth Reference Model
    The term Digital Earth Reference Model was coined by Tim Foresman in context with a vision for an all encompassing geospatial platform as an abstract for information flow in support of Al Gore’s vision for a Digital Earth...

     (DERM)
  • Geographic coordinate conversion
    Geographic coordinate conversion
    -Ways of writing coordinates:All of the following are valid and acceptable ways to write geographic coordinates:* 40:26:46N,79:56:55W* 40:26:46.302N 79:56:55.903W* 40°26′47″N 79°58′36″W* 40d 26′ 47″ N 79d 58′ 36″ W* 40.446195N 79.948862W...

  • Geocoding
    Geocoding
    Geocoding is the process of finding associated geographic coordinates from other geographic data, such as street addresses, or zip codes...

  • Geodetic system
    Geodetic system
    Geodetic systems or geodetic data are used in geodesy, navigation, surveying by cartographers and satellite navigation systems to translate positions indicated on their products to their real position on earth....

  • Geographical distance
    Geographical distance
    Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude.-An abstraction:...

  • Geotagging
    GeoTagging
    Geotagging is the process of adding geographical identification metadata to various media such as a geotagged photograph or video, websites, SMS messages, QR Codes or RSS feeds and is a form of geospatial metadata...

  • Great-circle distance
    Great-circle distance
    The great-circle distance or orthodromic distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere . Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a...

     the shortest distance between any two points on the surface of a sphere.
  • Lambert coordinate system
    Lambert conformal conic projection
    A Lambert conformal conic projection is a conic map projection, which is often used for aeronautical charts. In essence, the projection superimposes a cone over the sphere of the Earth, with two reference parallels secant to the globe and intersecting it. This minimizes distortion from projecting...

  • Map projection
    Map projection
    A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...

  • Tropic of Cancer
    Tropic of Cancer
    The Tropic of Cancer, also referred to as the Northern tropic, is the circle of latitude on the Earth that marks the most northerly position at which the Sun may appear directly overhead at its zenith...

  • Tropic of Capricorn
    Tropic of Capricorn
    The Tropic of Capricorn, or Southern tropic, marks the most southerly latitude on the Earth at which the Sun can be directly overhead. This event occurs at the December solstice, when the southern hemisphere is tilted towards the Sun to its maximum extent.Tropic of Capricorn is one of the five...


External links