Geographic coordinate system
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History[edit]
Geographic latitude and longitude[edit]
Main articles:
Latitude and
Longitude
The "latitude" (abbreviation: Lat.,
φ, or phi) of a point on the 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 3] Lines joining points of the same latitude trace circles on the surface of the 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 planeof 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 the Earth's surface is the angle east or west from a reference
meridian to another meridian that passes through that point. All meridians are halves of great
ellipses (often improperly called
great circles), which converge at the north and south poles. The meridian of the
British Royal Observatory in
Greenwich, a little east of 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 the Earth, without consideration of
altitude or depth. The grid thus formed by latitude and longitude is known as the "graticule". The zero/zero point of this system is located in the
Gulf of Guinea about 625 km (390 mi) south of
Tema,
Ghana.
Measuring height using datums[edit]
Complexity of the problem[edit]
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.
The Earth is not a sphere, but an irregular shape approximating a
biaxial ellipsoid. 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. Though early navigators thought of the sea as a flat surface that could be used as a vertical datum, this is not actually the case. The Earth has a series of layers of equal
potential energy within its
gravitational field. 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.
Common baselines[edit]
Common height baselines include
[2]
Along with the latitude
and longitude
, the height
provides the three-dimensional
geodetic coordinates or
geographic coordinates for a location.
[7]
In order to be unambiguous about the direction of "vertical" and the "surface" above which they are measuring, map-makers 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 systemor
geodetic datum. Datums may be
global, meaning that they represent the whole earth, or they may be
local, meaning that they represent a best-fit ellipsoid to only a portion of the earth.
Datums may be
global, meaning that they represent the whole earth, or they may be
local, meaning that they represent a best-fit ellipsoid 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 movement caused by the
Moon and the
tides. The 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 high-pressure area can cause a sinking of
5 mm.
Scandinaviais 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]
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, height systems in use. WGS 84 differs at Greenwich from the one used on published maps
OSGB36 by approximately 112m. The military system
ED50, used by
NATO, differs by about 120m to 180m.
[1]
The latitude and longitude on a map made against a local datum 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. 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.
[9] 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, three-dimensional
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'.
Map projection[edit]
Main article:
Map projection
Coordinates on a map are usually in terms
northing N and
easting E offsets relative to a specified origin. Usually associated with a map projection is a
natural origin at which the ellipsoid and flat map surfaces coincide.
[11]:9-10. To ensure that the northing and easting coordinates on a map are not negative, map projections may set up
false northing and
false easting values that offset the true northing and easting values.
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]:9-10 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]:45-54
UTM and UPS systems[edit]
Stereographic coordinate system[edit]
During medieval times, the stereographic coordinate system was used for navigation purposes.
[citation needed] The stereographic coordinate system was superseded by the latitude-longitude 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]
Cartesian coordinates[edit]
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]
Earth-centered, earth-fixed[edit]
Earth Centered, Earth Fixed coordinates in relation to latitude and longitude.
The
earth-centered earth-fixed (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 right-handed coordinate system puts:
- The origin at the center of mass of the earth, a point close to the Earth's center of figure
- The Z axis on the line between the north and south poles, with positive values increasing northward (but does not exactly coincide with the Earth's rotational axis[12])
- The X and Y axes in the plane of the equator
- The X axis passing through extending from 180 degrees longitude at the equator (negative) to 0 degrees longitude (prime meridian) at the equator (positive)
- The X axis passing through extending from 90 degrees west longitude at the equator (negative) to 90 degrees east longitude at the equator (positive)
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. 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.
Local east, north, up (ENU) coordinates[edit]
Earth Centred Earth Fixed and East, North, Up coordinates.
In many targeting and tracking applications the local
East, North, Up (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
.
Local north, east, down (NED) coordinates[edit]
In an airplane, most objects of interest are below the aircraft, so it is sensible to define down as a positive number. The
North, East, Down(NED) coordinates allow this as an alternative to the ENU local tangent plane. By convention, the north axis is labeled
, the east
and the down
. To avoid confusion between
and
, etc. in this web page we will restrict the local coordinate frame to ENU.
Expressing latitude and longitude as linear units[edit]
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 west-east width of a second naturally decreasing as latitude increases. On the
equator at sea level, one longitudinal
[contradiction] 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
-
-
-
-
- [13]
Similarly, the length in meters of a degree of longitude can be calculated as
-
-
-
-
- [13]
(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.
Longitudinal length equivalents at selected latitudes
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 |
Geostationary coordinates[edit]
Geostationary satellites (e.g., television satellites) are over the
equator at a specific point on Earth, so their position related to Earth is expressed in
longitude degrees only. Their
latitude is always zero, that is, over the equator.
On other celestial bodies[edit]
Similar coordinate systems are defined for other celestial bodies such as:
See also[edit]
- ^ In specialized works, "geographic coordinates" are distinguished from other similar coordinate systems, such as geocentric coordinates and geodetic coordinates. See, for example, Sean E. Urban and P. Kenneth Seidelmann, Explanatory Supplement to the Astronomical Almanac, 3rd. ed., (Mill Valley CA: University Science Books, 2013) p. 20–23.
- ^ The pair had accurate absolute distances within the Mediterranean but underestimated the circumference of the earth, causing their degree measurements to overstate its length west from Rhodes or Alexandria, respectively.
- ^ Alternative versions of latitude and longitude include geocentric coordinates, which measure with respect to the center of the earth, geodetic coordinates, which model the Earth as an ellipsoid, and geographic coordinates, which measure with respect to a plumb line at the location for which coordinates are given.
- ^ WGS 84 is the default datum used in most GPS equipment, but other datums can be selected.
References[edit]
- ^ a b c d e f A guide to coordinate systems in Great Britain (PDF), D00659 v2.3, Ordnance Survey, Mar 2015, retrieved 2015-06-22
- ^ a b c Taylor, Chuck. "Locating a Point On the Earth". Retrieved 4 March 2014.
- ^ McPhail, Cameron (2011), Reconstructing Eratosthenes' Map of the World (PDF),Dunedin: University of Otago, pp. 20–24.
- ^ Evans, James (1998), The History and Practice of Ancient Astronomy, Oxford: Oxford University Press, pp. 102–103, ISBN 9780199874453.
- ^ Greenwich 2000 Limited (9 June 2011). "The International Meridian Conference". Wwp.millennium-dome.com. Retrieved 31 October 2012.
- ^ DMA Technical Report Geodesy for the Layman, The Defense Mapping Agency, 1983