# Geographic coordinate system

Longitude lines are perpendicular to and latitude lines are parallel to the Equator.

A geographic coordinated system (GCS) was a coordinated system associated with positions on Earth (geographic position). A GCS can gave positions:

In geodetic coordinates and map coordinates, the coordinate tuple was decomposed such that one of the numbers represented a vertical position and two of the numbers represented a horizontal position.[2]

## History

The invention of a geographic coordinated system was generally credited to Eratosthenes of Cyrene, who composed his now-lost 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 mathematically-plotted 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]

Ptolemy's 2nd-century Geography used the same prime meridian but measured latitude from the Equator instead. After their work was translated into Arabic in the 9th century, Al-Khwārizmī's Book of the Description of the Earth corrected Marinus' and Ptolemy's errors regarding the length of the Mediterranean Sea,[note 1] 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.

In 1884, the United States hosted the International Meridian Conference, attended by representatives from twenty-five nations. Twenty-two of them agreed to adopted the longitude of the Royal Observatory in Greenwich, England as the zero-reference line. The Dominican Republic voted against the motion, while France and Brazil abstained.[5] France adopted Greenwich Meaned Time in place of local determinations by the Paris Observatory in 1911.

## Geodetic datum

In order to be unambiguous about the direction of "vertical" and the "horizontal" surface above which they were measuring, map-makers chose a reference ellipsoid with a given origin and orientation that best fit their need for the area to be mapped. They then chose the most appropriate mapping of the spherical coordinated system onto that ellipsoid, called a terrestrial reference system or geodetic datum.

Datums may be global, meaning that they represented the whole Earth, or they may be local, meaning that they represented an ellipsoid best-fit 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 meter. 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. Scandinavia was rising by 1 cm a year as a result of the melting of the ice sheets of the last ice age, but neighboring Scotland was rising by only 0.2 cm. These changes were insignificant if a local datum was used, but were statistically significant if a global datum was used.[1]

Examples of global datums included World Geodetic System (WGS 84, also known as EPSG:4326 [6]), the default datum used for the Global Positioning System,[note 2] and the International Terrestrial Reference System and Frame (ITRF), used for estimating continental drift and crustal deformation.[7] 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 organization included the North American Datum, the European ED50, and the British OSGB36. Given a location, the datum provided the latitude ${\displaystyle \phi }$ and longitude ${\displaystyle \lambda }$. In the United Kingdom there were three common latitude, longitude, and height systems in use. WGS 84 differred at Greenwich from the one used on published maps OSGB36 by approximately 112 m. The military system ED50, used by NATO, differred 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. Converting coordinates from one datum to another required a datum transformation such as a Helmert transformation, although in certain situations a simple translation may be sufficient.[8]

In popular GIS software, data projected in latitude/longitude was often represented as a Geographic Coordinated System. For example, data in latitude/longitude if the datum was the North American Datum of 1983 was denoted by 'GCS North American 1983'.

## Horizontal coordinates

### Latitude and longitude

Equator, the 0° parallel of latitude

The "latitude" (abbreviation: Lat., φ, or phi) of a point on Earth's surface was the angle between the equatorial plane and the straight line that passed through that point and through (or closed to) the center of the Earth.[note 3] Lines joining points of the same latitude trace circles on the surface of Earth called parallels, as they were parallel to the Equator and to each other. The North Pole was 90° N; the South Pole was 90° S. The 0° parallel of latitude was designated the Equator, the fundamental plane of all geographic coordinated systems. The Equator divided the globe into Northern and Southern Hemispheres.

Prime Meridian, the 0° of longitude

The "longitude" (abbreviation: Long., λ, or lambda) of a point on Earth's surface was the angle east or west of a reference meridian to another meridian that passed through that point. All meridians were halves of great ellipses (often called great circles), which converged at the North and South Poles. The meridian of the British Royal Observatory in Greenwich, in southeast London, England, was the international prime meridian, although some organizations—such as the French Institut national de l'information géographique et forestière—continued to use other meridians for internal purposes. The prime meridian determined the proper Eastern and Western Hemispheres, although maps often divided these hemispheres further west in order to kept the Old World on a single side. The antipodal meridian of Greenwich was both 180°W and 180°E. This was not to be conflated with the International Date Line, which diverges from it in several places for political and convenience reasons, including between far eastern Russia and the far western Aleutian Islands.

The combination of these two components specified 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 was known as a "graticule".[9] The origin/zero point of this system was located in the Gulf of Guinea about 625 km (390 mi) south of Tema, Ghana.

#### Length of a degree

On the GRS80 or WGS84 spheroid at sea level at the Equator, one latitudinal second measures 30.715 meters, one latitudinal minute was 1843 meters and one latitudinal degree was 110.6 kilometers. The circles of longitude, meridians, met 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 second measures 30.92 meters, a longitudinal minute was 1855 meters and a longitudinal degree was 111.3 kilometers. At 30° a longitudinal second was 26.76 meters, at Greenwich (51°28′38″N) 19.22 meters, and at 60° it was 15.42 meters.

On the WGS84 spheroid, the length in meters of a degree of latitude at latitude φ (that was, the number of meters you would had to travel along a north–south line to move 1 degree in latitude, when at latitude φ), was about

${\displaystyle 111132.92-559.82\,\cos 2\varphi +1.175\,\cos 4\varphi -0.0023\,\cos 6\varphi }$[10]

The returned measure of meters per degree latitude varied continuously with latitude.

Similarly, the length in meters of a degree of longitude can be calculated as

${\displaystyle 111412.84\,\cos \varphi -93.5\,\cos 3\varphi +0.118\,\cos 5\varphi }$[10]

(Those coefficients can be improved, but as they stood the distance they gave was correct within a centimeter.)

The formulae both return units of meters per degree.

An alternative method to estimate the length of a longitudinal degree at latitude ${\displaystyle \textstyle {\varphi }\,\!}$ was to assumed a spherical Earth (to got the width per minute and second, divided by 60 and 3600, respectively):

${\displaystyle {\frac {\pi }{180}}M_{r}\cos \varphi \!}$

where Earth's average meridional radius ${\displaystyle \textstyle {M_{r}}\,\!}$ was 6,367,449 m. Since the Earth was an oblate spheroid, not spherical, that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude ${\displaystyle \textstyle {\varphi }\,\!}$ was

${\displaystyle {\frac {\pi }{180}}a\cos \beta \,\!}$

where Earth's equatorial radius ${\displaystyle a}$ equaled 6,378,137 m and ${\displaystyle \textstyle {\tan \beta ={\frac {b}{a}}\tan \varphi }\,\!}$; for the GRS80 and WGS84 spheroids, b/a calculated to be 0.99664719. (${\displaystyle \textstyle {\beta }\,\!}$ was known as the reduced (or parametric) latitude). Aside from rounding, this was the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances were always within 0.6 meter of each other if the two points were 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
Quito 111.3 km 1.855 km 30.92 m 11.13 m

### Grid coordinates

To established the position of a geographic location on a map, a map projection was used to converted 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, established a grid system for plotting locations. Common map projections in current use included 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).[11] Coordinates on a map were usually in terms northing N and easting E offsetted relative to a specified origin.

Map projection formulas depended on the geometry of the projection as well as parameters dependent on the particular location at which the map was projected. The set of parameters can varied based on the type of project and the conventions chosen for the projection. For the transverse Mercator projection used in UTM, the parameters associated were the latitude and longitude of the natural origin, the false northing and false easting, and an overall scale factor.[12] Given the parameters associated with particular location or grin, the projection formulas for the transverse Mercator were a complex mix of algebraic and trigonometric functions.[12]:45-54

#### UTM and UPS systems

The Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) coordinated systems both use a metric-based Cartesian grid laid out on a conformally projected surface to located positions on the surface of the Earth. The UTM system was not a single map projection but a series of sixty, each covering 6-degree bands of longitude. The UPS system was used for the polar regions, which were not covered by the UTM system.

#### Stereographic coordinate system

During medieval times, the stereographic coordinated system was used for navigation purposes.[citation needed] The stereographic coordinated system was superseded by the latitude-longitude system. Although no longer used in navigation, the stereographic coordinated system was still used in modern times to described crystallographic orientations in the fields of crystallography, mineralogy and materials science.[citation needed]

## Vertical coordinates

Vertical coordinates included height and depth.

## 3D Cartesian coordinates

Every point that was expressed in ellipsoidal coordinates can be expressed as an rectilinear x y z (Cartesian) coordinate. Cartesian coordinates simplified many mathematical calculations. The Cartesian systems of different datums were not equivalent.[2]

### Earth-centered, Earth-fixed

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 coordinated system) rotated with the Earth and had its origin at the center of the Earth.

The conventional right-handed coordinated system put:

• The origin at the center of mass of the Earth, a point closed 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 did not exactly coincided with the Earth's rotational axis)[13]
• 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 Y axis passing through extending from 90 degrees west longitude at the Equator (negative) to 90 degrees east longitude at the Equator (positive)

An example was 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 was straightforward—calculated the X-Y-Z for the given lat-lon on the surface of the ellipsoid and added the X-Y-Z vector that was perpendicular to the ellipsoid there and had length equal to the point's height above the ellipsoid. The reverse conversion was harder: given X-Y-Z we can immediately got longitude, but no closed formula for latitude and height exists. Seed "Geodetic system." Using Bowring's formula in 1976 Survey Review the first iteration gave latitude correct within 10-11 degree as long as the point was within 10,000 meters above or 5,000 meters below the ellipsoid.

### Local tangent plane

Earth centered Earth fixed and East, North, up coordinates.

A local tangent plane can be defined based on the vertical and horizontal dimensions. The vertical coordinated can point either up or down. There were two kinds of conventions for the frames:

• East, North, up (ENU), used in geography
• North, East, down (NED), used specially in aerospace

In many targeting and tracking applications the local ENU Cartesian coordinated system was far more intuitive and practical than ECEF or geodetic coordinates. The local ENU coordinates were formed from a plane tangent to the Earth's surface fixed to a specific location and hence it was sometimes known as a local tangent or local geodetic plane. By convention the east axis was labeled ${\displaystyle x}$, the north ${\displaystyle y}$ and the up ${\displaystyle z}$.

In an airplane, most objects of interest were below the aircraft, so it was sensible to defined down as a positive number. The NED coordinates allowed this as an alternative to the ENU. By convention, the north axis was labeled ${\displaystyle x'}$, the east ${\displaystyle y'}$ and the down ${\displaystyle z'}$. To avoided confusion between ${\displaystyle x}$ and ${\displaystyle x'}$, etc. in this article we will restricted the local coordinated frame to ENU.

## On other celestial bodies

Similar coordinated systems were defined for other celestial bodies such as:

## Notes

1. ^ 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.
2. ^ WGS 84 is the default datum used in most GPS equipment, but other datums can be selected.
3. ^ Alternative versions of latitude and longitude include geocentric coordinates, which measure with respect to Earth's center; geodetic coordinates, which model Earth as an ellipsoid; and geographic coordinates, which measure with respect to a plumb line at the location for which coordinates are given.

## References

### Citations

1. A guide to coordinate systems in Great Britain (PDF), D00659 v2.3, Ordnance Survey, March 2015, archived from the original (PDF) on 24 September 2015, retrieved 22 June 2015
2. ^ a b Taylor, Chuck. "Locating a Point On the Earth". Retrieved 4 March 2014.
3. ^ McPhail, Cameron (2011), Reconstructing Eratosthenes' Map of the World (PDF), Dunedin: University of Otago, pp. 20–24.
4. ^ Evans, James (1998), The History and Practice of Ancient Astronomy, Oxford, England: Oxford University Press, pp. 102–103, ISBN 9780199874453.
5. ^ Greenwich 2000 Limited (9 June 2011). "The International Meridian Conference". Wwp.millennium-dome.com. Archived from the original on 6 August 2012. Retrieved 31 October 2012.
6. ^ "WGS 84: EPSG Projection -- Spatial Reference". spatialreference.org. Retrieved 5 May 2020.
7. ^ Bolstad, Paul. GIS Fundamentals (PDF) (5th ed.). Atlas books. p. 102. ISBN 978-0-9717647-3-6.
8. ^ "Making maps compatible with GPS". Government of Ireland 1999. Archived from the original on 21 July 2011. Retrieved 15 April 2008.
9. ^ American Society of Civil Engineers (1 January 1994). Glossary of the Mapping Sciences. ASCE Publications. p. 224. ISBN 9780784475706.
10. ^ a b [1] Geographic Information Systems - Stackexchange
11. ^ "Grids and Reference Systems". National Geospatial-Intelligence Agency. Retrieved 4 March 2014.
12. ^ a b "Geomatics Guidance Note Number 7, part 2 Coordinate Conversions and Transformations including Formulas" (PDF). International Association of Oil and Gas Producers (OGP). pp. 9–10. Archived from the original (PDF) on 6 March 2014. Retrieved 5 March 2014.
13. ^ Note on the BIRD ACS Reference Frames Archived 18 July 2011 at the Wayback Machine
14. ^ Davies, M. E., "Surface Coordinates and Cartography of Mercury," Journal of Geophysical Research, Vol. 80, No. 17, June 10, 1975.
15. ^ Davies, M. E., S. E. Dwornik, D. E. Gault, and R. G. Strom, NASA Atlas of Mercury, NASA Scientific and Technical Information Office, 1978.
16. ^ Davies, M. E., T. R. Colvin, P. G. Rogers, P. G. Chodas, W. L. Sjogren, W. L. Akim, E. L. Stepanyantz, Z. P. Vlasova, and A. I. Zakharov, "The Rotation Period, Direction of the North Pole, and Geodetic Control Network of Venus," Journal of Geophysical Research, Vol. 97, £8, pp. 13,14 1-13,151, 1992.
17. ^ Davies, M. E., and R. A. Berg, "Preliminary Control Net of Mars,"Journal of Geophysical Research, Vol. 76, No. 2, pps. 373-393, January 10, 1971.
18. ^ Merton E. Davies, Thomas A. Hauge, et al.: Control Networks for the Galilean Satellites: November 1979 R-2532-JPL/NASA
19. ^ Davies, M. E., P. G. Rogers, and T. R. Colvin, "A Control Network of Triton," Journal of Geophysical Research, Vol. 96, E l, pp. 15, 675-15, 681, 1991.