Geographic coordinate system

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 three-dimensional 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]

FedStats Lat long
Longitude lines are perpendicular to and latitude lines are parallel to the Equator.

History

The invention of a geographic coordinate system is 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,[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.

In 1884, the United States hosted the International Meridian Conference, attended by representatives from twenty-five nations. Twenty-two of them agreed to adopt 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 Mean 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 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 system or geodetic datum.

Datums may be global, meaning that they represent the whole Earth, or they may be local, meaning that they represent 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 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. 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]

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 112m. The military system ED50, used by NATO, differs from about 120m to 180m.[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, 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'.

Horizontal coordinates

Latitude and longitude

Line across the Earth
Equator, the 0° parallel of latitude

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.

Line across the Earth
Prime Meridian, the 0° of longitude

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 south-east 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.

Length of a degree

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 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

[9]

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

[9]

(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
Quito 111.3 km 1.855 km 30.92 m 11.13 m

Map projection

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]:45-54

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 sixty, each covering 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 navigation, the stereographic coordinate system is still used in modern times to describe crystallographic orientations in the fields of crystallography, mineralogy and materials science.

Vertical coordinates

Vertical coordinates include height and depth.

3D Cartesian coordinates

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

ECEF
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 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 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 tangent plane

ECEF ENU Longitude Latitude relationships
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 coordinate can point either up or down. There are 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 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.

On other celestial bodies

Similar coordinate systems are defined for other celestial bodies such as:

See also

Notes

  1. ^ 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.
  2. ^ 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.
  3. ^ WGS 84 is the default datum used in most GPS equipment, but other datums can be selected.
  4. ^ 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 b c d e A guide to coordinate systems in Great Britain (PDF), D00659 v2.3, Ordnance Survey, Mar 2015, retrieved 2015-06-22
  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: 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. ^ Bolstad, Paul. GIS Fundamentals, 5th Edition (PDF). Atlas books. p. 102. ISBN 978-0-9717647-3-6.
  7. ^ "Making maps compatible with GPS". Government of Ireland 1999. Archived from the original on 21 July 2011. Retrieved 15 April 2008.
  8. ^ American Society of Civil Engineers (1994-01-01). Glossary of the Mapping Sciences. ASCE Publications. p. 224. ISBN 9780784475706.
  9. ^ a b [1] Geographic Information Systems - Stackexchange
  10. ^ "Grids and Reference Systems". National Geospatial-Intelligence Agenc. Retrieved 4 March 2014.
  11. ^ 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.
  12. ^ Note on the BIRD ACS Reference Frames Archived 18 July 2011 at the Wayback Machine

Sources

External links

Celestial coordinate system

In astronomy, a celestial coordinate system (or celestial reference system) is a system for specifying positions of celestial objects: satellites, planets, stars, galaxies, and so on. Coordinate systems can specify an object's position in three-dimensional space or plot merely its direction on a celestial sphere, if the object's distance is unknown or trivial.

The coordinate systems are implemented in either spherical or rectangular coordinates. Spherical coordinates, projected on the celestial sphere, are analogous to the geographic coordinate system used on the surface of Earth. These differ in their choice of fundamental plane, which divides the celestial sphere into two equal hemispheres along a great circle. Rectangular coordinates, in appropriate units, are simply the cartesian equivalent of the spherical coordinates, with the same fundamental (x, y) plane and primary (x-axis) direction. Each coordinate system is named after its choice of fundamental plane.

Coordinate (disambiguation)

Coordinate may refer to:

An element of a coordinate system in geometry and related domains

Coordinate space in mathematics

Cartesian coordinate system

Coordinate (vector space)

Geographic coordinate system

Coordinate structure in linguistics

Coordinate covalent bond in chemistry

Coordinate descent, an algorithm

Fundamental plane (spherical coordinates)

The fundamental plane in a spherical coordinate system is a plane of reference that 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.For a geographic coordinate system of the Earth, the fundamental plane is the Equator. Celestial coordinate systems have varying fundamental planes:

The horizontal coordinate system uses the observer's horizon.

The Besselian coordinate system uses Earth's terminator (day/night boundary). This is a Cartesian coordinate system (x, y, z).

The equatorial coordinate system uses the celestial equator.

The ecliptic coordinate system uses the ecliptic.

The galactic coordinate system uses the Milky Way's galactic equator.

Graticule

Graticule may refer to:

An oscilloscope graticule scale

The reticle pattern in an optical instrument

The grid used in the geographic coordinate system

Horizontal position representation

A position representation is the parameters used to express a position relative to a reference. When representing positions relative to the Earth, it is often most convenient to represent vertical position (height or depth) separately, and to use some other parameters to represent horizontal position.

There are also several applications where only the horizontal position is of interest, this might e.g. be the case for ships and ground vehicles/cars.

It is a type of geographic coordinate system.

There are several options for horizontal position representations, each with different properties which makes them appropriate for different applications. Latitude/longitude and UTM are common horizontal position representations.

The horizontal position has two degrees of freedom, and thus two parameters are sufficient to uniquely describe such a position. However, similarly to the use of Euler angles as a formalism for representing rotations, using only the minimum number of parameters gives singularities, and thus three parameters are required for the horizontal position to avoid this.

Initial point

In surveying, an initial point is a datum (a specific point on the surface of the earth) that marks the beginning point for a cadastral survey. The initial point establishes a local geographic coordinate system for the surveys that refer to that point.

An initial point is defined by the intersection of a principal meridian and a base line.

Irish Transverse Mercator

Irish Transverse Mercator (ITM) is the geographic coordinate system for Ireland. It was implemented jointly by the Ordnance Survey Ireland (OSi) and the Ordnance Survey of Northern Ireland (OSNI) in 2001. The name is derived from the Transverse Mercator projection it uses and the fact that it is optimised for the island of Ireland.

Israeli Cassini Soldner

Israeli Cassini Soldner (Hebrew: רשת ישראל הישנה‎ Reshet Yisra'el Ha-Yeshana; ICS) is the old geographic coordinate system for Israel. The name is derived from the Cassini Soldner projection it uses and the fact that it is optimized for Israel. ICS has been mostly replaced by the new coordinate system Israeli Transverse Mercator (ITM), but still referenced by older books and navigation software.

Israeli Transverse Mercator

Israeli Transverse Mercator (Hebrew: רשת ישראל החדשה‎ Reshet Yisra'el Ha-Ḥadasha; ITM) is the new geographic coordinate system for Israel. The name is derived from the Transverse Mercator projection it uses and the fact that it is optimized for Israel. ITM has replaced the old coordinate system ICS. This coordinate system is sometimes also referred as the "New Israeli Grid". It has been use since January 1, 1994.

Jordan Transverse Mercator

Jordan Transverse Mercator (JTM) (Arabic: نظام تربيع ميركاتور الأردني المستعرض) is a grid system created by the Royal Jordan Geographic Center (RJGC). This system is based on 6° belts with a Central Meridian of 37° East and a Scale Factor at Origin (mo) = 0.9998. The JTM is based on the Hayford ellipsoid adopted by the IUGG in 1924. No transformation parameters are presently offered by the government. However, Prof. Stephen H. Savage of Arizona State University provides the following parameters for the projection:

Jordan Transverse Mercator

Geographic Coordinate System: GCS_International_1924

Datum: D:International_1924

Spheroid: International_1924

Axis: 6378388

Flattening: 297

Prime Meridian: Greenwich

Prime Meridian Longitude: 0

Units: Degree

Unit Scale Factor: 0.017453292519943295

Projection: Transverse Mercator

False Easting: 500,000

False Northing: -3,000,000

Central Meridian: 37

Scale Factor: 0.9998

Central Parallel: 0

Units: Meter

Scale Factor 1

Three-parameter transformation to WGS84 is:

ΔX = –86 meters

ΔY = –98 meters

ΔZ = –119 meters

Prof. Savage also offers software, ReprojectME!, which will convert coordinates between JTM and other systems. (See http://daahl.ucsd.edu/gaialab/# for more information.)

Palestine grid

The Palestine grid (Arabic: التربيع الفلسطيني) was the geographic coordinate system used in Mandatory Palestine.

The system was chosen by the Survey Department of the Government of Palestine in 1922. The projection used was the Cassini-Soldner projection. The central meridian (the line of longitude along which there is no local distortion) was chosen as that passing through a marker on the hill of Mar Elias Monastery south of Jerusalem. The false origin (zero point) of the grid was placed 100 km to the south and west of the Ali el-Muntar hill that overlooks Gaza city. The unit length for the grid was the kilometre; the British units were not even considered.At the time the grid was established, there was no intention of mapping the lower reaches of the Negev Desert, but this did not remain true. The fact that those southern regions would have negative north-south coordinate then became a source of confusion, which was solved by adding 1000 to the northern coordinate in that case. For some military purposes, 1000 was added to the north-south coordinates of all locations, so that they then ranged uniformly from about 900 to about 1300.

During World War II, a Military Palestine Grid was used that was similar to the Palestine Grid but used the transverse Mercator projection. The difference between the two projections was only a few meters.After the establishment of the State of Israel, the Palestine grid continued to be used under the name of the Israel Grid or the Israeli Cassini Soldner (ICS) grid, now called the "Old Israeli Grid", with 1000km added to the northing component to make the north-south range continuous. It was replaced by the Israeli Transverse Mercator grid in 1994. The Palestine grid is still commonly used to specify locations in the historical and archaeological literature.

Prime meridian

A prime meridian is a meridian (a line of longitude) in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian (the 180th meridian in a 360°-system) form a great circle. This great circle divides a spheroid, e.g., Earth, into two hemispheres. If one uses directions of East and West from a defined prime meridian, then they can be called the Eastern Hemisphere and the Western Hemisphere.

A prime meridian is ultimately arbitrary, unlike an equator, which is determined by the axis of rotation—and various conventions have been used or advocated in different regions and throughout history. The most widely used modern meridian is the IERS Reference Meridian. It is derived but deviates slightly from the Greenwich Meridian, which was selected as an international standard in 1884.

QRA locator

The QRA locator, also called QTH locator in some publications, is an obsolete geographic coordinate system used by amateur radio operators in Europe before the introduction of the Maidenhead Locator System. As a radio transmitter or receiver location system the QRA locator is considered defunct, but may be found in many older documents.

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.

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.

In the context of standardization and geographic applications, a geodesic reference ellipsoid is the mathematical model used as foundation by Spatial reference system or Geodetic datum definitions.

South American Datum

The South American Datum (SAD) is a regional geodesic datum for South America. It was established in Brazil by SIRGAS 2000, and was made official in 2005.

State Plane Coordinate System

The State Plane Coordinate System (SPS or SPCS) is a set of 124 geographic zones or coordinate systems designed for specific regions of the United States. Each state contains one or more state plane zones, the boundaries of which usually follow county lines. There are 110 zones in the contiguous US, with 10 more in Alaska, 5 in Hawaii, and one for Puerto Rico and US Virgin Islands. The system is widely used for geographic data by state and local governments. Its popularity is due to at least two factors. First, it uses a simple Cartesian coordinate system to specify locations rather than a more complex spherical coordinate system (the geographic coordinate system of latitude and longitude). By using the Cartesian coordinate system's simple XY coordinates, "plane surveying" methods can be used, speeding up and simplifying calculations. Second, the system is highly accurate within each zone (error less than 1:10,000). Outside a specific state plane zone accuracy rapidly declines, thus the system is not useful for regional or national mapping.

Most state plane zones are based on either a transverse Mercator projection or a Lambert conformal conic projection. The choice between the two map projections is based on the shape of the state and its zones. States that are long in the east–west direction are typically divided into zones that are also long east–west. These zones use the Lambert conformal conic projection, because it is good at maintaining accuracy along an east–west axis, due to the projection cone intersecting the earth's surface along two lines of latitude. Zones that are long in the north–south direction use the Transverse Mercator projection because it is better at maintaining accuracy along a north–south axis, due to the circumference of the projection cylinder being oriented along a meridian of longitude. The panhandle of Alaska, whose maximum dimension is on a diagonal, uses an Oblique Mercator projection, which minimizes the combined error in the X and Y directions.

Swiss coordinate system

The Swiss coordinate system (or Swiss grid) is a geographic coordinate system used in Switzerland for maps and surveying by the Swiss Federal Office of Topography (Swisstopo).

The map projection used is Oblique Mercator on an 1841 Bessel ellipsoid.

The geodetic datum CH1903 (SRID 21781) uses as fundamental point the old observatory of Bern (46°57′3.9″N 7°26′19.1″E (WGS84)), the current location of the Institut für exakte Wissenschaften of the University of Bern. In order to avoid errors during coordinate transmissions, the coordinates of this point are 600'000 m E / 200'000 m N. The 0 / 0 coordinate is located near Bordeaux, France. Though E coordinate is denoted as y and N coordinate x, E coordinate is the first axis of this Cartesian system, namely a point is denoted as (y, x). As an example, the observatory is (600'000, 200'000). This definition invokes the following effects:

All coordinates are always positive, since Switzerland is located in the 1st quadrant of the coordinate system.

Furthermore, the whole area of Switzerland is located below the y=x line of the coordinate system. Thus, all E-coordinates are always bigger than N-coordinates.The CH1903+ datum is a refinement and improvement of CH1903. It is based on WGS84, and it was devised for the national land survey of 1995 (LV 95 for Landesvermessung 1995). The coordinates of its new reference point, the Zimmerwald Observatory, should maintain the CH1903 coordinates as far as possible - the maximum shift between the two datums is 3 metres on the ground, so most map and GPS users will not notice any difference. The easting and northing is increased by 2 and 1 million, respectively, resulting in 14-digit rather than 12-digit metre coordinates.

Tonal system

The Tonal system is a base 16 system of notation (predating the widespread use of hexadecimal in computing), arithmetic, and metrology proposed in 1859 by John W. Nystrom. In addition to new weights and measures, his proposal included a new calendar with sixteen months, a new system of coinage, and a clock with sixteen major divisions of the day (called tims). Nystrom advocated his system thus:

I am not afraid, or do not hesitate, to advocate a binary system of arithmetic and metrology. I know I have nature on my side; if I do not succeed to impress upon you its utility and great importance to mankind, it will reflect that much less credit upon our generation, upon scientific men and philosophers.

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