Map of Earth showing lines of latitude (horizontally) and longitude (vertically), Eckert VI projection; large version (pdf, 3.12MB)

Geodesy
Fundamentals
Geodesy · Geodynamics Geomatics · Cartography
Concepts
Datum · Distance · Geoid Fig. Earth · Geodetic sys. Geog. coord. system Hor. pos. represent. Lat./Long. · Map proj. Ref. ellipsoid · Sat. geodesy Spatial ref. sys.
Technologies
GNSS · GPS · GLONASS · IRNSS
Standards
ED50 · ETRS89 · GRS 80 NAD83 · NAVD88 · SAD69 SRID · UTM · WGS84
History
History of geodesy NAVD29
v t e

A geographic coordinate system is a coordinate system 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, and two or three of the numbers represent horizontal position. A common choice of coordinates is latitude, longitude and elevation.^[1]

Geographic latitude and longitude[link]

Latitude phi (φ) and Longitude lambda (λ)

Latitude and Longitude of the Earth

The latitude (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 to the surface of a reference ellipsoid which approximates the shape of the Earth.^{[n 1]} 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.^{[n 2]} 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 plane of all geographic coordinate systems. The equator divides the globe into Northern and Southern Hemispheres.

The Longitude (abbreviation: Long., λ, or lambda) of a point on 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.

A line passing near the Royal Observatory, Greenwich (near London in the UK) has been chosen as the international zero-longitude reference line, the Prime Meridian. Places to the east are in the eastern hemisphere, and places to the west are in the western hemisphere. The antipodal 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 and twenty-five nations attended. Twenty-two of them agreed to adopt the location of Greenwich as the zero-reference line. The Dominican Republic voted against the adoption of that motion, while France and Brazil abstained.^[2] 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.^{[n 3]}

The combination of these two components specifies the position of any location on the planet, but does not consider altitude nor depth.

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

In defining an ellipse, 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.^[3] With a sphere or ellipsoid, the conjugate diameter is known as the polar axis and the transverse as the equatorial axis. The graticule perspective 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 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[link]

This section's tone or style may not reflect the encyclopedic tone used on Wikipedia. Specific concerns may be found on the talk page. See Wikipedia's guide to writing better articles for suggestions. (December 2011)

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. National Geodetic Survey (NGS) data predicts you will find the tank to be at 19.7323 deg North, 155.0412 deg West.^[4]

You cross the island and set the T4 next to the Keahole Point lighthouse; the NGS estimates that by the stars the lighthouse will turn out to be 19.7244 N 156.0787 W. ^[5] 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"^[6] 21 deg 18 min 13.889 sec North, 157 deg 50 min 55.796 sec West, and define the lat-lon of every other point by its distance and direction from there.^[7] The 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. The NGS now figures those two points are 339192.8 meters apart.

Similarly in North America. If in 1980 you had asked the 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[link]

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[link]

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 navigations, 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]}

Geodetic height[link]

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 the reference point is Newlyn, while in Canada, Mexico and the United States, the point is near Rimouski, Quebec, Canada. The distance to Earth's centre can be used both for very deep positions and for positions in space.^[1]

Cartesian coordinates[link]

Every point that is expressed in ellipsoidal coordinates can be expressed as an x y z (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.

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 Failed to parse (Missing texvc executable; please see math/README to configure.): 10^{-11}

degree as long as the point is within 10000 meters above or 5000 meters below the ellipsoid.

Shape of the Earth[link]

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. 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, 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]

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 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. The reference height we have chosen is the one closest to the average height of the world's oceans. This is called the geoid.^[1][8]

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 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. 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 the global GPS datum is used.^[1]

Expressing latitude and longitude as linear units[link]

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

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\phi}\,\!

we can assume a spherical Earth (to get the width per minute and second, divide by 60 and 3600, respectively):

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\pi}{180}M_r\cos \phi \!

where Earth's average meridional radius Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{M_r}\,\!

is 6,367,449 m. 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\phi}\,\!
is

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\pi}{180}a \cos \beta \,\!

where Earth's equatorial radius Failed to parse (Missing texvc executable; please see math/README to configure.): a

equals 6,378,137 m and Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\tan \beta = \frac{b}{a}\tan\phi}\,\!

for the GRS80 and WGS84 spheroids, b/a calculates to be 0.99664719. (Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\beta}\,\!

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.

Datums often encountered[link]

Latitude and longitude values can be based on different geodetic systems or datums, the most common being WGS 84, a global datum used by all GPS equipment.^{[n 4]} 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. 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'.

Geostationary coordinates[link]

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.

See also[link]

Atlas portal

Notes[link]

1. ^ The surface of the Earth is closer to an ellipsoid than to a sphere, as its equatorial diameter is larger than its north-south diameter.
2. ^ 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: (6367.5 km)×tan(11.67')=21.9 km.
3. ^ The French Institut Géographique National (IGN) maps still use longitude from a meridian passing through Paris, along with longitude from Greenwich.
4. ^ WGS 84 is the default datum used in most GPS equipment, but other datums can be selected.

References[link]

• Portions of this article are from Jason Harris' "Astroinfo" which is distributed with KStars, a desktop planetarium for Linux/KDE. See [5]

External links[link]

• Mathematics Topics-Coordinate Systems
• Geographic coordinates of countries (CIA World Factbook)
• FCC coordinates conversion tool (DD to DMS/DMS to DD)
• Coordinate converter, formats: DD, DMS, DM
• Latitude and Longitude
• Find the GPS coordinates for a location or a place in several geocoding formats, which can be entered into a GPS device
• Convert Address to Coordinates and vice-versa
• Get the decimal coordinates, sexagesimal and UTM of any location

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 on a manifold such as Euclidean space.^[1][2] The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in 'the x-coordinate'. In elementary mathematics the coordinates are taken to be real numbers, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.^[3]

An example in everyday use is the system of assigning longitude and latitude to geographical locations. In physics, a coordinate system used to describe points in space is called a frame of reference.

Number line[link]

The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O (the origin) is chosen on a given line. The coordinate of a point P is defined as the signed distance from O to P, where the signed distance is the distance taken as positive or negative depending on which side of the line P lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.^[4]

Cartesian coordinate system[link]

The Cartesian coordinate system in the plane.

The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines.

In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space.

Polar coordinate system[link]

The Polar coordinate system in the plane.

Another common coordinate system for the plane is the Polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates (r, θ) there is a single point, but any point is represented by many pairs of coordinates. For example (r, θ), (r, θ+2π) and (−r, θ+π) are all polar coordinates for the same point. The pole is represented by (0, θ) for any value of θ.

Cylindrical and spherical coordinate systems[link]

There are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates. Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ)

Homogeneous coordinate system[link]

A point in the plane may be represented in homogeneous coordinates by a triple (x, y, z) where x/z and y/z are the Cartesian coordinates of the point. This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane without the use of infinity. In general, a homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values.

Coordinates of other elements[link]

Coordinates systems are often used to specify the position of a point, but they may also be used to specify the position of more complex figures such as lines, planes, circles or spheres. For example Plücker coordinates are used to determine the position of a line in space. When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term line coordinates is used for any coordinate system that specifies the position of a line.

It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis. An example of this is the systems of homogeneous coordinates for points and lines in the projective plane. The two systems in a case like this are said to be dualistic. Dualistic systems have the property that results from one system can be carried over to the other since these results are only different interpretations of the same analytical result; this is known as the principle of duality.^[5]

Transformations between coordinate systems[link]

Because there are often many different possible coordinate systems for describing geometrical figures, it is important to understand how they are related. Such relations are described by coordinate transformations which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (x, y) and polar coordinates (r, θ) have the same origin, and the polar axis is the positive x axis, then the coordinate transformation from polar to Cartesian coordinates is given by x = r cosθ and y = r sinθ.

[edit] Coordinate curves and surfaces

Coordinate surfaces in the Spherical coordinate system

In two dimensions if all but one coordinate in a point coordinate system is held constant and the remaining coordinate is allowed to vary, then the resulting curve is called a coordinate curve (some authors use the phrase "coordinate line"). This procedure does not always make sense, for example there are no coordinate curves in a homogeneous coordinate system. In the Cartesian coordinate system the coordinate curves are, in fact, lines. Specifically, they are the lines parallel to one of the coordinate axes. For other coordinate systems the coordinates curves may be general curves. For example the coordinate curves in polar coordinates obtained by holding r constant are the circles with center at the origin. Coordinates systems for Euclidean space other than the Cartesian coordinate system is called curvilinear coordinate systems.^[6]

In three dimensional space, if one coordinate is held constant and the remaining coordinates are allowed to vary, then the resulting surface is called a coordinate surface. For example the coordinate surfaces obtained by holding ρ constant in the spherical coordinate system are the spheres with center at the origin. In three dimensional space the intersection of two coordinate surfaces is a coordinate curve. Coordinate hypersurfaces are defined similarly in higher dimensions.^[7]

Coordinate maps[link]

The concept of a coordinate map, or chart is central to the theory of manifolds. A coordinate map is essentially a coordinate system for a subset of a given space with the property that each point has exactly one set of coordinates. More precisely, a coordinate map is a homeomorphism from an open subset of a space X to an open subset of Rⁿ. It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form an atlas covering the space. A space equipped with such an atlas is called a manifold and additional structure can be defined on a manifold if the structure is consistent where the coordinate maps overlap. For example a differentiable manifold is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function.

Change of coordinates[link]

In geometry and kinematics, coordinate systems are used not only to describe the (linear) position of points, but also to describe the angular position of axes, planes, and rigid bodies. In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation matrix, which includes, in its three columns, the Cartesian coordinates of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three unit vectors aligned with those axes.

Transformations[link]

A coordinate transformation is a conversion from one system to another, to describe the same space.

With every bijection from the space to itself two coordinate transformations can be associated:

• such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
• such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)

For example, in 1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to -3, so that the coordinate of each point becomes 3 more.

Systems commonly used[link]

Some coordinate systems are the following:

• The Cartesian coordinate system (also called the "rectangular coordinate system"), which, for two- and three-dimensional spaces, uses two and three numbers (respectively) representing distances from the origin in three mutually perpendicular directions.
• Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
• Polar coordinate system represents a point in the plane by a distance from the origin and an angle measured from a reference line intersecting the origin.
• Log-polar coordinate system represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference line intersecting the origin.
• Cylindrical coordinate system represents a point in three-space using two perpendicular axes; distance is measured along one axis, while the other axis formes the reference line for a polar coordinate representation of the remaining two components.
• Spherical coordinate system represents a point in three space by the distance from the origin and two angles measured from two reference lines which intersect the origin.
• Plücker coordinates are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates.
• Generalized coordinates are used in the Lagrangian treatment of mechanics.
• Canonical coordinates are used in the Hamiltonian treatment of mechanics.
• Parallel coordinates visualise a point in n-dimensional space as a polyline connecting points on n vertical lines.
• Barycentric coordinates (mathematics) as used for Ternary_plot

There are ways of describing curves without coordinates, using intrinsic equations that use invariant quantities such as curvature and arc length. These include:

• Whewell equation relates arc length and tangential angle.
• Cesàro equation relates arc length and curvature.

List of orthogonal coordinate systems[link]

In mathematics, two vectors are orthogonal if they are perpendicular. The following coordinate systems all have the properties of being orthogonal coordinate systems, that is the coordinate surfaces meet at right angles.

See also[link]

• Active and passive transformation
• Alpha-numeric grid
• Frame of reference
• Galilean transformation
• Coordinate-free
• Nomogram, graphical representations of different coordinate systems
• Geographic coordinate system
• Astronomical coordinate systems
• Axes conventions in engineering

References[link]

1. ^ Woods p. 1
2. ^ Weisstein, Eric W., "Coordinate System" from MathWorld.
3. ^ Weisstein, Eric W., "Coordinates" from MathWorld.
4. ^ Woods p. 8
5. ^ Woods p. 2
6. ^ Tang, K. T. (2006). Mathematical Methods for Engineers and Scientists. 2. Springer. p. 13. ISBN 3-540-30268-9. 
7. ^ Liseikin, Vladimir D. (2007). A Computational Differential Geometry Approach to Grid Generation. Springer. p. 38. ISBN 3-540-34235-4. 

• Voitsekhovskii, M.I.; Ivanov, A.B. (2001), "Coordinates", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=C/c026470 
• Woods, Frederick S. (1922). Higher Geometry. Ginn and Co.. pp. 1ff. http://books.google.com/books?id=3ZULAAAAYAAJ&pg=PA1#v=onepage&q&f=false. 
• Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. AMS Bookstore. p. 12. ISBN 0-8218-1045-6. http://books.google.com/books?id=5N33Of2RzjsC&pg=PA12. 

External links[link]

Look up coordinate in Wiktionary, the free dictionary.

• Hexagonal Coordinate System