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# Coordinate Reference System in Theory

This page documents the basic concepts and theory about coordinate system, datum and projection.

For use and application of Coordinate Reference Systems in Orbit, see Coordinate Reference Systems.

© This is a summary of the available documentation on Wikipedia and EPSG Registry supplemented with own interpretations.

# Concepts and Theory

A **Coordinate System** is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point. Coordinates are expressed as measurements along two or more axis. The units for this measurement, and indeed the units of the axis, are defined as part of the Coordinate System.

When coordinates are used to describe a position on the earth, they belong to a Coordinate Reference System.

A **Spatial Reference System (SRS) or Coordinate Reference System (CRS)** is a coordinate-based local, regional or global system used to locate geographical entities. It is a Coordinate System which is referenced to the earth (or other horizontal and or vertical reference frame) The referencing is achieved through a Reference Ellipsoid.

A coordinate reference system defines (inter alia) the representative volume of the earth, in case of projected systems the map projection method and the transformation between different spatial reference systems.

Coordinate reference systems can be referred to using a SRID integer, including EPSG codes defined by the International Association of Oil and Gas Producers.

A **reference ellipsoid** aka spheroids, the mathematical representation of the earth.

The surface of the earth with its topography is far too irregular to be a convenient basis for computing position, it is impossible to calculate on directly. To simplify computing of position, the shape of the earth is approximated by the nearest mathematically definable figure, the ellipsoid. This is due to the fact that the Earth is not a perfect sphere, but flattened at the poles and bulging at the equator.

The ellipsoid is effectively a 'best fit'. However, there are numerous ellipsoids available, each of them uniquely named and defined by a set of parameters.

A satisfying approximation of the shape of the earth by a reference ellipsoid could traditionally only be done locally, not globally, and this limitation led to the existence of many ellipsoids, each with a different size and shape. Some of these ellipsoids approximated different parts of the surface of the earth, whereas others expressed the increasing knowledge about the earth's shape and size over time.

Using a CRS every position on the Earth can be defined by a set of numbers. Often, two of the numbers represent horizontal position and another number represents vertical position.

A CRS defines a specific map projection, as well as transformations between different spatial reference systems. Spatial reference systems are defined by the Open Geospatial Consortium. Spatial reference systems can be referred to using a Spatial Reference System Identifier (SRID), including EPSG codes defined by the International Association of Oil and Gas Producers.

A **Spatial Reference System Identifier (SRID)** is a unique value used to unambiguously identify projected, unprojected, and local spatial coordinate system definitions. These coordinate systems form the heart of all GIS applications.
Virtually all major spatial vendors have created their own SRID implementation or refer to those of an authority, such as the European Petroleum Survey Group (EPSG). (Note Since 2005 the EPSG SRID values are maintained by the International Association of Oil & Gas Producers Surveying & Positioning Committee).

SRIDs are typically associated with a well known text string definition of the coordinate system. This string describes the datum, geoid, coordinate system, and map projection of the spatial objects.

An example of a SRID description

PROJCS["NAD27(76) / UTM zone 17N", GEOGCS["NAD27(76)", DATUM["North_American_Datum_1927_1976", SPHEROID["Clarke 1866",6378206.4,294.9786982138982, AUTHORITY["EPSG","7008"]], AUTHORITY["EPSG","6608"]], PRIMEM["Greenwich",0, AUTHORITY["EPSG","8901"]], UNIT["degree",0.01745329251994328, AUTHORITY["EPSG","9122"]], AUTHORITY["EPSG","4608"]], UNIT["metre",1, AUTHORITY["EPSG","9001"]], PROJECTION["Transverse_Mercator"], PARAMETER["latitude_of_origin",0], PARAMETER["central_meridian",-81], PARAMETER["scale_factor",0.9996], PARAMETER["false_easting",500000], PARAMETER["false_northing",0], AUTHORITY["EPSG","2029"], AXIS["Easting",EAST], AXIS["Northing",NORTH]]

# From Concepts to Coordinates

- A Coordinate Reference System is used to express or define a location on the earth.

- The mathematical representation of the earth is an ellipsoid.
- There are two different kind of coordinate systems to express a location on the earth.
- A location is expressed as a point on an ellipsoid

= Spherical coordinate systems. - A location is expressed as a point on an ellipsoid that has been projected on a plane.

= Cartesian coordinate systems

## Datum

Datum = reference ellipsoid + prime meridian

a Geodetic datum defines the position and orientation of the reference ellipsoid relative to the center of the earth, and the meridian used as zero longitude - the prime meridian. The size and shape of the ellipsoid are traditionally chosen to best fit the shape of the earth in your area of interest. A local best fit will attempt to align the minor axis of the ellipsoid with the earth's rotational axis. It will also ensure that the zerolongitude of the ellipsoid coincides with a defined prime meridian.

The prime meridian is usually that through Greenwich, England, but historically, countries used the meridian through their national astronomic observatory. The best fit is centered on a position on the earth's surface within the area of interest eg the Helmert Tower at Potsdam, near Berlin, was used for the European Datum 1950 (ED 50). The WGS84 ellipsoïd and the Hayford (sometimes referred as International 1924) ellipsoid are amongst the most widely used reference ellipsoids.

A geodetic datum is inextricably linked to the generation of geographical coordinates.

## Spherical coordinate systems

- A spherical coordinate system is a coordinate system for a three-dimensional space.

- A location on the earth is expressed as a point on an ellipsoid.

- There are two different ways to define a point on an ellipsoid :
- Geocentric coordinate systems
- Geographic coordinate systems

- the radial distance of that point from a fixed origin,
- its inclination angle measured from a fixed zenith direction

(the inclination angle is often replaced by the elevation angle measured from the reference plane), - and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.

More information : http://en.wikipedia.org/wiki/Spherical_coordinate_system

### Geocentric coordinate systems

Many nations have defined **grid systems** based on coordinates that cover their territory. Australia, Belgium, Great Britain, Finland , Ireland, Italy, The Netherlands, New Zealand, and Sweden are a examples of nations that have defined a National Grid System.

### Geographic coordinate systems

One speaks of a Geographic coordinate systems when the position of a point is described on the CRS ellipsoid and is expressed by means of geographical coordinates : latitude (Φ)and longitude (λ). These are angular expressions related to the equator and the prime meridian, usually, but not always, the meridian passing through Greenwich, London.

It is very important to appreciate that latitude and longitude are not unique and are therefore entirely dependent on the chosen geodetic datum.

There are several formats for the angular units of geographic coordinates :

- Degrees:Minutes:Seconds : 41°30'00”N, 87°30'00”W
- Decimal Degrees : 41.5000°, -87.5000° (generally with 4-6 decimal numbers)

More information : http://en.wikipedia.org/wiki/Geographic_coordinate_system

## Projected Coordinate Reference Systems

Projected CRS provide Cartesian or grid coordinates

A projected coordinate reference system is a flat, two-dimensional representation of the earth. It is a mathematical map of geographical coordinates onto a plane surface, so that calculations of distance and area are more easily performed. These projected CRS's are usually expressed in meters as Easting and Northing.

A graticule (lattice of lines of equal latitude and longitude) cannot be projected onto a plane without distortion. This is similar to removing an orange peel and trying to force it onto a flat surface - it tears, bends and distorts. When one property is preserved, others are distorted. Map projection methods have been formulated so that distortion of one or more characteristics (area, shape, direction, distance, etc) is controlled.

The most commonly encountered map projections methods preserve shape (the technical term is 'conformal'). Several map projection methods such as the Lambert Conic Conformal and Transverse Mercator have this property. Projected coordinate reference systems incorporating these map projection methods contain distortion in distance and area. A Lambert Conic Conformal projection uses a cone, whereas a Transverse Mercator Projection is a conformal cylindrical projection. The latter can be visualized by imagining the map plane wrapped around the earth representing ellipsoid in the form of a cylinder tangential to the equator. One meridian, usually at the center of the mapped area, will be defined to be the longitude of the projection origin. It is easily seen that only closely around this meridian the projection is reasonably distortion free.

It is important to note that any one map projection, including UTM, may be applied to any geodetic datum. Therefore, projected CRS must be properly identified to avoid any ambiguity. As grid coordinates are derived from geographical coordinates, they too describe location uniquely only when the geodetic datum is identified.

More information :

### Universal Transverse Mercator Projection

The most common map projection is the Universal Transverse Mercator (UTM) projection. Introduced by the US Army Map Service in the 1950s, it uses a series of 60 individual zones. Each zone is defined to be 6 degrees of longitude wide to cover the world.

The point at which the central meridian in each zone intersects the equator has been given the coordinates Easting = 500 000, Northing = 0 meters. For the southern hemisphere the same point has the coordinates : Easting = 500 000, Northing = 10 000 000 meters. This prevents map coordinates from ever be coming negative, which reduces potential errors in its use.

Many people mistakenly believe UTM to be superior or a synonym for other map projections. This is a common misconception. UTM is simply an internationally agreed map projection system covering the whole world except the poles.

### Local Grids

In case of small scale projects, a few qsuare kilometers, it is sometimes convenient to use an Engineering Coordinate Reference System, aka Plant Grid or Local Grid. The application areas of such grids are mainly production plant, industrial or residential compounds. Engineering CRS are commonly based on a simple flat-earth approximation of the earth's surface, and the effect of earth curvature on feature geometry is ignored.

## Coordinate transformations

In order to merge points such as surface well locations, whose geographical coordinates are referenced to one particular CRS, with other points on a different CRS, one of the two datasets must be transformed. It is possible to measure and calculate the displacements, rotations and scale difference between them. There are numerous different methods of transforming coordinates.

Colloquially a coordinate transformation may be referred to as a datum transformation. This usage is not strictly correct: it are the coordinates not the datum that are being transformed.

## What about the Z

Two types of height are recognized: ellipsoidal height and gravity-related height. Heights may be referenced to the geoid or, as part of a 3-dimensional geographic CRS, to an ellipsoid.

**Ellipsoidal heights** cannot exist by themselves; they form the vertical component of a 3-dimensional geographical CRS that has a defined geodetic datum. Ellipsoidal heights are measured perpendicularly to the surface of the relevant ellipsoid.

**Gravity-related heights** or heights above the geoid are measured in the direction of the earth's gravity field. They are referenced to a vertical coordinate system and measured perpendicular from a reference surface defined by the earth gravity. Typically, the reference surface will be associated with sea level. Because ocean tides cause water levels to change constantly, the sea level is generally taken to be the average of the tide heights at some particular place over some specified period.

The surface of the earth with its topography is far too irregular to be a convenient basis for computing position or height. Observations are reduce to the gravitational surface, which approximates mean sea level. This equipotential surface is known as the geoid. It is approximately spherical, but because of the rotation of the earth, there is a slight bulge at the equator and flattening at the poles. In addition, because of the variations in rock density that impact the gravitational field, there are many local irregularities. These factors make the geoid a complex surface.

## Summarized

A Coordinate Reference System combines a coordinate system with a datum, which gives the relationship of the coordinate system to the surface and shape of the Earth.

# EPSG

The European Petroleum Survey Group or EPSG (1986–2005) was a scientific organization with ties to the European petroleum industry consisting of specialists working in applied geodesy, surveying, and cartography related to oil exploration. EPSG compiled and disseminated the EPSG Geodetic Parameter Set, a widely used database of Earth ellipsoids, geodetic datums, geographic and projected coordinate systems, units of measurement, etc.

In 2005 the International Association of Oil & Gas Producers (OGP) absorbed the EPSG into its structure.

© http://en.wikipedia.org/wiki/EPSG

The EPSG Geodetic Parameter Dataset is a structured database of Coordinate Reference Systems and Coordinate Transformations. The geographic coverage of the data is worldwide, but it is stressed that the database does not and cannot record all possible geodetic parameters in use around the world. The EPSG Geodetic Parameter Dataset is maintained by the Geodesy Subcommittee of OGP.

© http://www.epsg-registry.org

EPSG assembles different kind of objects referenced by one number : crs, datum, projection, projection method, area, units :

- 4326 : WGS84, a CRS
- 6230 : European Datum 1950, a datum
- 7022 : International 1924, an ellipsoid
- 8901 : Greenwich, the Prime meridian
- 9001 : meter, a unit
- 9102 : degree, a unit

# Examples

### CRS WGS-84

- Datum : World Geodetic System 1984
- Prime Meridian : Greenwich

- Spherical coordinate system
- Geographical coordinates
- latitude, longitude in degrees

- EPSG : 4326
- Use : CRS used by the Global Positioning System (GPS)

### CRS Belgian Lambert72

- Datum : Belgian Datum 1972
- Cartesian coordinate system
- National Grid coordinates
- X,Y in meters

- EPSG : 31370
- Use : National coordinate system of Belgium