Powered by
JSPWiki v2.8.2
g2gp 17-01-2009
View PDF
This is version . It is not the current version, and thus it cannot be edited.
[Back to current version]   [Restore this version]
Coordinate Systems, including GPS, PostCodes & site grids, and Projection and Projections
'Coordinate Systems and Projections' is equally necessary, especially now that consumer grade GPS are virtually ubiquitous, as is the use of postal codes. I am not sure whether there is a need for a dedicated GPS section somewhere, to discuss GPS and Galileo, EGNOS, usage and survey techniques, sources of error and differential processing, etc. Some mention will be essential in the GIS Guide, certainly, but we need to decide where the detail will be provided. Once GPS are mentioned, projection and projections become vital, as does a brief discussion of the concept of Datums. In any case, a knowledge of projections and projection will be necessary for those seeking to use archive data in new work. With the growing interest in GPS, it may also be necessary to include a pointer to information on the impact of plate tectonics and the rate of movement - for Great Britain and NW Europe, the north-easterly rotation is measurable with survey grade GPS over as little as ten years; other locations on the planet are moving faster, others slower. This will inevitably link to Uncertainty and Error Propagation. Point to EH and other guides where possible.
All new content

Projections and Coordinate Systems#

These matter because at some point there is a need to integrate information from different projects in order to study the results together or in the context of something new.  Projections are increasingly important because of the widespread availability of affordable satellite navigation devices that enable one or more locations to be accurately recorded and the data retrieved for later use.  Coordinate systems are important because these provide the means of specifying a location.  This section aims to introduce these topics but cannot be a replacement for the excellent texts on the subjects, to which reference is made in the text and where the necessary detail will be found.

Why do we need Projections?#

Our planet Earth is not in fact a sphere, although it is often most convenient to treat it as such, as indeed is done by the various satellite positioning systems.  Even if the Earth were a perfect sphere, there would remain a problem, because spheres make rather inconvenient notebooks or publication media.  Imagine a set of paper maps replaced with set of balls or curved plates!  Unfortunately it is not possible to accurately and completely represent the surface of a sphere on any other type of surface: for example, having peeled an orange, the peel only joins up again when back in the orange shape.  The process of rendering the surface of a sphere onto another type of surface is likened to placing a light source within the sphere and projecting the surface as an image onto the destination medium as though that were a screen.  How this works is a matter for the textbooks; however, because the process inevitably introduces elements of distortion into the result, because the result is not on the surface of a sphere, it is necessary to understand something of the implications.

There is an additional complication: not only is the Earth not a sphere, it is not actually a regular geometric shape: it is flattened at the poles and over the oceans and bulges at the Equator and over the continents.  Sometimes this shape is treated as an ellipsoid or an oblate spheroid, but the term most generally used is Geoid - which simply means Earth-shaped.  As a result, there is no single elegant solution but many alternatives, as different parts of the Earth's surface require different types of solution.  Sadly, the choice of the wrong projection for a particular purpose or location can have very serious consequences.

There are four key characteristics of any representation of a three dimensional phenomenon or of a planimetric representation: shape, scale, direction and area.  These are also characteristics of projections but, with the source being a transformation from the curved surface of the Geoid, it is not possible to preserve all four - except on the Geoid.  The loss of one or more of these characteristics means that the result is a compromise between fidelity to the surface of the Earth and the flat represntation on paper or the computer screen.  This also means that there are different types of projection for different applications.

Properties and Types of projection#

There are four classes of projection properties and three classes of type.  It is theoretically possible for there to be projections of each type and property, and combinations of up to three of each of the properties but there are combinations that are uncommon.  This guide can only explore some of the most common types of projection, the literature must be consulted for others.

Conformal projections#

Conformal projections preserve shape.  However, especially where the coordinate system is Cartesian, area will be distorted; it is only possible to preserve shape for small parts of the Earth's surface and no projection can preserve shape at a regional or global scale.

Equal-area projections#

Equal-area projections preserve the area of the represented part of the Earth's surface but at the expense of distortion to shape and possibly angle and or scale.  For very small areas of the Earth's surface, the extent of distortion to shape may be difficult to measure.  Some equal-area projections do not yield Cartesian coordinate systems.

Equidistant projections#

Equidistant projections maintain correct scaled distance between selected points but, as with all other projections, no projection can preserve true distance everywhere on the map.

Azimuthal projections#

Azimuthal projections preserve direction between all points represented and are sometimes called true-direction projections.  Azimuthal projections are typically used for maps used to support navigation.

Conic projections#

Conic projections work as though a cone of paper has been placed on the Earth.  Conic projections where the cone only touches the surface along a single circle are said to have one standard parallel, whilst those which intersect the surface are said to have two.  The simplest conic projections sit on a pole, polar conic projections, with the others are termed oblique.  In conic projections, the meridians, lines of longitude, meet at the tip of the cone and the parallels, lines of latitude, are parallel to each other, but not necessarily equally spaced.  The cone is generally opened out along the line opposite the central meridian.  Conic projections are sometimes used for polar regions, but are more often used for regions that have a large horizontal, East to West, extent: the Lambert Conic Conformal projection is often used for representing the co-terminal states of the USA.

Cylindrical projections#

Cylindrical projections can be envisaged as a sheet of paper wrapped around the Earth to form a cylinder.  In cylindrical projections, the meridians are all parallel and equally spaced and the parallels intersect the meridians at right angles but need not be equally spaced.   In Normal cylindrical projections, the paper is wrapped around the equator and is open at the poles.  In Transverse cylindrical projections, the paper is wrapped around the poles and open at two sides to the Equator.   The well known Mercator projection is a normal cylindrical projection, whilst the Transverse Mercator projection is the transverse form.  The British National Grid system, introduced in 1936, is a Transverse Mercator projection, as are UTM (which is widely used in the US) and the Gauss Conformal or Gauss-Krüger projection widely used in mainland Europe.  The Cassini projection, used for the early County Series mapping of Great Britain, is also a transverse cylindrical projection and is well suited for mapping regions that have limited horizontal extent but extend considerably North to South.

Planar projections#

Planar projections work as though a flat piece of paper is placed tangential to the Earth's surface and can take polar, where the point of tangency is a pole, equatorial, where the point of tangency is on the Equator,  or oblique forms.  The polar form is the most common planar projection, in azimuthal form being often used for mapping polar regions.

Geographic projection#

The literature also often refers to a concept of a geographic projection yet, properly, this is not a projection at all.  In the geographic projection position is represented by spherical coordinates of angles relative to the centre of the Earth, degrees of Latitude, and angles around the Equator relative to a reference Meridian, degrees of Longitude.  The reference, prime, meridian, zero degrees, is arbitrarily chosen to be the Greenwich Meridian.  There is usually no geoid associated with this concept, thus it is not possible to measure distance or area but it is possible to estimate direction.   When assoicated with the WGS84 spheroid, this equates to the positioning system used by satellite navigation systems.

Coordinate Systems#

Coordinate systems provide a means of communicating the location of some feature represented on a map in terms of that map representation.  The most commonly used are termed Cartesian, after Descartes, and comprise a regular grid relative to two or three axes at right angles to each other, as in a graph or chessboard.  There are many types of Cartesian coordinate systems, from mixtures of letters and digits, as with a chessboard, to numeric offsets from an origin, as with a graph.  It is not strictly necessary for a Cartesian coordinate system to use the same units or unit interval on each axis, although using different units or intervals for the horizontal axes is liable to lead to considerable confusion for anyone seeking to read the map or plan.  Some of the impacts on end users of the choice of projection and coordinate system are discussed by Monmonier (1996).

Cartesian coordinates#

These are familiar to most people and are those used in archaeological site grids, comprising a set of measurements along axes set at right angles to each other.  Typically these measurements are in metres, as in the British National Grid, but feet are used in the USA.

Spherical coordinates#

These are measurements of angles relative to the centre of a sphere and relative to a central prime meridian.  The Greenwich Meridian, the meridian passing through Greenwich, in south west London and as measured by Sir George Airy in 1851 was adopted internationally in 1884.  More recently and based on measurements from satellite observation, an International Earth Rotation and Reference Systems  meridian, the IERS Reference Meridian (IRM), was established for use by satellite navigation systems.  The IRM is very slightly to the East of Airy's Greenwich Meridian.

Other coordinate systems#

Postal codes provide a means of referencing the location of properties, or small groups of properties, and have given rise to a geography based on these references.  In the UK, Postcodes are widely used for social science applications, such as Census work and the unit is also used by emergency services and planners.

Datums#

Every projection requires a Datum, which provides the reference grounding to the Earth's surface.  The datum provides the reference parameters for the coordinate system to lock it to the surface of the Earth.  All datums define the geoid to be used for approximating the shape of the Earth when calculating position and it is critically important to use the correct datum, or positional calculations can become wildly inaccurate.  The datum also defines the zero point for measuring altitude or depth.  Ordnance Survey define a number of datums for use for Great Britain, based on the Airy Spheroid; the choice of datum is dependent on scale and location within Great Britain and there is a general purpose datum for use with small scale mapping, eg 1:50000.

Bibliography#

This guide is not the place to teach map projections, coordinate systems or datums.  Some of the following publications assume the reader to have some understanding of the topics concerned, although Kennedy and Kopp (2000), Monmonier (1996) and Snyder (1987) all provide at least some introductory material.  Snyder (1987) is the key authority, especially should formulae for individual projections be required.  Bugayevsiy and Snyder (1995) and Yang, Snyder and Tobler (2000) also include discussions and formulae relating to mapping in Russia and the former USSR and the Peoples Republic of China respectively.

Bugayevskiy, Lev M., Snyder, John P., 1995, Map Projections.  London: Taylor & Francis.
Kennedy, Melita, and Kopp, Steve, 2000, Understanding Map Projections.  Redlands Cf: ESRI Press.
Monmonier, Mark, 1996, How to Lie with Maps.  Chicago: University of Chicago Press.
Snyder, John P., 1987, Map Projections - A Working Manual.  USGS Professional Paper  1395.  Also available in PDF form for download from USGS
Yang, Qihe, Snyder, John P., and Tobler, Waldo R., 2000, Map Projection Transofrormation: Principles and Applications. London: Taylor & Francis.

Previous Content to cf

3.3 Generic Issues#

3.3.1 Projections and co-ordinate systems#

Projection is the process by which the irregular three-dimensional form of the earth's surface is represented systematically on a two-dimensional plane, most commonly in the form of a map. Closely linked to the topic of projection is that of co-ordinate systems, which enable us to locate objects correctly on the resulting flat maps. Although we can locate objects on the globe using geographical co-ordinates expressed in units of latitude and longitude, most commonly we utilise a Cartesian or planar co-ordinate system with a fixed origin, a uniform distance unit of measure, and a pair of perpendicular axes usually termed Easting and Northing. Identification of the projection that was used in the creation of a data source is an essential first step in incorporating it into a spatial database.

For very small study areas, it is sometimes acceptable to ignore projection, and to assume that the region of interest is comprised of a flat, two-dimensional surface. However, if the study region is larger than a few kilometres, or if information is to be included from data sources, e.g. mapsheets, which have been constructed with different projections, then a GIS needs to understand the projection used for each layer in order to avoid inaccuracies.

Projection consists of two main stages: first the surface of the earth is estimated through the use of a geometric description called an ellipsoid (sometimes, though not always correctly, referred to as a spheroid), and secondly the surface of this ellipsoid is projected on to a flat surface to generate the map. Ellipsoids are defined in terms of their equatorial radius (the semi-major axis of the ellipse) and by another parameter, such as the flattening, reciprocal flattening or eccentricity. However, as a user of GIS, the ellipsoid definition is usually uncomplicated to incorporate. Most of the ellipsoids which have been used to generate maps have names such as the 1830 'Airy' spheroid (used by the Ordnance Survey) or the 'International' or 'Hayford' ellipsoid of 1909, and it should be sufficient to provide the full name of the ellipsoid used (for a simple introduction to ellipsoids see Defence Mapping Agency 1984).

3.3.2 Projection methods#

There are a huge variety of methods available for undertaking the projection itself. Since by their very nature projections are a compromise, each method produces a map with different properties. In a cylindrical projection, for example, the lines of latitude (parallels) of the selected ellipsoid are simply drawn as straight, parallel lines. In the resulting map the parallels become shorter with distance from the equator, and to maintain the right-angled intersections of the lines of latitude and longitude, the lines of longitude (meridians) are also drawn as parallel lines. This maintains the correct length of the meridians, at the expense of areas close to the poles which become greatly exaggerated in an east-west direction. A transverse cylindrical projection is created in the same manner, but the cylinder is rotated with respect to the parallels and is then defined by the meridian at which the cylinder touches the spheroid rather than the parallel.

The Mercator projection exaggerates the distance between meridians by the same degree as the lengths of the parallels, in order to obtain an orthomorphic projection. A transverse Mercator is similar, but based on the transverse cylindrical projection. There are also many other forms of projection which are not based on the cylinder, including conical projections (based on the model of a cone, placed with its vertex immediately above one of the poles) and entirely separate families of projections such as two-world equal area projections and zenithal projections.

Depending on the projection used, different parameters will need to be specified in order to define it. Basic projections are often modified through the use of correction factors. In transverse projections, for example, it is not uncommon for a scaling factor to be applied to the central meridian to correct for the east-west distortion of the projection itself. A projection may also use a false origin, which arbitrarily defines a point on the projection plane to be the point 0,0. False origins are normally used to ensure that all co-ordinates in the area of the projection have positive values.

All projections have limits beyond which one or more of their attributes will become too distorted. For example, the Transverse and Universal Transverse Mercator projections work well only for a narrow east-west width - around 6 degrees of longitude - beyond this limit the distortion increases rapidly. When choosing a map projection it is essential to check the details of both the capabilities and the limitations of any given projection method against the nature of the area of interest: size, extents, nature of use, etc.

Details of projection procedures can be found in a variety of standard texts, for example Bugayevsky and Snyder 1995; Snyder 1987; Evenden 1983; 1990. Standard software for specifying and undertaking cartographic projection is available for a variety of platforms. Probably the most flexible is the PROJ 4 product of the USGS discussed by Evenden 1990. 3.3.3 Co-ordinate systems

Geographical co-ordinates, expressed in angles of latitude and longitude are used to locate features upon the globe whereas planar Cartesian co-ordinate systems are used to locate features upon projected maps.

In a planar co-ordinate system, the relative positions of objects represented upon a given mapsheet can be specified using standard units of distance measured with respect to a fixed origin point. The precise characteristics of a given co-ordinate system depend heavily upon the projection used to generate the 2-dimensional representation; as a result coordinate systems are as numerous as projection systems.

For example, there are five primary coordinate systems in use in the US, a country with a very broad east-west extent. Some of these are based upon the properties of specific map projections and others on historical land division strategies (DeMers 1997: 63).

By contrast, a single planar co-ordinate system based upon the transverse Mercator projection is used for all recent mapping for Great Britain, which is elongated north-south but narrow east-west. The British National Grid uses the Airy spheroid for its datum, and its origin is located at 49 degrees North, 2 degrees West. There is a false origin defined at 400,000, -100,000 such that the central meridian is the Zero easting of the National Grid - which is also aligned to the 2 degree West meridian of Longitude. Additionally, the scale along the central meridian is 0.9996012717 (as opposed to the more normal scale of 0.9996 for the Transverse Mercator projection): this results in distances along the northing 180,000, almost exactly half way across Great Britain, being to true scale. The units of the National Grid are metres measured east and north from the origin. A similar Transverse Mercator projection, but with an origin further to the west, is used for Ireland.

It is important to realise that projections and the resultant planar co-ordinate systems vary across nations and through time. The British National Grid has only been in use since the 1940s. British mapping prior to this time was based upon the Cassini projection and used an independent datum for each county. This mapping is often termed the County Series. A problem common to all situations where multiple datums are in use for any one country is that features which cross the borders of the different datums may not match when the respective map sheets are brought together. This problem certainly exists in the pre-war British mapping and is also present in the State-Plane system in use in the US today.

As with the British Ordnance Survey, many national mapping agencies use local projections designed to suit the size and shape of the area covered by their maps.

The Universal Transverse Mercator (UTM) system is a planar projection where degrees of longitude and latitude form a rectangular grid. Since distortion tends to increase most markedly on either side of the central meridian with this projection, UTM is used for narrow north-south oriented zones.

The world is divided into zones each covering 6 degrees of longitude and numbered in an easterly direction from 1, centred on 177 degrees west, to 60, centred on 177 degrees east. Within each zone, a transverse Mercator projection is established with its origin at the intersection of the central meridian with the equator. The false origin is offset so that the central meridian is at 500,000 metres east. The false Northing is zero metres in the northern hemisphere and 10,000,000 metres in the southern hemisphere. The scaling factor on the central meridian is 0.99960.

Despite its world-wide applicability, UTM has some disadvantages. Different ellipsoids may be required in different parts of world, and transformations between zones are required when the area of interest covers more than one zone. The UK, for example, is mostly in zone 30, but areas east of the Greenwich meridian fall into zone 31. The US State-Plane co-ordinate system is an application of UTM.

The use of three-dimensional absolute co-ordinates from satellite positioning systems introduces further complications. These systems measure the relative positions of receiver and satellites using an 'Earth-Centred, Earth-Fixed' Cartesian co-ordinate system (ECEF). This system, which is aligned with the World Geodetic System 1984 (WGS 84) reference ellipsoid, has its origin close to the earth's centre of mass, its z axis parallel with the direction of the conventional terrestrial pole and its x axis passing through the intersection of the equator and the Greenwich meridian. Fortunately most receivers convert ECEF co-ordinates to WGS 84 latitude, longitude and height for output, and some will also perform transformations to other datums and co-ordinate systems, for example to the Great Britain National Grid.