An introduction to Coordinate Reference Systems An introduction to Coordinate Reference Systems 27/11/2009 1. Introduction A coordinate that defines a position on the Earth is ambiguous unless its coordinate reference system (CRS) is defined. This ambiguity arises from the fact that the Earth is a complex shape and that, over time, many different methods of defining position have been developed. Consequently, it is essential that anyone involved in the management of spatial data has at least a basic grasp of the concepts of CRSs and knows where to find out further information. The purpose of this paper is to provide an introduction to CRS concepts and to provide links to further reference material should the reader wish to read around the subject in more depth. One of the issues faced by anyone wanting to find out more about the subject of CRSs is that many different terms are used to describe CRS entities. To make matters worse, some terms mean different things within different domains. In an attempt to bring a level of clarity to this, the terms used in this paper are taken from ISO 19111 Spatial Referencing by Coordinates. A useful guide to terminology, including an ISO 19111 reference table, is provided in Iliffe and Lott (2008), Appendix A. Where appropriate, identifiers are used to specify CRSs and their properties. The identifiers are drawn from the EPSG Geodetic Parameter Dataset.1 2. The Earth The Earth is a complex body. Its surface, the ‘topographic surface’, is irregular and encompasses mountains, valleys and ocean trenches ranging in height between about 9km above, and 11km below, sea level. Relative to the overall size of the Earth these distances represent only a thin veneer (Smith 1997). However, for geometric calculations and mapping the topographic surface is too Geoid… an equipotential complex. surface that corresponds to mean sea level An alternative surface is represented by the Geoid. This is an equipotential surface which is defined as the “equipotential surface that most closely corresponds to mean sea level” (Iliffe and Lott 2008). When travelling across an equipotential surface no work is done against gravity which suggests that the Geoid is a level surface with no hills or valleys. The traveller will experience a flat surface but in fact the Geoid undulates. Whilst the Geoid is a useful reference surface for elevation the undulations mean that it too is a complex surface for mapping. To simplify the situation mathematical models of the Earth are derived which provide a smooth surface for mapping. These models fit the Geoid partially or wholly and are known as ellipsoids. 1 http://www.epsg-registry.org/ 27/11/2009 1 An introduction to Coordinate Reference Systems 3. Ellipsoid Geodetic applications that have low demands in terms of accuracy can use a spherical model of the Earth. However, the geoid is slightly flattened at the poles and bulges slightly at the equator so applications requiring higher accuracy should use an ellipsoid. An ellipsoid is a simple mathematical model of the geoid. Illustrations of the ellipsoid are usually greatly exaggerated. In truth, the flattening of the Earth at the poles is marginal and measures approximately 22km in 6378km (Iliffe and Lott 2008) Ellipsoid… a simplified which is just 0.34%. Hence it is not detectable to the naked eye and this is mathematical model of borne out by the famous Earthrise photograph taken during the Apollo 11 2 mission in 1969. the geoid The defining parameters of an ellipsoid are the semi-major axis length (a) and the semi-minor axis length (b) (Figure 3.1). In the early days of geodesy, the approximation of the geoid by an ellipsoid could only be done over relatively small extents and certainly not globally. This has lead to a proliferation of different ellipsoids each with a different size and shape. While some of these ellipsoids are now obsolete others are still used in national mapping systems. The national standard topographic mapping system for Britain, British National Grid, uses an ellipsoid determined in 1830, known as Airy 1830. Figure 3.1: Ellipsoidal parameters 2 http://nssdc.gsfc.nasa.gov/planetary/lunar/images/as11_44_6552.jpg 27/11/2009 2 19 80 An introduction to Coordinate Reference Systems H ug he s 6378500 6378300 Semi-Major-Axis Length (m) 6378100 6377900 6377700 6377500 6377300 6377100 6376900 6376700 Ev er es t( Pl es si s 18 A 18 i r 17 30 y 1 D 830 ef in B i Ai ry ess tion ) M el od 18 i fi e d 41 C la 184 rk 9 St e 1 ru 85 ve 8 18 C la rk 60 D e1 an 8 i s 66 h C la 18 7 r H ke 6 e 1 In l te me 880 rn at rt 1 9 io na 06 l Kr W 19 as 2 a so r O 4 w s k ffice y Av H 19 ou 40 er gh ag e G 196 Te R rre S 0 st 19 ria 67 W lS G ys S te 7 m 2 1 G 97 R S 7 Xi 198 an 0 H ug 19 he 80 s 19 W 80 G S O 84 SU 86 F PZ 9 O SU 0 91 A 6376500 Ellipsoid Name (Ordered by Realisation Date) Figure 3.2: Evolution of estimates of the Length of Semi-major Axis of Ellipsoids Values are taken from the EPSG Geodetic Parameter Dataset Figure 3.2 shows how estimates of the length of the semi-major axis of reference ellipsoids have evolved over time. It can be seen that since 1980 there has been broad agreement on the appropriate value. 4. Geodetic datum A datum fixes a coordinate system to an object. A geodetic datum fixes an ellipsoid to the Earth and defines the meridian that is used for zero longitude. Historically, the method by which a geodetic datum was defined was to first choose an ellipsoid. Typically the chosen ellipsoid was either known to fit the local geoid well or was acknowledged as the best definition of the shape of the Earth. A grossly exaggerated diagram of this is shown in Figure 4.1. Geodetic Datum… fixes an ellipsoid to the Earth and defines the meridian that is used for zero longitude Modern techniques for deriving geodetic datums rely on artificial satellites and the adoption of the International Terrestrial Reference System (ITRS). In addition, the understanding of the size and shape of the Earth is more refined (see Figure 3.2) and GRS 19803 is accepted as the best model (Iliffe and Lott 2008). 3 EPSG Identifier: urn:ogc:def:ellipsoid:EPSG::7019 27/11/2009 3 An introduction to Coordinate Reference Systems However, there is a legacy of different geodetic datum definitions, still in use today, with which we must contend. Figure 4.1: Ellipsoids and the Geoid 5. Coordinate system A coordinate system is expressed by a set of properties that enable the meaning of the coordinates within the system to be determined. The properties that define a coordinate system are (Iliffe and Lott 2008): The dimension – defines the number of axes associated with the system For each axis: The name of the axis (e.g. Easting, Geodetic Latitude) The sequence of the axis (i.e. the position of the axis in an ordered list) The direction in which coordinates increase along the axis (e.g. positive up for vertical coordinate systems) The units of measure of the axis It is important to note that the axes of a coordinate system are listed in order and that coordinates within the coordinate system follow the order. That is to say that if the axis order of a coordinate system is 1) Geodetic latitude 2) Geodetic longitude and a coordinate is (50, 10) it is known that the value 50 refers to geodetic latitude while 10 refers to geodetic longitude. This follows the OGC Axis Order Policy Guidance.4 This is particularly important when encoding data using modern formats such as GML5. In other spatial data formats the order of coordinates may be part of the data specification and cannot be changed. 4 OGC Axis Order Policy Guidance - http://www.ogcnetwork.net/node/491 5 OGC Geography Markup Language - http://www.opengeospatial.org/standards/gml 27/11/2009 4 An introduction to Coordinate Reference Systems 6. Ellipsoidal coordinate system Pe rpe the ndicu ellip lar soid to Ellipsoidal coordinates are measured in terms of geodetic latitude and geodetic longitude and, in the three dimensional case, ellipsoidal height (Figure 6.1). The coordinates define position relative to an ellipsoid. Geodetic longitude (λ) is the angle from the prime meridian to the Coordinates in an meridian plane of a given point. Geodetic latitude (φ) is the angle from the ellipsoidal coordinate equatorial plane to the perpendicular to the ellipsoid through a given system define the point. Note that the perpendicular to the ellipsoid does not necessarily intersect the centre of the ellipsoid, as can be seen in Figure 6.1. If a position of a point spherical model is used then the perpendicular will always intersect the relative to an ellipsoid centre. The convention is that values of longitude are positive in an easterly direction and that values of geodetic latitude are positive in a northerly direction. Figure 6.1: Ellipsoidal coordinates The fact that different ellipsoids and different geodetic datums have been used over time raises an important point: the geodetic latitude and longitude coordinate of a position on the Earth’s surface is NOT unique The numeric values of geodetic latitude and longitude for a given position on the Earth’s surface will depend entirely on the geodetic datum in use. If the geodetic datum is not defined, a coordinate will have an inherent ambiguity of up to 1500m (OGP Surveying and Positioning Guidance Note 1). In practical terms the geodetic datum is expressed as part of the CRS definition. 7. Cartesian coordinate system Hitherto we have considered systems and parameters for mapping on the curved surface of the Earth. However, when presenting spatial data on a flat surface, be it a piece of paper or a computer 27/11/2009 Projections always result in Cartesian coordinate systems 5 An introduction to Coordinate Reference Systems screen6, the coordinates must be projected. Projections always result in a Cartesian coordinate system which is a coordinate system that has axes that are straight and mutually perpendicular (Figure 7.1). Projected coordinate systems are derived from ellipsoidal coordinate systems by means of map projections. A map projection is often explained conceptually in diagrammatic terms such as, in the case of the Mercator projection, wrapping a piece of paper around the Earth (Figure 7.2 but in practice the process involves the application of mathematical formulae to ellipsoidal coordinates. When spatial data are drawn within a computer mapping system such as a GIS a projection is always employed, regardless of whether the CRS of the underlying data is projected or geodetic. If the data are referenced to a geodetic CRS then typically the Plate Carée projection is used (Figure 7.3). The data are projected on-the-fly and the underlying coordinates are not changed. The coordinates presented to the user as the mouse pointer is moved across the map display will be those of the geodetic CRS In a Cartesian coordinate system axes are straight and mutually perpendicular (e, n) Northing Easting Map projections are also used in some mapping systems to give Figure 7.1: Cartesian coordinate system (two dimensional case) a three dimensional appearance to data that are otherwise two dimensional (Figure 7.4). The same two dimensional data were used in the creation of Figure 7.3 and Figure 7.4. Cartesian coordinate systems are not only associated with projections. A Cartesian coordinate system can be used at a global scale, instead of an ellipsoidal coordinate system. These are termed geocentric coordinate systems. An example is shown in Figure 7.5. The origin of the coordinate system is placed to coincide with the Earth’s Geocentric coordinate systems are globally applicable Cartesian coordinate systems where the origin is the Earth’s centre of mass Figure 7.2: Diagrammatic representation of the Mercator projection centre of mass, or an assumed centre implied by a geodetic datum (OSNI 1999), the X axis is aligned with the prime meridian in the equatorial plane, the Y axis is aligned perpendicular to the X axis in the equatorial plane and the Z axis is aligned with the minor axis of the ellipsoid. 6 In spite of the fact that a projection has been applied in this case, coordinates may still be expressed as ellipsoidal coordinates as the user moves the mouse pointer across the map. 27/11/2009 6 An introduction to Coordinate Reference Systems Readers are likely only to come into direct contact with geocentric coordinate systems if they undertake complex geodetic computations. For most though it will not be obvious when a computer mapping system makes use of a geocentric coordinate system.7 Some coordinates are published using a geodetic CRS with a geocentric coordinate system. An example is the International Reference Frame (ITRF)8 and positional solutions from NAVSTAR GPS are fundamentally computed within the ITRF using geocentric coordinates (Blewitt 2009). Figure 7.3: Plate Carée map projection Figure 7.4: Vertical Perspective projection 7 Some coordinate transformations are computed using geocentric coordinates. 8 ITRF is a realisation of the International Terrestrial Reference System (ITRS). The current (at the time of writing) ITRF realisation is ITRF2005 – urn:ogc:def:crs:EPSG::4896 27/11/2009 7 An introduction to Coordinate Reference Systems Figure 7.5: Geocentric Cartesian Coordinate System Strictly, the coordinate system is geocentric only if the origin of the coordinate system coincides with the Earth’s centre of mass. This is only true in a few cases such as ITRS where the origin is considered to be within 0.1m of the Earth’s centre of mass (Smith 1997). However, the term geocentric is widely used. 8. Vertical coordinate system Vertical coordinates are measured along a single axis of a vertical coordinate system. For gravity related heights the axis is aligned with the Earth’s gravity field. Earlier in this paper it was stated that the geoid was a complex surface for horizontal computation. However, it provides a suitable surface to which height, or depth, can be related because it is Vertical coordinates are equipotential – the surface is perpendicular to the direction of gravity. measured along a single Since mean sea level is a close approximation of the geoid it is a axis of a vertical coordinate common starting point for the development of vertical coordinate system reference systems. A value of mean sea level is derived by observation at a certain point (or in some cases more than one point). For the British mainland the value for mean sea level was observed at Newlyn. This value becomes the vertical datum for the vertical CRS. The height at the datum is transferred to other positions across the area of interest through surveying techniques. Nautical charts present safety critical information. Consequently, it is standard practice when compiling charts and presenting the results from hydrographic surveys to remove the effect of the tide and to present all depths (and inter-tidal heights) to a datum below which the water level does not normally, or rarely, falls. This datum is known as Chart Datum. Chart Datum defines the seabed relative to a tidal surface which is not an equipotential surface. Consequently it is not appropriate to use Chart Datum for offshore engineering without further work. The vertical datum chosen for nautical charts is referenced to the level of water below which the tide does not normally fall – known as Chart Datum Vertical coordinates may be measured in an upwards direction along the axis, in which case they are termed heights, or downwards along the axis, in which case 27/11/2009 8 An introduction to Coordinate Reference Systems they are termed depths. The convention on nautical charts is that vertical coordinates are presented as depths. Therefore, depths for areas that dry (i.e. are above the vertical datum) will be negative while those for areas that are always wet will be positive. The convention for topographic maps is that vertical coordinates are presented as heights. 9. Coordinate reference system A CRS associates a coordinate system with an object by means of a datum. Therefore, a CRS definition must encompass a definition of a coordinate system and a datum. In the context of ISO 19111 a CRS is defined by its datum. Three types are important in the context of this paper: Geodetic CRS – a CRS based on a geodetic datum (e.g. WGS 849 or OSGB 193610 - note that both these names are also names of geodetic datums) Projected CRS – a CRS derived from a geodetic CRS by means of a map projection. The datum is expressed by the geodetic datum of the base geodetic CRS from which the projected CRS is derived (e.g. British National Grid11) Vertical CRS – a one dimensional CRS based on a vertical datum (e.g. MSL Depth12). It is the definition of the CRS that must be supplied with a spatial dataset in order that a full understanding of the meaning of the coordinates in the data can be gained. By extension the CRS should also be expressed in full on any map products that are produced. 10. Common coordinate reference systems in the UK 10.1.British National Grid British National Grid (BNG) is the national standard CRS for land mapping in the British Isles. The components of the CRS are as follows: Base Geodetic CRS – OSGB 1936 (urn:ogc:def:crs:EPSG::4277) 13 Geodetic Datum – OSGB 1936 (urn:ogc:def:datum:EPSG::6277) – Ellipsoid – Airy 1830 (urn:ogc:def:ellipsoid:EPSG::7001) – Prime Meridian – Greenwich (urn:ogc:def:meridian:EPSG::8901) Projection Parameters – BNG (urn:ogc:def:coordinateOperation:EPSG::19916) Projection Method – Transverse Mercator (urn:ogc:def:operationMethod:EPSG:9807) Coordinate System – Cartesian (urn:ogc:def:cs:EPSG::4400). 9 EPSG Identifier: urn:ogc:def:crs:EPSG::4326 10 EPSG Identifier: urn:ogc:def:crs:EPSG::4277 11 EPSG Identifier: urn:ogc:def:crs:EPSG::27700 12 EPSG Identifier: urn:ogc:def:crs:EPSG::5715 13 References in brackets refer to the EPSG Registry Database 27/11/2009 9 An introduction to Coordinate Reference Systems BNG was originally created using classical techniques. The geodetic datum was realised by triangulation using physical monuments (Trig Points). However, the realisation of the datum is now performed through the use of the OSTN02 coordinate transformation.14 10.2.WGS 1984 The World Geodetic System of 1984 (WGS 1984) is a globally applicable geodetic CRS. It is the CRS that is employed by NAVSTAR GPS.15 The CRS is defined so that the origin of its coordinate system is approximately coincident with the Earth’s centre of mass (i.e. it is geocentric).16 The components of the CRS (when it defines ellipsoidal coordinates) are as follows: Geodetic Datum – World Geodetic System 1984 (urn:ogc:def:datum:EPSG::6326) Ellipsoid – WGS 84 (urn:ogc:def:ellipsoid:EPSG::7030) Prime Meridian – Greenwich (urn:ogc:def:meridian:EPSG::8901) Coordinate System – Ellipsoidal (urn:ogc:def:cs:EPSG::6422) This is the CRS in use in SeaZone’s spatial data products. 10.3.ETRS89 The European Terrestrial Reference System 1989 (ETRS89) was related to ITRF at epoch 1989.0 (i.e. 1st January 1989) and then fixed to the Eurasian continental plate. Since both WGS 84 and ETRS89 are related to ITRF they can, for most practical purposes where high levels of accuracy are not required, be considered equivalent. Since ETRS89 is fixed to the Eurasian plate it has diverged from the ITRF (and hence WGS 84) as a result of plate tectonic motion at about 2.5cm per year. The total difference (2007) is about 50cm (Iliffe and Lott 2008). The components of the CRS (when it defines ellipsoidal coordinates) are as follows: Geodetic Datum – European Terrestrial Reference System 1989 (urn:ogc:def:datum:EPSG::6258) Ellipsoid – GRS 1980 (urn:ogc:def:ellipsoid:EPSG::7019) Prime Meridian – Greenwich (urn:ogc:def:meridian:EPSG::8901) Coordinate System – Ellipsoidal (urn:ogc:def:cs:EPSG::6422). 11. Coordinate operations Coordinate operations are defined as methods that change coordinates from one CRS to another. A coordinate conversion is the general case of two specific operations: a coordinate conversion and a coordinate transformation. 14 http://www.ordnancesurvey.co.uk/oswebsite/gps/docs/geodesy_and_positioning.pdf 15 http://www.gps.gov/ 16 Measurement instruments are now so sensitive that it is known that the Earth’s centre of mass is dynamic. It responds to mass movements, notably atmospheric and hydrological, at or near the Earth’s surface. Movements in the centre of mass are in the order of 1 to 2cm (Herring 2009). 27/11/2009 10 An introduction to Coordinate Reference Systems A coordinate conversion changes coordinates from one CRS to another where both CRSs are referenced to the same datum A coordinate conversion changes coordinates from one CRS to another where both CRSs are referenced to the same datum. A map projection is an example of a coordinate conversion. Coordinate conversion methods are defined and consequently considered to be exact (Iliffe and Lott 2008). Therefore, coordinate conversions result in no loss in positional accuracy. A coordinate transformation changes coordinates from one CRS to another where the resulting CRS is referenced to a different datum from the source CRS. The realisation of a datum, through surveying techniques, is inherently error prone although great effort is of course taken to minimise error. These errors affect coordinate transformations so that a coordinate transformation will always introduce error. In addition the parameters for a coordinate transformation are empirically derived. Different sets of observations will result in different parameters so that any two CRSs may have more than one A coordinate transformation changes coordinate transformation defined for transforming coordinates from one CRS to another between them. Observations are restricted to certain where the resulting CRS is referenced to a extents so that the resulting transformation different datum from the source CRS parameters should only be used for datasets within that extent. 12. Issues 12.1.Terminology One of the problems faced by anyone new to geodesy is the range of terminology that is in use in the science. Terminology from ISO 19111 has been used in this paper in order to attempt to achieve clarity. However, desktop GIS systems use colloquial terms to name geodetic entities in their graphical interfaces and system documentation. Table 12.1 gives cross reference of terminology in use in ESRI and Cadcorp desktop GIS. Empty cells indicate that no direct equivalent exists or is evident. 27/11/2009 11 An introduction to Coordinate Reference Systems Table 12.1: Terminology ISO 19111 ESRI ArcGIS 9.3 Cadcorp SIS 6 Cadcorp SIS 7 Coordinate Reference System Coordinate System Projection Coordinate Reference System Geodetic Coordinate Reference System Geographic Coordinate System Projected Coordinate Reference System Projected Coordinate System Geoid Datum17 Geodetic Datum Semi-major axis (of an ellipsoid) Equator Equator Semi-minor axis (of an ellipsoid) Pole Pole Coordinate transformation parameters for a 7parameter transformation method 7 WGS84 conversion parameters 7 WGS84 conversion parameters Geodetic Datum Coordinate Transformation Ellipsoidal Coordinate System Transformation Spherical Coordinate System 12.2.Coordinate operations in Geographic Information Systems Most desktop mapping software applications provide mechanisms for implementing coordinate operations. They give the user powerful tools for manipulating data but users must ensure that the appropriate operations are applied for their particular application. This section outlines some of the common problems than can occur in invoking coordinate operations. It highlights the need for care in implementing coordinate operations. Software can implement coordinate operations only if it is provided with a definition of the CRS of the source data, encoded in an appropriate way. This is one reason why it is important to provide a definition of the CRS in use in a dataset. 17 The approach employed in Cadcorp SIS is that a geoid datum entity is defined in a Library (NOL) file with a set of coordinate transformation parameters which define the transformation to WGS84. Thus if more than one coordinate transformation exists a geoid datum will need to be entered more than once in a NOL. This gives the impression that a geodetic datum is defined by its coordinate transformation to WGS84. This is not the case. A geodetic datum, in fact, exists independently of any coordinate transformations which may be defined for it. Cadcorp SIS is not the only software which adopts this approach. An additional facet of Cadcorp’s approach is that there is a one-to-one relationship between a projection and its geoid datum (using Cadcorp SIS 6.2 terminology) which is of course conceptually correct (a CRS can have only one datum). However, since a datum needs to be defined more than once, in cases where there is more than one coordinate transformation available, the projection must be defined more than once too. The correct projection must be selected in order to invoke the desired coordinate transformation. 27/11/2009 12 An introduction to Coordinate Reference Systems Software may, in some cases, not apply the full set of steps needed for an accurate coordinate operation. ESRI ArcGIS, for example, will not apply a coordinate transformation by default when one is required. It will apply the necessary coordinate conversions so that data are loaded and displayed – to the uninformed user everything may appear correct. However, it is necessary for the user to select a particular coordinate transformation in order for it to be applied. Not doing so will lead to errors in position of several hundred metres but in extreme cases, up to 1500m.18 Figure 12.1 shows the steps that must be implemented when changing the CRS of a dataset from British National Grid to WGS 84 / UTM Zone 30 using a 7 parameter coordinate transformation. By default, ArcGIS follows the blue arrows, missing the important coordinate transformation step. The user must specify the coordinate transformation in order that it is implemented. It cannot be guaranteed that a software application will include definitions for all possible coordinate operations. However, most applications provide a user interface to allow the entry of bespoke parameters. Extreme care must be taken when doing this. Common errors are misinterpreting the units of measure that are expected by the software application for certain parameters. For example, the Cadcorp SIS 7-WGS84 conversion19 parameters expect X, Y and Z values in metres, rotation values in arc-seconds and the scale difference (“correction” in Cadcorp terms) in parts per million. Parameters may be published in units different from these. To complicate matters further coordinate transformations are directional: they are expressed in terms of “from CRS A to CRS B”. Users must ensure that parameters are entered such that coordinate transformations are performed correctly in line with this. Cadcorp SIS expects parameters appropriate for the “…to WGS84” direction. If that is not enough, there are two methods for implementing a 7-parameter coordinate transformation: position vector and coordinate frame. It is not possible to judge from a parameter set alone which method it is appropriate for (the numeric values will be identical but the signs of the rotation parameters are reversed). ESRI ArcGIS allows the user to select either method from a drop down list (which is more extensive than simply 7-parameter coordinate transformations) while Cadcorp SIS implements the coordinate frame method alone. 18 This is not a fault of the software and no criticism is intended. We have seen that any two CRSs may have more than one coordinate transformation defined. There is no way of the software knowing which should be applied and it is for the user to decide. From ArcGIS version 9.2 onwards the user is warned that a coordinate transformation has not been applied. 19 The parameters are for a coordinate transformation, in ISO 19111 terms. 27/11/2009 13 An introduction to Coordinate Reference Systems Figure 12.1: OSGB36 / BNG to WGS 84 / UTM Zone 30N Users are advised to check the effect of user defined parameters. The best way of checking user defined parameters for any coordinate operation is to obtain coordinates in source and target CRS by independent means and to compare these with coordinates generated using the user defined parameters. 13. Further reading The OGP Surveying and Positioning Committee publish a series of Guidance Notes on geodesy which are well worth reading. They can be found at: http://info.ogp.org.uk/geodesy/ Particular attention is drawn to Guidance Note 1 (Geodetic Awareness), which provides an introduction to geodesy, and Guidance Note 5 (Coordinate Reference System Definition) which describes the parameters which are necessary to define geodetic entities in full. Advanced readers are directed to Guidance Note 7 27/11/2009 14 An introduction to Coordinate Reference Systems Part 2 (Coordinate Conversions and Transformations including Formulas) which presents mathematical formulae for projections, coordinate conversions and coordinate transformations. Attention is also drawn to the UK Offshore Operators Association Guidance Notes on the Use of Coordinate Systems in Data Management on the UKCS which can be found at: http://info.ogp.org.uk/geodesy/Exchange/1065.pdf The Ordnance Survey has published a useful guide to coordinate reference systems in use in Great Britain which can be found at: http://www.ordnancesurvey.co.uk/oswebsite/gps/docs/A_Guide_to_Coordinate_Systems_in_Great_Britai n.pdf Ordnance Survey Ireland and Ordnance Survey of Northern Ireland also publish useful guides: http://www.osni.gov.uk/2.2_a_new_coordinate_system_for_ireland.pdf http://www.osni.gov.uk/5.3_making_maps_compatible_with_gps.pdf 14. References and other literature Allan, A.L. (2004) Maths for Map Makers, Whittles Publishing, 2nd Edition Blewitt, G. (2009) GPS and Space-Based Geodetic Methods, in Herring, T.A. and Schubert, G. (eds.) Geodesy, Treatise on Geophysics, Volume 3, Elsevier Bugayevskiy, L.M. and Snyder, J (1995) Map Projections: A Reference Manual, Taylor and Francis Herring, T.A. (2009) Overview, in Herring, T.A. and Schubert, G. (eds.) Geodesy, Treatise on Geophysics, Volume 3, Elsevier Iliffe, J. and Lott, R. (2008) Datums and Map Projections for Remote Sensing, GIS and Surveying, Whittles Publishing, 2nd Edition OSNI (1999) Making maps compatible with GPS, Director, Ordnance Survey Ireland and Director and Chief Executive of Northern Ireland20 Smith, J.R. (1997) Introduction to Geodesy: The History and Concepts of Modern Geodesy, John Wiley and Sons 20 http://www.osni.gov.uk/5.3_making_maps_compatible_with_gps.pdf 27/11/2009 15
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