contents section 1 2 3 4 5 6 7 8 a b c d 3.2 page no introduction .................................................................................................................................. 3 1.1 who should read this booklet? ....................................................................................... 3 1.2 a few myths about coordinate systems .......................................................................... 4 the shape of the earth ................................................................................................................ 6 the first geodetic question .............................................................................................. 6 2.1 ellipsoids ......................................................................................................................... 6 2.2 2.3 the geoid ....................................................................................................................... 7 local geoids ..................................................................................................... 8 2.3.1 what is position? ......................................................................................................................... 9 types of coordinates ...................................................................................................... 9 3.1 3.1.1 latitude, longitude and ellipsoid height ............................................................ 9 3.1.2 cartesian coordinates .................................................................................... 10 3.1.3 geoid height (also known as orthometric height) ........................................... 11 3.1.4 mean sea level height..................................................................................... 12 3.1.5 eastings and northings ................................................................................... 14 we need a datum.......................................................................................................... 15 3.2.1 datum definition before the space age ........................................................... 16 realising the datum definition with a terrestrial reference frame .............................. 16 3.3 3.4 summary ....................................................................................................................... 17 modern gps coordinate systems .............................................................................................. 18 world geodetic system 1984 (wgs84) ....................................................................... 18 4.1 realising wgs84 with a trf ....................................................................................... 19 4.2 the wgs84 broadcast trf ........................................................................... 19 4.2.1 the international terrestrial reference frame (itrf) ................................... 20 4.2.2 4.2.3 the international gnss service (igs) ........................................................... 20 european terrestrial reference system 1989 (etrs89) .............................. 21 4.2.4 ordnance survey coordinate systems ....................................................................................... 22 os net .......................................................................................................................... 22 5.1 5.2 national grid and the osgb36 trf ............................................................................ 23 the osgb36 datum ....................................................................................... 23 5.2.1 the osgb36 trf .......................................................................................... 24 5.2.2 5.2.3 relative accuracy of osgb36 control points ................................................. 25 ordnance datum newlyn .............................................................................................. 25 the odn datum ............................................................................................. 25 5.3.1 the odn trf ................................................................................................ 25 5.3.2 5.3.3 relative accuracy of odn bench marks ........................................................ 26 5.4 other height datums in use across great britain .......................................................... 26 5.5 the future of british mapping coordinate systems ....................................................... 27 from one coordinate system to another: geodetic transformations .......................................... 27 what is a geodetic transformation? .............................................................................. 27 6.1 helmert datum transformations .................................................................................... 29 6.2 6.3 national grid transformation ostn02 (etrs89–osgb36) ....................................... 31 national geoid model osgm02 (etrs89-orthometric height) ................................... 31 6.4 etrs89 to and from itrs ............................................................................................ 32 6.5 6.6 approximate wgs84 to osgb36/odn transformation ............................................... 33 transverse mercator map projections ....................................................................................... 33 7.1 the national grid reference convention ....................................................................... 35 further information .................................................................................................................... 36 ellipsoid and projection constants ............................................................................................. 37 a.1 shape and size of biaxial ellipsoids used in the uk ..................................................... 37 a.2 transverse mercator projections used in the uk ......................................................... 37 converting between 3d cartesian and ellipsoidal latitude, longitude and height coordinates .. 38 converting latitude, longitude and ellipsoid height to 3d cartesian coordinates ......... 38 b.1 b.2 converting 3d cartesian coordinates to latitude, longitude and ellipsoid height ......... 39 converting between grid eastings and northings and ellipsoidal latitude and longitude ........... 40 converting latitude and longitude to eastings and northings ........................................ 40 c.1 c.2 converting eastings and northings to latitude and longitude ........................................ 41 glossary ..................................................................................................................................... 43 5.3 a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 2 of 43

1.2 a few myths about coordinate systems myth 1: ‘a point on the ground has a unique latitude and longitude’ for reasons that are a mixture of valid science and historical accident, there is no one agreed ‘latitude and longitude’ coordinate system. there are many different meridians of zero longitude (prime meridians) and many different circles of zero latitude (equators), although the former generally pass somewhere near greenwich, and the latter is always somewhere near the rotational equator. there are also more subtle differences between different systems of latitude and longitude which are explained in this booklet. the result is that different systems of latitude and longitude in common use today can disagree on the coordinates of a point by more than 200 metres. for any application where an error of this size would be significant, it’s important to know which system is being used and exactly how it is defined. the figure below shows three points that all have the same latitude and longitude, in three different coordinate systems (osgb36, wgs84 and ed50). each one of these coordinate systems is widely used in britain and fit for its purpose, and none of them is wrong. the differences between them are just a result of the fact that any system of ‘absolute coordinates’ is always arbitrary. standard conventions ensure only that different coordinate systems tend to agree to within half a kilometre or so, but there is no fundamental reason why they should agree at all. figure 1: three points with the same latitude and longitude in three different coordinate systems. the map extract is 200 metres square. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 4 of 43

the result is that the relationship between two coordinate systems at the present time must also be observed on the ground, and this observation too is subject to error. therefore only approximate models can ever exist to transform (convert) coordinates from one coordinate system to another. the first question to answer realistically is ‘what accuracy do i really require?’ in general, if the accuracy requirements are low (5 to 10 metres, say) then transforming a set of coordinates from one coordinate system to another is simple and easy. if the accuracy requirements are higher (anywhere from 1 centimetre to half a metre, say), a more involved transformation process will be required. in both cases, the transformation procedure should have a stated accuracy level. 2 the shape of the earth 2.1 the first geodetic question when you look at all the topographic and oceanographic details, the earth is a very irregular and complex shape. if you want to map the positions of those details, you need a simpler model of the basic shape of the earth, sometimes called the ‘figure of the earth’, on which the coordinate system will be based. the details can then be added by determining their coordinates relative to the simplified shape, to build up the full picture. the science of geodesy, on which all mapping and navigation is based, aims firstly to determine the shape and size of the simplified ‘figure of the earth’ and goes on to determine the location of the features of the earth’s land surface – from tectonic plates, coastlines and mountain ranges down to the control marks used for surveying and making maps. hence geodesists provide the fundamental ‘points of known coordinates’ that cartographers and navigators take as their starting point. the first question of geodesy, then, is ‘what is the best basic, simplified shape of the earth?’ having established this, we can use it as a reference surface, with respect to which we measure the topography. geodesists have two very useful answers to this question: ellipsoids and the geoid. to really understand coordinate systems, you need to understand these concepts first. 2.2 ellipsoids the earth is very nearly spherical. however, it has a tiny equatorial bulge making the radius at the equator about one third of one percent bigger than the radius at the poles. therefore the simple geometric shape which most closely approximates the shape of the earth is a biaxial ellipsoid, which is the three-dimensional figure generated by rotating an ellipse about its shorter axis (less exactly, it is the shape obtained by squashing a sphere slightly along one axis). the shorter axis of the ellipsoid approximately coincides with the rotation axis of the earth. because the ellipsoid shape doesn’t fit the earth perfectly, there are lots of different ellipsoids in use, some of which are designed to best fit the whole earth, and some to best fit just one region. for instance, the coordinate system used with the global positioning system (gps) uses an ellipsoid called grs80 (geodetic reference system 1980) which is designed to best-fit the whole earth. the ellipsoid used for mapping in britain, the airy 1830 ellipsoid, is designed to best-fit britain only, which it does better than grs80, but it is not useful in other parts of the world. so various ellipsoids used in different regions differ in size and shape, and also in orientation and position relative to each other and to the earth. the modern trend is to use grs80 everywhere for reasons of global compatibility. hence the local best-fitting ellipsoid is now rather an old-fashioned idea, but it is still important because many such ellipsoids are built into national mapping coordinate systems. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 6 of 43

1 up level down 2 up level down g r a v i t y f i e l d g r a v i t y f i e l d figure 3: why the gravity field is important in height measurement. on the left is a hill in a uniform gravity field. on the right is a flat surface in a non-uniform gravity field. the white dotted lines are level surfaces. the experience of the blindfolded stick-men is the same in both cases. from this it is clear that the gravity field must be considered in our definition of height. this is why the geoid is the fundamental reference surface for vertical measurements, not an ellipsoid. the geoid is very nearly an ellipsoid shape – we can define a best-fitting ellipsoid which matches the geoid to better than two hundred metres everywhere on its surface. however, that is the best we can do with an ellipsoid, and usually we want to know our height much better than that. the geoid has the property that every point on it has exactly the same height, throughout the world, and it is never more than a couple of metres from local mean sea level. this makes it the ideal reference surface on which to base a global coordinate system for vertical positioning. the geoid is in many ways the true ‘figure of the earth’ that we introduced in section 2.1, because a fundamental level surface is intrinsic to our view of the world, living as we do in a powerful gravity field. if you like, the next step on from understanding that the shape of the earth is ‘round’ rather that ‘flat’, is to understand that actually it is the complex geoid shape rather than simply ‘round’. for more explanation of this subject, see section 3.1 below. a colour image of the global geoid relative to a best-fitting ellipsoid surface is available on the internet – see the further information list in section 8. 2.3.1 local geoids height measurements on maps are usually stated to be height above mean sea level. this means that a different level surface has been used as the ‘zero height’ reference surface – one based on a tide-gauge local to the mapping region rather than the average of global ocean levels. for most purposes, these local reference surfaces can be considered to be parallel to the geoid but offset from it, sometimes by as much as two metres. for instance, heights in mainland britain are measured relative to the tide-gauge in newlyn, cornwall, giving a reference surface which is about 80 centimetres below the geoid. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 8 of 43

with the coordinate triplet of latitude, longitude and ellipsoid height, we can unambiguously position a point with respect to a stated ellipsoid. to translate this into an unambiguous position on the ground, we need to know accurately where the ellipsoid is relative to the piece of ground we are interested in. how we do this is discussed in sections 3.2 and 3.3 below. figure 4: an ellipsoid with graticule of latitude and longitude and the associated 3-d cartesian axes. this example puts the origin at the geocentre (the centre of mass of the earth) but this is not always the case. this system allows position of point p to be stated as either latitude(cid:73), longitude (cid:79), and ellipsoid height h or cartesian coordinates x, y and z – the two types of coordinates give the same information. 3.1.2 cartesian coordinates rectangular cartesian coordinates are a very simple system of describing position in three dimensions, using three perpendicular axes x, y and z. three coordinates unambiguously locate any point in this system. we can use it as a very useful alternative to latitude, longitude and ellipsoid height to convey exactly the same information. we use three cartesian axes aligned with the latitude and longitude system (see figure 4). the origin (centre) of the cartesian system is at the centre of the ellipsoid. the x axis lies in the equator of the ellipsoid and passes through the prime meridian (0 degrees longitude). the negative side of the x axis passes through 180 degrees longitude. the y axis also lies in the equator but passes through the meridian of 90 degrees east, and hence is at right angles to the x axis. obviously, the negative side of the y axis passes through 90 degrees west. the z axis coincides with the polar axis of the ellipsoid; the positive side passes through the north pole and the negative side through the south pole. hence it is at right angles to both x and y axes. it is clear that any position uniquely described by latitude, longitude and ellipsoid height can also be described by a unique triplet of 3-d cartesian coordinates, and vice versa. the formulae for converting between these two equivalent systems are given in annexe b. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 10 of 43

a ha na ha b hb nb topography geoid model hb true geoid ellipsoid figure 5: ellipsoid height h and orthometric height h of two points a and b related by a model of geoid-ellipsoid separation n the development of precise geoid models is very important to increasing the accuracy of height coordinate systems. a good geoid model allows us to determine orthometric heights using gps (which yields ellipsoid height) and equation (1) (to convert to orthometric height). the gps ellipsoid height alone gives us the geometric information we need, but does not give real height because it tells us nothing about the gravity field. different geoid models will give different orthometric heights for a point, even though the ellipsoid height (determined by gps) might be very accurate. therefore orthometric height should never be given without also stating the geoid model used. as we will now see, even height coordinate systems set up and used exclusively by the method of spirit levelling involve a geoid model, although this might not be stated explicitly. 3.1.4 mean sea level height we will now take a look at the geoid model used in os mapping on the british mainland, although most surveyors might not immediately think of it as such. this is the ordnance datum newlyn (odn) vertical coordinate system. ordnance survey maps state that heights are given above ‘mean sea level’. if we’re looking for sub-metre accuracy in heighting this is a vague statement, since mean sea level (msl) varies over time and from place to place, as we noted in section 2.3. odn corresponds to the average sea level measured by the tide-gauge at newlyn, cornwall between 1915 and 1921. heights that refer to this particular msl as the point of zero height are called odn heights. odn is therefore a ‘local geoid’ definition as discussed in section 2.3. odn heights are used for all british mainland ordnance survey contours, spot heights and bench mark heights. odn heights are unavailable on many offshore islands, which have their own msl based on a local tide-gauge. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 12 of 43

thirdly and most importantly, we have the errors that can be incurred when using bench marks to obtain odn heights. some osbms were surveyed as long ago as 1912 and the majority have not been rigorously checked since the 1970s. therefore, we must beware of errors due to the limitations in the original computations, and due to possible movement of the bench mark since it was observed. there is occasional anecdotal evidence of bench mark subsidence errors of several metres where mining has caused collapse of the ground. this type of error can affect even local height surveys, and individual bench marks should not be trusted for high-precision work. however, odn can now be used entirely without reference to bench marks, by precise gps survey using os net® in conjunction with the national geoid model osgm02(cid:165). this is the method now recommended by ordnance survey for the establishment of all high-precision height control. see section 5 for more details. height above true geoid local orthometric height ellipsoid height mean sea level sst h h topography local geoid model true global geoid n ellipsoid figure 7: the relationship between the geoid, a local geoid model (based on a tide-gauge datum), mean sea level, and a reference ellipsoid. the odn geoid model is an example of a local geoid model. 3.1.5 eastings and northings the last type of coordinates we need to consider is eastings and northings, also called plane coordinates, grid coordinates or map coordinates. these coordinates are used to locate position with respect to a map, which is a two-dimensional plane surface depicting features on the curved surface of the earth. these days, the ‘map’ might be a computerised geographical information system (gis) but the principle is exactly the same. map coordinates use a simple 2-d cartesian system, in which the two axes are known as eastings and northings. map coordinates of a point are computed from its ellipsoidal latitude and longitude by a standard formula known as a map projection. this is the coordinate type most often associated with the ordnance survey national grid. a map projection cannot be a perfect representation, because it is not possible to show a curved surface on a flat map without creating distortions and discontinuities. therefore, different map projections are used for different applications. the map projections commonly used in britain are the ordnance survey national grid projection, and the universal transverse mercator projection. these are both projections of the transverse mercator type. any coordinates stated as eastings and northings should be accompanied by an exact statement of the map projection used to create them. the formulae for the transverse mercator projection are given in annexe c, and the parameters used in britain are in annexe a. there is more about map projections and the national grid in section 7. in geodesy, map coordinates tend only to be used for visual display purposes. when we need to do computations with coordinates, we use latitude and longitude or cartesian coordinates, then convert the results to map coordinates as a final step if needed. this working procedure is in contrast to the practice in geographical information systems, where map coordinates are used directly for many computational tasks. the ordnance survey transformation between the gps coordinate system wgs84 and the national grid works directly with map coordinates – more about this in section 6.3. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 14 of 43

3.2.1 datum definition before the space age how do we specify the position and orientation of a set of cartesian axes in relation to the ground, when the origin of the axes, and the surface of the associated ellipsoid, are within the earth? the way it used to be done before the days of satellite positioning (gps and so on) was to use a particular ground mark as the initial point of the coordinate system. this ground mark is assigned coordinates that are essentially arbitrary, but fit for the purpose of the coordinate system. also, the direction towards the origin of the cartesian axes from that point was chosen. this was expressed as the difference between the direction of gravity at that point and the direction towards the origin of the coordinate system. because that is a three-dimensional direction, three parameters were required to define it. so we have six conventional parameters of the initial point in all, which correspond to choosing the centre (in three dimensions) and the orientation (in three dimensions) of the cartesian axes. to enable us to use the latitude, longitude and ellipsoid height coordinate type, we also choose the ellipsoid shape and size, which are a further two parameters. once the initial point is assigned arbitrary parameters in this way, we have defined a coordinate system in which all other points on the earth have unique coordinates – we just need a way to measure them! we will look at the principles of doing this in the next section. although it is a good example of defining a datum, these days, a single initial point would never be used for this purpose. instead, we define the datum ‘implicitly’ by applying certain conditions to the computed coordinates of a whole set of points, no one of which has special importance. this method has become common in global gps coordinate systems since the 1980s. avoiding reliance on a single point gives practical robustness to the datum definition and makes error analysis more straightforward. 3.3 realising the datum definition with a terrestrial reference frame with our datum definition we have located the origin, axes and ellipsoid of the coordinate system with respect to the earth’s surface, on which are the features we want to measure and describe. we now come to the problem of making that coordinate system available for use in practice. if the coordinate system is going to be used consistently over a large area, this is a big task. it involves setting up some infrastructure of points to which users can have access, the coordinates of which are known at the time of measurement. these reference points are typically either on the ground, or on satellites orbiting the earth. all positioning methods rely on line-of-sight from an observing instrument to reference points of known coordinates. putting some of the reference points on orbiting satellites has the advantage that any one satellite is visible to a large area of the earth’s surface at any one time. this is the idea of satellite positioning. the network of reference points with known coordinates is called the coordinate terrestrial reference frame (trf), and its purpose is to realise the coordinate system by providing accessible points of known coordinates. examples of trfs are the network of ordnance survey triangulation pillars seen on hilltops across britain, and the constellation of 24 gps satellites operated by the united states department of defense. both these trfs serve exactly the same purpose: they are highly visible points of known position in particular coordinate systems (in the case of satellites, the points move so the ‘known position’ changes as a function of time). users can observe these trf points using a positioning tool (a theodolite or a gps receiver in these examples) and hence obtain new positions of previously unknown points in the coordinate system. a vital conceptual difference between a datum and a trf is that the former is errorless while the latter is subject to error. a datum might be unsuitable for a certain application, but it cannot contain errors because it is simply a set of conventionally adopted parameters – they are correct by definition! a trf, on the other hand, involves the physical observation of the coordinates of many points – and wherever physical observations are involved, errors are inevitably introduced. therefore it is quite wrong to talk of errors in a datum: the errors occur in the realisation of that datum by a trf to make it accessible to users. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 16 of 43

4 modern gps coordinate systems 4.1 world geodetic system 1984 (wgs84) in this section, we look at the coordinate systems used in gps positioning, starting with wgs84. we’ll discuss gps coordinate systems in terms of the coordinate system concepts summarised in table 1. the datum used for gps positioning is called wgs84 (world geodetic system 1984). it consists of a three-dimensional cartesian coordinate system and an associated ellipsoid so that wgs84 positions can be described as either xyz cartesian coordinates or latitude, longitude and ellipsoid height coordinates. the origin of the datum is the geocentre (the centre of mass of the earth) and it is designed for positioning anywhere on earth. in line with the definition of a datum given in section 3.2, the wgs84 datum is nothing more than a set of conventions, adopted constants and formulae. no physical infrastructure is included, and the definition does not indicate how you might position yourself in this system. the wgs84 definition includes the following items: (cid:120) the wgs84 cartesian axes and ellipsoid are geocentric; that is, their origin is the centre of mass of the whole earth including oceans and atmosphere. (cid:120) (cid:120) (cid:120) (cid:120) the scale of the axes is that of the local earth frame, in the sense of the relativistic theory of gravitation. their orientation (that is, the directions of the axes, and hence the orientation of the ellipsoid equator and prime meridian of zero longitude) coincided with the equator and prime meridian of the bureau internationale de l’heure at the moment in time 1984.0 (that is, midnight on new year’s eve 1983). since 1984.0, the orientation of the axes and ellipsoid has changed such that the average motion of the crustal plates relative to the ellipsoid is zero. this ensures that the z-axis of the wgs84 datum coincides with the international reference pole, and that the prime meridian of the ellipsoid (that is, the plane containing the z and x cartesian axes) coincides with the international reference meridian. the shape and size of the wgs84 biaxial ellipsoid is defined by the semi-major axis length a (cid:32) 6378137 0. very, very close in shape and size to the grs80 ellipsoid. metres, and the reciprocal of flattening 1 . this ellipsoid is 298 257223563 . f (cid:32) / (cid:120) conventional values are also adopted for the standard angular velocity of the earth, and for the earth gravitational constant. the first is needed for time measurement, and the second to define the scale of the system in a relativistic sense. we will not consider these parameters further here. there are a couple of points to note about this definition. firstly, the ellipsoid is designed to best-fit the geoid of the earth as a whole. this means it generally doesn’t fit the geoid in a particular country as well as the non-geocentric ellipsoid used for mapping that country. in britain, grs80 lies about 50 metres below the geoid and slopes from east to west relative to the geoid, so the geoid-ellipsoid separation is 10 metres greater in the west than in the east. our local mapping ellipsoid (the airy 1830 ellipsoid) is a much better fit. secondly, note that the axes of the wgs84 cartesian system, and hence all lines of latitude and longitude in the wgs84 datum, are not stationary with respect to any particular country. due to tectonic plate motion, different parts of the world move relative to each other with velocities of the order of ten centimetres per year. the international reference meridian and pole, and hence the wgs84 datum, are stationary with respect to the average of all these motions. but this means they are in motion relative to any particular region or country. in britain, all wgs84 latitudes and longitudes are changing at a constant rate of about 2.5 centimetres per year in a north-easterly direction. over the course of a decade or so, this effect becomes noticeable in large-scale mapping. some parts of the world (for example hawaii and australia) are moving at up to one metre per decade relative to wgs84. the full definition of wgs84 is available on the internet – see section 8 for the address. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 18 of 43

4.2.2 the international terrestrial reference frame (itrf) the itrf is an alternative realisation of wgs84 that is produced by the international earth rotation and reference system service (iers) based in paris, france. it includes many more stations than the broadcast wgs84 trf – more than 500 stations at 290 sites all over the world. four different space positioning methods contribute to the itrf – very long baseline interferometry (vlbi), satellite laser ranging (slr), global positioning system (gps) and doppler ranging integrated on satellite (doris). each has strengths and weaknesses – their combination produces a strong multi-purpose trf. itrf was created by the civil gps community, quite independently of the us military organisations that operate the broadcast trf. each version of the itrf is given a year code to identify it – the current version is itrf2008. itrf2008 is simply a list of coordinates (x y and z in metres) and velocities (dx, dy and dz in metres per year) of each station in the trf, together with the estimated level of error in those values. the coordinates relate to the time 2008.0 – to obtain the coordinates of a station at any other time, the station velocity is applied appropriately. this is to cope with the effects of tectonic plate motion. itrf2008 is available as a sinex format text file from the iers internet website – see section 8 for the addresses. the datum realised by the itrf is actually called itrs (international terrestrial reference system) rather than wgs84. there used to be a difference between the two, but they have been progressively brought together and are now so similar that they can be assumed identical for almost all purposes. because the itrf is of higher quality than the military wgs84 trf, the wgs84 datum now effectively takes its definition from itrs. therefore, although in principle the broadcast trf is the principal realisation of wgs84, in practice itrf has become the more important trf because it has proven to be the most accurate global trf ever constructed. the defining conventions of itrs are identical to those of wgs84 given in section 4.1. the itrf is important to us for two reasons. firstly, we can use itrf stations equipped with permanent gps receivers as reference points of known coordinates to precisely coordinate our own gps stations, using gps data downloaded from the internet. this procedure is known as ‘fiducial gps analysis’. secondly, we can obtain precise satellite positions (known, rather bizarrely, as ephemerides) in the itrf2008 trf that are more accurate than the ephemerides transmitted by the gps satellites. both these vital geodetic services are provided free by the international gnss service on the internet. 4.2.3 the international gnss service (igs) igs tracking station dual-frequency gps data. precise gps satellite orbits (ephemerides). the igs is essential for anybody requiring high accuracy wgs84 positions. the igs operates a global trf of 368 gps stations (2 december 2013) and from these produces the following free products, distributed via the internet: (cid:120) (cid:120) (cid:120) gps satellite clock parameters. (cid:120) (cid:120) (cid:120) igs tracking station coordinates and velocities; many of these stations are also listed in the itrf. earth orientation parameters. zenith path delay estimates. of these, the first two are commonly used for general-purpose high-accuracy positioning, and the third is becoming increasingly important. the satellite ephemerides, clock parameters and earth rotation parameters are available two days after the time of observation, and also in advance of the observation in a less accurate predicted version. the igs products give us access to a high-accuracy realisation based on itrf (currently, itrf2008). used in conjunction with the itrf2008 coordinates of nearby igs tracking stations and the dual-frequency gps data from those stations, a user can position a single geodetic-quality gps receiver to within a few millimetres. hence the igs products are a vital part of the civil gps community’s access to the itrf. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 20 of 43

5 ordnance survey coordinate systems let’s now take a look at the coordinate systems used by os for mapping great britain. there are three coordinate systems to consider: (cid:120) os net, a modern 3-d trf using the etrs89 datum (as just described in section 4.2.4). this coordinate system is the basis of modern ordnance survey ‘control survey’ (the surveyor’s jargon for adding local points to a trf for mapping purposes), and is the basis of definition of all ordnance survey coordinates. a subset of os net (see section 5.1) has been ratified as the official densification of etrf89 in gb. (cid:120) the national grid, a ‘traditional’ horizontal coordinate system, which consists of: a traditional geodetic datum (see section 3.2) using the airy ellipsoid; a trf called osgb36 (ordnance survey great britain 1936) which was observed by theodolite triangulation of trig pillars; and a transverse mercator map projection (see section 7 below) allowing the use of easting and northing coordinates. this coordinate system is important because it is used to describe the horizontal positions of features on british maps. however, its historical origins and observation methods are not of interest to most users and will be skipped over in this booklet. national grid coordinates are nowadays determined by gps plus a transformation rather than theodolite triangulation. (cid:120) ordnance datum newlyn (odn), a ‘traditional’ vertical coordinate system, consisting of a tide-gauge datum with initial point at newlyn (cornwall), and a trf observed by spirit levelling between 200 fundamental bench marks (fbms) across britain. the trf is densified by more than half a million lower-accuracy bench marks. each bench mark has an orthometric height only (not ellipsoid height or accurate horizontal position). this coordinate system is important because it is used to describe vertical positions of features on british maps (for example, spot heights and contours) in terms of height above ‘mean sea level’. again, its historical origins and observation methods are not of interest to most users. the word ‘datum’ in the title refers, strictly speaking, to the tide-gauge initial point only, not to the national trf of levelled bench marks. because triangulation networks need hilltop stations whereas levelling networks need low-lying, easily accessible routes, there are hardly any common points between the height bench mark network and the osgb36 horizontal network. os net provides a single 3-d trf that unifies odn and osgb36 via transformation software (see section 6 below). using transformation techniques, precise positions can be determined by gps in etrs89 using os net and then converted to national grid and odn coordinates. this is the approach used today by ordnance survey. 5.1 os net gps is the standard tool for precise surveying and mapping, used for all os precise surveying work. some characteristics of gps as a surveying tool are: (cid:120) gps is a three-dimensional positioning system: a precise gps fix yields latitude, longitude and ellipsoid height. (cid:120) the highest precision of gps positions is at the 2 mm level horizontally relative to a global datum. to achieve this requires networks of permanently-installed gps receivers. typical field gps survey gives accuracies of a few centimetres relative to a global datum. vertical position quality is generally about 2.5 times worse than horizontal. (cid:120) gps is a purely geometric positioning tool; that is, gps coordinates do not give you any information in relation to level surfaces, only in relation to the geometric elements of coordinate system axes and ellipsoid. for this reason, gps does not give orthometric height information. (cid:120) gps does not require intervisibility between ground reference points; neither is the geometric arrangement of the ground network crucial to the results as it is in theodolite triangulation survey. (cid:120) with care, gps can be used very accurately for terrestrial survey over any distance – even between points on opposite sides of the world. for this reason, global datums are used for gps positioning. this feature makes gps vastly more powerful than traditional survey techniques. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 22 of 43

the ellipsoid used in the osgb36 datum is that defined by sir george airy in 1830 (later astronomer royal). its defining constants are a (cid:32) 6377563 396. summary of datum constants). hence the airy 1830 ellipsoid is a bit smaller than grs80 and a slightly different shape. also, it is not geocentric as grs80 is: it is designed to lie close to the geoid beneath the british isles. hence only a tiny fraction of the surface of the ellipsoid has ever been used – the part lying beneath britain. the rest is not useful. so the airy ellipsoid differs from grs80 in size, shape, position and orientation, and this is generally true of any pair of geodetic ellipsoids9. m (see annexe a for a m and (cid:32)b 6356256 909. 5.2.2 the osgb36 trf before the 1950s, coordinate system trfs contained many angle measurements but very few distance measurements. this is because angles could be measured relatively easily between hilltop primary control stations with a theodolite, but distance measurement was very difficult. a consequence of this was that the shape of the trf was well known, but its size was poorly known. the distance between primary control stations was established by measuring just one or two such distances, then propagating these through the network of angles by trigonometry (hence the name ‘trig pillars’). when the osgb36 triangulation trf was established, no new distance measurements were used. instead, the overall size of the network was made to agree with that of the old 18th century principal triangulation using the old coordinates of the 11 control stations mentioned in the previous section. hence the overall size of the trf still used for british mapping came to be derived from the measurement of a single distance between two stations on hounslow heath in 1784 using eighteen- foot glass rods! the error thus incurred in osgb36 is surprisingly low – only about 20 metres in the length of the country. the 326 primary control stations of the osgb36 trf cover the whole of great britain, including orkney and shetland but not the scilly isles, rockall or the channel islands. it is also connected to northern ireland, the irish republic, and france. this primary control network was densified by the addition of 2 695 secondary control stations, 12 236 tertiary control stations and 7 032 fourth order control stations. hence until recently the ‘official osgb36’ included more than 22 000 reference points. in continuous survey work over the decades, ordnance survey surveyors have added many tens of thousands more ‘minor control’ points, which are used in map revision. osgb36 coordinates are latitudes and longitudes on the airy ellipsoid. ideally, ellipsoid heights relative to the airy ellipsoid would be added to give complete three-dimensional information. unfortunately, no easy method of measuring ellipsoid height was available before the late 1980s, so we only have horizontal information in osgb36. the coordinates of all osgb36 control points are available from ordnance survey. in 2002 the definition of osgb36 underwent a fundamental change with the release of the definitive transformation ostn02™. ostn02 in combination with the etrs89 coordinates of the os net stations, rather than the fixed triangulation network, now define the national grid. this is a subtle change in definition only and does not mean that existing osgb36 coordinates need to be changed in any way. 9 if we want to convert latitudes and longitudes on one ellipsoid to those quantities on another ellipsoid, we need to take account of the differences in position and orientation of the ellipsoids as well as the differences in size and shape. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 24 of 43

the orthometric height of each fundamental bench mark relative to newlyn was determined by a network of precise levelling lines across britain. you can find descriptions of the precise levelling technique in land surveying textbooks. this technique is capable of transferring orthometric height from one point to another with an error of less than 2 (cid:117) d millimetres, where d is the distance levelled in kilometres. today, the accuracy of the precise levelling technique is rivalled by the combination of gps ellipsoid heighting with a precise gravimetric geoid model that allows the ellipsoid height difference between two points to be easily converted to an orthometric height difference. the odn trf was subsequently densified by less precise levelling to obtain the heights of about three-quarters of a million ordnance survey bench marks all over britain. these lower-order bench marks are often seen cut into stone at the base of a building, church or bridge. about half a million of them are in existence and usable today. an important error source to bear in mind is possible subsidence of ordnance survey bench marks, especially in areas where mining has caused collapse of the ground. in these areas, cases of bench mark subsidence of several metres have been reported. ordnance survey from time-to-time installs new bench marks that are needed for surveying projects requiring accurate height control over a large area – precise gps survey and the national geoid model osgm02 (see section 6.4) are used for this task. 5.3.3 relative accuracy of odn bench marks the following table is a statistical statement of odn bench mark accuracy relative to other bench marks in the same area. the primary layer of the odn trf consists of the fundamental bench marks, which are considered error-free within the odn coordinate system. the ordnance survey national geoid model (see section 6.4 below) relates odn orthometric heights to the gps ellipsoid grs80. table 3 shows the expected maximum levelling error between bench marks on the same levelling line. this only describes the original measurement error. the ordnance survey bench mark network has not been maintained since the 1970s, so beware of possible bench mark subsidence. type of bench mark maximum error fundamental geodetic secondary tertiary error free (cid:114)2mm (cid:117) d (cid:114)5mm (cid:117) d (cid:114)12mm (cid:117) d table 3: accuracy of ordnance survey bench mark levelling, where d is distance levelled in km. 5.4 other height datums in use across great britain as discussed in section 5.3, odn is the national height datum used across mainland great britain. there are, however, a number of other british height datums that are used on the surrounding islands. the main ones being; lerwick on the shetland islands, stornoway on the outer hebrides, st. kilda, douglas on the isle of man and st. marys on the scilly isles. all of these height datums are incorporated within the national geoid model osgm02. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 26 of 43

to clarify this idea, let’s look at an example. to keep things simple, we’ll consider one-dimensional coordinate systems (ordnance datum newlyn in section 5.3 is an instance of a real one-dimensional coordinate system). imagine we have two coordinate systems: a and b, each of which state coordinates for three points called white, grey and black. so here we have 100% overlap between the trfs of the two coordinate systems – all three points appear in both trfs. 0 0 5 coordinate system a 10 5 coordinate system b 10 15 15 point white grey black coordinate system a coordinate system b 4.1 9.8 12.2 7.3 12.7 15.0 we want to work out a transformation from the datum of coordinate system a to the datum of coordinate system b. this transformation will be a single number t that can be added to any coordinate in system a to give the equivalent coordinate in system b. we don’t have any theoretical datum definitions of these two systems to work from, and even if we did we would not use them (in this example, we don’t even know what the coordinates represent – it doesn’t matter). we want a practical transformation that will convert any coordinate in coordinate system a to coordinate system b. so we use points that have known coordinates in both systems – in other words, we relate the two datums by using their trfs. in this example, we have three points that are coordinated in both trfs – white, grey and black. the best estimate of the transformation t between the frames is simply the average of the transformation at each of these points: (cid:32) white b (cid:16) white a 2.3(cid:32) , white (cid:32) grey b (cid:16) grey a 9.2(cid:32) , grey (cid:32) black b (cid:16) black a 8.2(cid:32) , black t t t t white (cid:14) t so t (cid:32) grey 3 (cid:14) t black (cid:32) 97.2 a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 28 of 43

when this transformation is applied to a trf, it has the effect of rotating and translating the network of points relative to the cartesian axes and applying an overall scale factor (that is, a change of size), while leaving the shape of the figure unchanged. equivalently, we can think of the network of points remaining still while the cartesian axes are rotated, translated and rescaled. since our application of this transformation is to ‘change datum’ with the points firmly fixed on the ground, we usually use the second mental picture. the helmert transformation can only provide a perfect transformation between perfect mathematical reference systems. in practice, we determine and use transformations between sets of real ground points, that is, between trfs that are subject to the errors of the real world. the more distortion, subsidence, and so on present in the trf, the worse the helmert transformation will perform. both the parameters of the transformation and the output coordinates will contain errors. bear this in mind especially when working with pre-gps coordinate sets and orthometric heights over large areas. the usual mathematical form of the transformation is a linear formula which assumes that the rotation parameters are ‘small’. rotation parameters between geodetic cartesian systems are usually less than five seconds of arc, because the axes are conventionally aligned to the greenwich meridian and the pole. do not use this formula for larger angles – instead apply the full rotation matrices, which can be found in most photogrammetry and geodesy textbooks. in matrix notation, the coordinates of each point in reference system b are computed from those in reference system a by: b (cid:170) (cid:171) (cid:171) (cid:171) (cid:172) x y z (cid:186) (cid:187) (cid:187) (cid:187) (cid:188) (cid:32) (cid:170) t (cid:171) (cid:171) (cid:171) (cid:172) t t (cid:186) (cid:187) (cid:187) (cid:187) (cid:188) x y z (cid:14) s (cid:14) r z (cid:16) r y (cid:170) 1 (cid:171) (cid:171) (cid:171) (cid:172) (cid:16) r z (cid:14) 1 s r x a r y (cid:16) r x (cid:14) 1 s (cid:186) (cid:187) (cid:187) (cid:187) (cid:188) (cid:152) (cid:170) (cid:171) (cid:171) (cid:171) (cid:172) x y z (cid:186) (cid:187) (cid:187) (cid:187) (cid:188) (3) , yt and zt are the translations along the x, y and z axes respectively in metres, xr , yr and where xt zr are the rotations about the x, y and z axes respectively in radians, and s is the scale factor (unitless) minus one. this definition of s is convenient, since its deviation from unity is usually very small. s is often stated in parts per million (ppm) – remember to divide this by a million before using equation 3. rotations are often given in seconds of arc, which must be converted to radians. beware that two opposite conventions are in use regarding the rotation parameters of the helmert transformation. equation (3) uses the ‘right-hand-rule’ rotation convention: with the positive end of an axis pointing towards you, a positive rotation about that axis is anticlockwise. a way to remember this is: if your right-hand thumb in hitch-hiking pose is the positive end of an axis, the curl of the fingers indicates positive rotation. some published transformations use the opposite convention. in this case, reverse the signs of the three rotation parameters before applying equation (3). for safety, always include a test point with coordinates in systems a and b when stating a transformation. sometimes, three-parameter (translation only) or six-parameter (translation and rotation only) transformations are encountered, which omit some of the seven parameters used here. the same formula can be used to apply these transformations, setting parameters not used to zero. any ‘small’ helmert transformation can be reversed by simply changing the signs of all the parameters and applying equation (3). this is valid only when the changes in coordinates due to the transformation are very small compared to the size of the network. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 30 of 43

osgm02 also covers areas of britain not related to odn such as the shetland isles, outer hebrides and the isle of man. orthometric heights in these areas are related to specific local vertical datums. the national geoid model osgm02 is a grid of geoid-ellipsoid separation values at 1 kilometre resolution, from which a value is obtained for any location by bi-linear interpolation. the nominal accuracy of the geoid model is 2 cm rms (root mean square) in mainland uk and less than 4 cm rms for other areas. this transformation is freely available from the os gps website, www.ordnancesurvey.co.uk/gps. os recommends the use of the national geoid model osgm02 and os net for high-accuracy orthometric heighting, in preference to ordnance survey bench marks. 6.5 etrs89 to and from itrs the transformations between etrs89 (european trs 1989, the europe-fixed version of wgs84) and the various versions of the itrs (international terrestrial reference system) is published by iers (international earth rotation service) via their internet site. the easiest way to carry out these transformations is using the etrs89/itrs transformation tool on the euref permanent network (epn) website – http://www.epncb.oma.be/_productsservices/coord_trans/. the usual application of these transformations is in dealing with gnss coordinates determined in a ‘fiducial network’ using igs permanent stations as fixed control points and using igs products such as satellite ephemerides, satellite clock solutions and earth orientation parameters. in this application, the coordinates adopted for the fixed igs stations should be itrs coordinates in the same realisation of itrs as used to determine the igs precise ephemerides and other products. that realisation is currently itrf2008. the itrf2008 coordinates of the igs reference stations at the epoch of observation are obtained by multiplying the itrf2008 station velocity by the time difference between the itrf2008 reference epoch and epoch of observation and adding this position update to the itrf2008 reference epoch station positions. the resulting coordinates of the unknown stations are in itrs at the epoch of observation. the transformations above can then be used to obtain coordinates in the etrs89. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 32 of 43

the utm projections are a way of mapping the whole world in a systematic way by dividing the earth by longitude into 60 zones, each 6 degrees of longitude wide. the 60 utm zones each have a different central meridian. the zones relevant to british mapping are 29 (central meridian 9(cid:113) w), 30 (central meridian 3(cid:113) w) and 31 (central meridian 3(cid:113) e). the scale on the central meridian is 0.9996 for all utm zones. the international 1924 ellipsoid is usually used in the utm projection in europe, but other ellipsoids can also be used. when working with utm coordinates, check which ellipsoid is being used. annexe c gives formulae for conversion between latitude and longitude and tm eastings and northings. annexe a gives the tm parameters for the national grid and the british utm zones. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 34 of 43

8 further information quite a lot of information on this subject is available on the internet. you might find online implementations of the formulae given in the annexes, although beware that os does not check or endorse these. here are a few addresses directly related to the information in this booklet: (cid:120) http://www.ordnancesurvey.co.uk/gps this is the os gps website and contains useful information for all users of gps and os mapping, both recreational and professional. (cid:120) (cid:120) (cid:120) (cid:120) (cid:120) (cid:120) http://earth-info.nga.mil/gandg/coordsys/csat_pubs.html is the us nga geodesy and geophysics department publications page. this has lots of information on the wgs84 datum and the egm96 global geoid model. http://igscb.jpl.nasa.gov/ is the igs home page. this gives access to all the igs products and information about them. http://www.euref-iag.net/ is the euref home page. euref is the governing body of the etrs89 coordinate system. http://www.epncb.oma.be/ is the euref permanent gps network information system. http://etrs89.ensg.ign.fr/memo-v7.pdf is the text file from which the etrs89-itrs transformation given in section 6.5 is taken. http://www.ordnancesurvey.co.uk/oswebsite/gi/nationalgrid/nghelp1.html is an interactive guide to the national grid on the os website. a detailed online guide to the national grid can be found on the os gps website at http://www.ordnancesurvey.co.uk/gps. here are a few book titles on geodesy, gps surveying, and map projections: (cid:120) introduction to geodesy: the history and concepts of modern geodesy by james r smith, published by john wiley & sons, 1997. (cid:120) geodesy: the concepts by petr vanicek and edward krakiwsky, published by north-holland 1982. (cid:120) gps satellite surveying 2nd edition by alfred leick, published by john wiley and sons 1995 (cid:120) gps theory and practice, 4th edition, by b hofmann-wellenhof, h lichtenegger and j collins, published by springer-verlag 1997. (cid:120) map projections: theory and applications by frederick pearson, published by crc press 1990. (cid:120) map projections: a reference manual by lev bugayevskiy and john snyder, published by taylor and francis 1995. specific enquiries concerning the os coordinate systems, os net, the national grid transformation and the national geoid model can be answered by os customer service centre, phone +44 (0) 3456 050505. the os gps website http://www.os.uk/gps will feature an increasing amount of information and online services related to coordinate systems and gps positioning; it also gives email contact information. a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 36 of 43

b converting between 3d cartesian and ellipsoidal latitude, longitude and height coordinates b.1 converting latitude, longitude and ellipsoid height to 3d cartesian coordinates note: these formulae are used to convert the format of coordinates between 3d earth centred earth fixed (ecef) cartesian coordinates and latitude, longitude and ellipsoidal height. this is not a transformation (see section 6 for more information). values are required for the following ellipsoid constants: the semi-major axis length a and eccentricity squared a 2 e (cid:32) 2e . the latter can be calculated from a and b or the flattening f by (cid:16) a (cid:16) (cid:32) 2 b f f 2 2 2 2 b1 the cartesian coordinates x y and z of a point are obtained from the latitude (cid:73), longitude (cid:79) and ellipsoid height h by (cid:81) (cid:32) (cid:32) (cid:32) (cid:32) x y z a 2 (cid:16) 1 e (cid:81) (cid:14) ( (cid:81) (cid:14) ( (cid:16) 1(( h h 2 2 sin ) cos cos ) (cid:14) (cid:81) ) e (cid:73) (cid:79)(cid:73) cos (cid:79)(cid:73) sin (cid:73) h sin) b2 b3 b4 b5 here’s a worked example using the airy 1830 ellipsoid. intermediate values are shown here to 10 decimal places. compute all values using double-precision arithmetic. latitude, (cid:73) longitude, (cid:79) ellipsoidal height, h e² (cid:81) x y z 52(cid:113) 39(cid:99) 27.2531(cid:115) n 1(cid:113) 43(cid:99) 4.5177(cid:115) e 24.700 m 6.6705400741e-03 6.3910506267e+06 3874938.850 m 116218.624 m 5 047168.207 m a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 38 of 43

c converting between grid eastings and northings and ellipsoidal latitude and longitude c.1 converting latitude and longitude to eastings and northings note: these formulae are used to convert the format of coordinates between grid eastings and northings and ellipsoidal coordinates (latitude and longitude) on the same datum. this is not a transformation. if you need to transform between gps coordinates and osgb36 national grid coordinates, you need to apply either the ostn02 transformation (see section 6.3) or the approximate helmert type transformation (see section 6.6). to convert a position from the graticule of latitude and longitude coordinates ((cid:73),(cid:79)) to a grid of easting and northing coordinates (e, n) using a transverse mercator projection (for example, national grid or utm), compute the following formulae. remember to express all angles in radians. you will need the ellipsoid constants a , b and e² and the following projection constants. annexe a gives values of these constants for the ellipsoids and projections usually used in britain. 0n 0e 0f 0(cid:73) 0(cid:79) longitude of true origin and central meridian. scale factor on central meridian northing of true origin easting of true origin latitude of true origin (cid:14)(cid:14) n 1 2 n (cid:14) 5 4 3 n (cid:183) (cid:16)(cid:184) (cid:73)(cid:73) ( (cid:185) 0 ) (cid:167) (cid:16) (cid:168) (cid:169) 3 n (cid:14) 2 3 n (cid:14) 21 8 n 3 n sin( (2 (cid:73)(cid:73) (cid:16) 0 )) cos( (2 (cid:73)(cid:73) (cid:14) (cid:16) )) 0 5 4 (cid:183) (cid:184) (cid:185) 3 (cid:183) (cid:184) (cid:185) 35 24 sin( (cid:73)(cid:73) (cid:16) 0 ) cos( (cid:73)(cid:73) (cid:14) ) 0 3 n sin( (cid:73)(cid:73) (cid:16) (3 0 )) cos( (3 (cid:73)(cid:73) (cid:14) 0 (cid:32) n ba ba (cid:16) (cid:14) af 0 (cid:32) (cid:81) (cid:32) (cid:85) af 0 (cid:75) v (cid:85) (cid:32) 2 (cid:16) 1 (cid:16) (cid:16) 2 e 2 e 2 (cid:73) ) sin (cid:16) 2 1)( e 1( 1( (cid:16) 5.0 (cid:73) ) 2 sin (cid:16) 5.1 (cid:32) fbm 0 (cid:167) (cid:168) (cid:168) (cid:168) (cid:168)(cid:168) (cid:169) (cid:167) (cid:168) (cid:169) (cid:167) (cid:14) (cid:168) (cid:169) ii (cid:32) (cid:73)(cid:73) cos 15 8 (cid:32) i 0 15 2 n (cid:14) sin 8 nm (cid:14) (cid:81) 2 (cid:81) (cid:32) 24 (cid:81) (cid:32) 720 iiia iv (cid:32) v (cid:32) (cid:73)(cid:81)cos (cid:81) 6 cos 3 (cid:32) vi (cid:81) 120 (cid:14)(cid:32)n i (cid:32) ee 0 (ii (cid:14) 5 (cid:16) cos (cid:167) (cid:168)(cid:168) (cid:73) (cid:169) (cid:81) (cid:85) (cid:11) (cid:73) 5 (cid:79)(cid:79) (cid:16) (cid:16) ) (cid:79)(cid:79) ( iv 2 0 0 iii sin (cid:73)(cid:73) 5( cos 3 (cid:16) 2 tan (cid:75)(cid:73) (cid:14) 9 2 ) sin (cid:73)(cid:73) cos 5 61( (cid:16) 58 tan 2 (cid:73) (cid:14) tan 4 (cid:73) ) c1 c2 c3 (cid:183) (cid:184) (cid:184) (cid:184) (cid:184)(cid:184) (cid:185) )) c4 c5 2 tan (cid:183) (cid:184)(cid:184) (cid:73) (cid:185) (cid:16) 18 tan 2 (cid:14) (cid:73) 4 tan (cid:75)(cid:73) 14 (cid:14) 2 (cid:16) 58 (tan 2 (cid:12)2 (cid:75)(cid:73) ) 4 (cid:14) iii( (cid:14) (v) (cid:79)(cid:79) (cid:16) ) (cid:79)(cid:79) (cid:16) 0 3 ) 0 (cid:79)(cid:79) (cid:14) (cid:16) iiia ( (cid:79)(cid:79) (cid:14) (cid:16) (vi ) 0 0 ) 5 6 a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 40 of 43

here’s a worked example using the airy 1830 ellipsoid and national grid. intermediate values are shown here to 10 decimal places. compute all values using double-precision arithmetic. 651409.903 m 313177.270 m e n (cid:73)(cid:99) #1 9.2002324604e-01 rad 4.1290347143e+05 m #1 n-n0-m#1 2.7379857228e+02 (cid:73)(cid:99) #2 9.2006619470e-01 rad m #2 4.1317717541e+05 n-n0-m#2 9.4594338385e-02 (cid:73)(cid:99) #3 9.2006620954e-01 rad m #3 4.1317726997e+05 n-n0-m#3 3.2661366276e-05 (cid:73)(cid:99) #4 9.2006620954e-01 rad m #4 4.1317727000e+05 n-n0-m#4 1.1350493878e-08 final (cid:73)(cid:99) (cid:81) (cid:85) 9.2006620954e-01 rad 6.3885233415e+06 6.3728193094e+06 2.4642206357e-03 1.6130562489e-14 3.3395547427e-28 9.4198561675e-42 2.5840062507e-07 4.6985969956e-21 1.6124316614e-34 6.6577316285e-48 52(cid:113) 39(cid:99) 27.2531(cid:115) n 1(cid:113) 43(cid:99) 4.5177(cid:115) e (cid:75)2 vii viii ix x xi xii xiia (cid:73) (cid:79) a guide to coordinate systems in great britain d00659 v2.3 mar 2015 © crown copyright page 42 of 43